1 Introduction

In probability theory it is well known that martingale inequalities can be used to prove and extend classical inequalities such as Riesz transforms and Poincaré inequalities to larger setting. Moreover, once a true probabilistic argument has been found, it is then often easier to prove dimension free estimates. This applies in particular to Riesz transforms; see Gundy [23], Pisier [50]. The impressive work of Lust-Piquard shows that, whenever the method applies, it provides the optimal constant for Riesz transforms and Poincaré inequalities. The only drawback here is that the setup for Pisier’s method is so special that it requires ingenuity to establish it in every single case.

Our aim here is to establish a method which applies in a fairly general noncommutative situations. Let us first set up the framework. Let \({\mathcal {N}}\) be a finite von Neumann algebra equipped with a normal faithful tracial state \(\tau :{\mathcal {N}}\rightarrow {\mathbb {C}}\), i.e. \(\tau (1)=1\) and \(\tau (xy)=\tau (yx)\). Let \(({\mathcal {N}}_k)_{k=1,\ldots ,n}\subset {\mathcal {N}}\) be a filtration of von Neumann subalgebras with conditional expectation \(E_k: {\mathcal {N}}\rightarrow {\mathcal {N}}_k\). For a martingale sequence \((x_k)\) with \(x_k\in {\mathcal {N}}_k\), we write \(dx_k=x_k-x_{k-1}\) for the martingale differences. The starting point here is the Burkholder inequality first proved in [34]

$$\begin{aligned} \Bigg \Vert \sum _k dx_k\Bigg \Vert _p \le c(p) \Bigg (\Bigg (\sum _k \Vert dx_k\Vert _p^p\Bigg )^{\frac{1}{p}}+\Bigg \Vert \Bigg (\sum _k E_{k-1} (dx_k^*dx_k + dx_kdx_k^*)\Bigg )^{1/2}\Bigg \Vert _p\Bigg ) \,. \end{aligned}$$
(1.1)

The optimal order of constant here is \(c(p)\sim cp\) in the noncommutative setting, and is due to Randrianantoanina [51]. In the commutative, dissipative setting, Barlow and Yor [6] showed a better constant in Burkholder–Davis–Gundy inequality

$$\begin{aligned} \Vert X_T\Vert _p \le c\sqrt{p} \,\Vert [ X,X ]_T^{1/2}\Vert _p \,\end{aligned}$$
(1.2)

by reducing it with a time change to the case of Brownian motion. Here \([X,X]\) is the quadratic variation of the continuous (local) martingale \(X\); see e.g. [52]. Since the nature of the Brownian motion in the noncommutative setting is so vast (see [13]), it is inconceivable that such an easy argument could work in a noncommutative situation. In fact stationarity of the Brownian motion is certainly required to perform a time change, and can no longer be guaranteed for noncommutative martingales. For many applications towards deviation inequalities it is enough to use the \(L_\infty \) norm on the right hand side of (1.2). Following Oliveira’s idea [26], we are able to use Golden–Thompson inequality to prove the following result. Throughout this paper \(c,C\) and \(C'\) will always denote positive constants which may vary from line to line.

Theorem 1.1

Let \(2\le p<\infty \) and \(x_n=\sum _{k=1}^n dx_k\) be a discrete mean zero martingale. Then

$$\begin{aligned} \Vert x_n\Vert _p \le C \sqrt{p} \Bigg \Vert \left( \sum _{k=1}^n E_{k-1}(dx_k^*dx_k+dx_kdx_k^*)\right) ^{1/2}\Bigg \Vert _{\infty }+ C p \sup _k \Vert dx_k\Vert _{\infty } \,\!. \end{aligned}$$

If \(x_n\) is self-adjoint, then

$$\begin{aligned} \tau \left( 1_{[t,\infty )}\left( x_n\right) \right) \le \exp \left( -\frac{t^2}{8\Vert \sum _{k=1}^n E_{k-1}(dx_k^2)\Vert _\infty +{4t \sup _k\Vert dx_k\Vert _\infty }}\right) \,. \end{aligned}$$

Note that in the commutative context

$$\begin{aligned} \tau (1_{[t,\infty )}(x))=\mathrm{Prob }(x\ge t) \,\!. \end{aligned}$$

In the future we will simply take this formula as a definition. The deviation inequality is a martingale version of noncommutative Bernstein inequality proved in [36]. Tropp [59] obtained better constants for tail estimate of random matrix martingales by using Lieb’s concavity theorem. However, it seems Lieb’s result is only applicable for commutative randomness.

Let us now indicate how to use this result in connection with curvature condition, more precisely the \(\Gamma _2\)-condition introduced by Bakry–Emery [5]. Here we assume that \((T_t)_{t\ge 0}\) is a semigroup of completely positive trace preserving maps on a finite von Neumann algebra \({\mathcal {N}}\) with infinitesimal generator \(A\ge 0\). We assume in addition that \((T_t)\) is a noncommutative diffusion process (in short \((T_t)\) is nc-diffusion), namely that Meyer’s famous gradient form

$$\begin{aligned} 2\Gamma (x,x) =A(x^*)x+x^*A(x)-A(x^*x) \in L_1(N) \end{aligned}$$
(1.3)

for all \(x\in \mathrm{Dom }(A)\,\,\cap \,\, {\mathcal {N}}\) (To be more precise, for all \(x\in \mathrm{Dom }(A^{1/2})\,\,\cap \,\,{\mathcal {N}}\) by extension). In this paper, \(\mathrm{Dom }(A)\) denotes the domain of the operator \(A\) in the underlying Hilbert space. Recall that according to [12, Section 9], we always have \(\Gamma (x,x)\in {\mathcal {B}}^*\), where \({\mathcal {B}}=\mathrm{Dom }(A^{1/2})\,\,\cap \,\,{\mathcal {N}}\) is a \(*\)-algebra by Davies and Lindsay [16, Proposition 2.8]. Thus (1.3) is a regularity assumption, which in general is much weaker than assuming that \((T_t)\) is a usual diffusion semigroup. The standard example for a nc-diffusion semigroup which is not a diffusion is the Poisson semigroup on the circle. Let \(\alpha >0\) be a constant in what follows.

Theorem 1.2

Let \((T_t)\) be a nc-diffusion semigroup on a von Neumann algebra \({\mathcal {N}}\) satisfying the \(\Gamma _2\)-condition

$$\begin{aligned} \tau ( \Gamma (T_tx,T_tx)y)\le C e^{-2\alpha t} \tau ( T_t\Gamma (x,x)y) \end{aligned}$$
(1.4)

for all \(x\in \mathrm{Dom }(A^{1/2})\) and all positive \(y\in {\mathcal {N}}\). Then for self-adjoint \(x\) we have

$$\begin{aligned} \Vert x-E_{\mathrm{Fix}}(x)\Vert _p \le C' {\alpha ^{-1/2}}\min \left\{ \sqrt{{p}} \,\Vert \Gamma (x,x)^{1/2}\Vert _\infty , ~p \Vert \Gamma (x,x)^{1/2}\Vert _p \right\} \,\!. \end{aligned}$$

Here \(\mathrm{Fix}=\{x: T_tx=x\}\) is the fixed point von Neumann subalgebra given by \(T_t\) and \(E_{\mathrm{Fix}}\) the corresponding conditional expectation.

Condition (1.4) is usually formulated in the form \(\Gamma _2(x,x)\ge \alpha \Gamma (x,x)\) where

$$\begin{aligned} 2\Gamma _2(x,y) =\Gamma (Ax,y)+\Gamma (x,Ay)-A\Gamma (x,y)\,\!. \end{aligned}$$

As in the commutative case, this result implies the deviation inequality and exponential integrability.

Corollary 1.3

Under the hypotheses above, we have for \(t>0\),

$$\begin{aligned} \tau (e^{t(x-E_\mathrm{Fix}x})\le \exp \Big (c(\alpha ) t^2 \Vert \Gamma (x,x)\Vert _\infty \Big ). \end{aligned}$$

and

$$\begin{aligned} \mathrm{Prob }(x-E_\mathrm{Fix} x\ge t) \le \exp \Big (-\frac{t^2}{C(\alpha )\Vert \Gamma (x,x)\Vert _\infty }\Big )\,. \end{aligned}$$

Quite surprisingly, these results apply to many commutative and noncommutative examples which cannot be treated with the usual commutative diffusion semigroup approach. Moreover, Bobkov and Götze’s [8] application to the \(L_1\) Wasserstein distance

$$\begin{aligned} W_1(f,h) =\sup \{ |\phi _f(x)-\phi _h(x)|: x \,\mathrm{self}\hbox {-}\mathrm{adjoint},\,\Vert \Gamma (x,x)\Vert _\infty \le 1\} \end{aligned}$$

for normal state \(\phi _f(x)=\tau (fx)\), \(\phi _h(x)=\tau (hx)\), remains applicable in the noncommutative setting. This leads to the transportation inequality.

Corollary 1.4

Under the assumptions above

$$\begin{aligned} W_1(f,E_\mathrm{Fix} f) \le C(\alpha ) \sqrt{\tau (f\ln f)} \end{aligned}$$

for all normal states \(\phi _f(x)=\tau (fx)\).

The Wasserstein distance has been extensively studied in the noncommutative setting; see [7, 45, 53]. This probabilistic connection which provides universal upper bound given by the entropy functional, however, seems to be new. Our definition of Wasserstein distance in the noncommutative setting is closely related to the metric used by Rieffel to define his quantum metric space. Inspired by Connes’ work in noncommutative geometry [14], Rieffel defined the metric on the state space of a \(*\)-algebra \(A\)

$$\begin{aligned} \rho _L(\phi ,\psi )=\sup \{|\phi (a)-\psi (a)|: L(a)\le 1, a\in A\} \end{aligned}$$

where \(\phi \),\(\psi \) are states and \(L(a)\) is a seminorm. For Connes’ spectral triple, \(L(a)=\Vert [D,a]\Vert \) where \(D\) is a self-adjoint operator; see [53] and the references therein for more details. It is an interesting question but beyond the scope of this paper to determine whether the transportation inequality is possible for \(\rho _L\).

At this point it seems helpful to compare our approach with previous ones. Using classical diffusion theory, it is proved by Bakry and Emery [5] that the \(\Gamma _2\)-criterion implies the logarithmic Sobolev inequality (LSI). Bobkov and Götze [8] deduced an exponential integrability (EI) result based on a variant of LSI and showed that the EI is equivalent to the transportation inequality (TI). The relation can be illustrated by the following

We refer the reader to the lecture notes [22] for more details on this subject and its applications to random matrices. In the noncommutative setting, however, the \(\Gamma _2\)-criterion no longer implies LSI; see Example 4.11 below. But we can still use our \(L_p\) Poincaré inequality (\(L_p\mathrm{PI}\)) to deduce EI and TI. In particular, this gives an alternative proof in the commutative case. Our approach is illustrated as follows

At the time of this writing, we are not sure whether this alternative is known or not in the commutative case. In addition, a simple argument shows that EI would hold (thus TI follows) provided the space has finite diameter. This gives a criterion for the validity of TI when we only have \(\Gamma _2(x,x)\ge 0\) instead of \(\Gamma _2(x,x)\ge \alpha \Gamma (x,x)\).

At the end of the paper we consider an algebraic version of the \(\Gamma _2\)-condition and say that \(\Gamma _2\ge \alpha \Gamma \) (in the form sense) if

$$\begin{aligned} {[}\Gamma _2(x_j,x_k)]_{j,k} \ge \alpha [\Gamma (x_j,x_k)]_{j,k} \end{aligned}$$

for all finite families in a weakly dense \(A\) invariant algebra \({\mathcal {A}}\subset {\mathcal {N}}\). Then we collect/prove the following facts

  • \(\Gamma _2\ge \Gamma \) for a suitable semigroup on group von Neumann algebra of the free group \({\mathbb {F}}_n\) and the noncommutative tori.

  • \(\Gamma _2\ge \frac{n+2}{2n} \Gamma \) for suitable semigroups on group von Neumann algebra of the discrete Heisenberg group and the hyperfinite \(II_1\) factor.

  • Let \({\mathcal {N}}=L_{\infty }(\{-1,1\})\) and \(T_t(1)=1\), \(T_t(\varepsilon )=e^{-t}\varepsilon \) where \((1,\varepsilon )\) is the orthonormal basis of \(L_2(\{-1,1\})\). Then \(\Gamma _2\ge \Gamma \).

  • Let \({\mathcal {N}}=L_{\infty }(\{1,\ldots ,n\})\) and \(T_t(e^{\frac{2\pi ik}{n}})=e^{-t(1-\delta _{k,0})} e^{\frac{2\pi ik}{n}}\). Then \(\Gamma _2\ge \frac{n+2}{2n}\Gamma \).

  • \(\Gamma _2\ge \Gamma \) for all \(q\)-Gaussian random variables and the number operator.

  • \(\Gamma _2\ge \alpha \Gamma \) for compact Riemannian manifolds with strictly positive Ricci curvature.

  • \(\Gamma _2\ge \alpha \Gamma \) is stable under tensor products.

  • \(\Gamma _2\ge \alpha \Gamma \) is stable under free products.

  • \(\Gamma _2\ge \frac{n+2}{2n}\Gamma \) for a suitable semigroup on random matrices \(M_n\).

We hope that this ample evidence that Poincaré type inequalities occur frequently in the commutative and the noncommutative setting even without assuming the strong diffusion assumption used in the Bakry–Emery theory justifies our new noncommutative theory. As special cases of our general theory, previous results obtained by Efraim and Lust-Piquard [19] and Li [41] are generalized or improved and many new inequalities are established in different contexts. For instance, the following deviation inequality for product probability spaces is an easy consequence of these examples: for \(f\in L_\infty (\Omega _1\times \cdots \times \Omega _n, {\mathbb {P}}_1\otimes \cdots \otimes {\mathbb {P}}_n)\),

$$\begin{aligned}&{\mathbb {P}}(f-{\mathbb {E}}\,(f)\ge t)\\&\quad \le \exp \left( -\frac{ct^2}{\sum _{i=1}^n\Vert f-\int fd{\mathbb {P}}_i\Vert _\infty ^2+\Vert \int (|f|^2-|\int f d{\mathbb {P}}_i |^2 ) d{\mathbb {P}}_i\Vert _\infty }\right) \,. \end{aligned}$$

where \({\mathbb {P}}={\mathbb {P}}_1\otimes \cdots \otimes {\mathbb {P}}_n\) and \({\mathbb {E}}\,\) is the corresponding expectation operator.

The paper is organized as follows. The martingale deviation inequality and Burkholder type inequality are proved in Sect. 2. After recalling some results in the continuous filtration in von Neumann algebras, we deduce two BDG type inequalities in Sect. 3. The Poincaré type inequalities and the transportation inequalities are proved in Sect. 4, which is also the most technical section. Then the group von Neumann algebras are considered in Sect. 5. In Sect. 6 we prove that the \(\Gamma _2\)-criterion is stable under tensor products and free products with amalgamation. The general theory is applied to classical diffusion processes in Sect. 7.

2 Noncommutative martingale deviation inequality

Our proof of the martingale deviation inequality relies on the well known Golden–Thompson inequality. The fully general case is due to Araki [3]. The version for semifinite von Neumann algebras we used here was proved by Ruskai in [55, Theorem 4].

Lemma 2.1

(Golden–Thompson inequality) Suppose that \(a,b\) are self-adjoint operators, bounded above and that \(a+b\) are essentially self-adjoint (i.e. the closure of \(a+b\) is self-adjoint). Then

$$\begin{aligned} \tau (e^{a+b})\le \tau (e^{a/2}e^be^{a/2})\,\!\!. \end{aligned}$$

Furthermore, if \(\tau (e^a)<\infty \) or \(\tau (e^b)<\infty \) then

$$\begin{aligned} \tau (e^{a+b})\le \tau (e^ae^b)\,\!\!. \end{aligned}$$
(2.1)

Lemma 2.2

Let \((x_k)\) be a self-adjoint martingale sequence with respect to the filtration \(({\mathcal {N}}_k, E_k)\) and \(d_k:=dx_k=x_k-x_{k-1}\) be the associated martingale differences such that

(i):

\(\tau (x_k)=x_0=0\); (ii) \(\Vert d_k\Vert \le M\); (iii) \(\sum _{k=1}^n E_{k-1}(d_k^2) \le D^2 1\).

Then

$$\begin{aligned} \tau (e^{\lambda x_n}) \le \exp [(1+\varepsilon )\lambda ^2 D^2] \end{aligned}$$

for all \(\varepsilon \in (0,1]\) and all \(\lambda \in [0, \sqrt{\varepsilon }/(M+M\varepsilon )]\).

Proof

We follow Oliveira’s original proof for matrix martingales [26] and generalize it to the fully noncommutative setting. With the help of functional calculus, we actually have fewer technical issues. Let \(\varepsilon \in (0,1]\). Put \(y_n=\sum _{k=1}^n E_{k-1}(d_k^2)\). Then \(y_n\le D^21\). We simply write \(D^2\) for the operator \(D^2 1\in {\mathcal {N}}\) in the following. Let us first assume \(M=1\). Since \(e^{-((1+\varepsilon )\lambda ^2D^2-(1+\varepsilon )\lambda ^2y_n)}\le 1\), it follows from (2.1) that

$$\begin{aligned} \tau \big (e^{\lambda x_n}\big )&\le \tau \big (\exp [\lambda x_n+(1+\varepsilon )\lambda ^2D^2-(1+\varepsilon )\lambda ^2y_n]\exp [-((1+\varepsilon )\lambda ^2D^2\\&\quad -(1+\varepsilon )\lambda ^2y_n)]\big )\\&\le \tau \big (\exp [\lambda x_n+(1+\varepsilon )\lambda ^2D^2-(1+\varepsilon )\lambda ^2y_n]\big ) \,. \end{aligned}$$

Put \(r_n=E_{n-1}d_n^2\). Then \(y_n=y_{n-1}+r_n\). Using (2.1) again we find

$$\begin{aligned}&\tau \big (\exp [\lambda x_n+(1+\varepsilon )\lambda ^2D^2-(1+\varepsilon )\lambda ^2y_n]\big )\\&\quad =\tau \big (\exp [\lambda x_{n-1}+\lambda d_n+(1+\varepsilon )\lambda ^2D^2-(1+\varepsilon )\lambda ^2y_{n-1}-(1+\varepsilon )\lambda ^2r_n]\big )\\&\quad \le \tau \big (\exp [\lambda d_n-(1+\varepsilon )\lambda ^2 r_n]\exp [\lambda x_{n-1}+(1+\varepsilon )\lambda ^2D^2-(1+\varepsilon )\lambda ^2y_{n-1}]\big )\,\!. \end{aligned}$$

Since \(x_{n-1},y_{n-1}\in {\mathcal {N}}_{n-1}\) and \(E_{n-1}\) is trace preserving, we obtain

$$\begin{aligned}&\tau \big (\exp [\lambda d_n-(1+\varepsilon )\lambda ^2 r_n]\exp [\lambda x_{n-1}+(1+\varepsilon )\lambda ^2D^2-(1+\varepsilon )\lambda ^2y_{n-1}]\big )\nonumber \\&\quad =\tau \big (E_{n-1}[\exp (\lambda d_n-(1+\varepsilon )\lambda ^2 r_n)]\exp [\lambda x_{n-1}+(1+\varepsilon )\lambda ^2D^2\nonumber \\&\qquad -(1+\varepsilon )\lambda ^2y_{n-1}]\big )\,. \end{aligned}$$
(2.2)

We claim that \(E_{k-1}[\exp (\lambda d_k-(1+\varepsilon )\lambda ^2 r_k)]\le 1\) for all \(k=1,\ldots ,n\) and \(0\le \lambda \le \sqrt{\varepsilon }/(1+\varepsilon )\). Indeed,

$$\begin{aligned} \Vert \lambda d_k-(1+\varepsilon )\lambda ^2 r_k\Vert \le \frac{\sqrt{\varepsilon }}{1+\varepsilon }+(1+\varepsilon )\frac{\varepsilon }{(1+\varepsilon )^2}=\frac{\sqrt{\varepsilon }+\varepsilon }{1+\varepsilon }\le 1. \end{aligned}$$

Note that \(e^x\le 1+x+x^2\) for \(|x|\le 1\). It follows from functional calculus that \(e^A\le 1+A+A^2\) for any self-adjoint operator \(A\) with \(\Vert A\Vert \le 1\). Plugging in \(A=\lambda d_k-(1+\varepsilon )\lambda ^2 r_k\) and using \(r_k\in {\mathcal {N}}_{k-1}\) and \(E_{k-1}d_k=0\) we obtain

$$\begin{aligned}&E_{k-1}[\exp (\lambda d_k-(1+\varepsilon )\lambda ^2 r_k)]\\&\quad \le E_{k-1} [ 1+\lambda d_k-(1+\varepsilon )\lambda ^2 r_k + \lambda ^2 d_k^2-(1+\varepsilon )\lambda ^3 d_k r_k\\&\qquad - (1+\varepsilon )\lambda ^3 r_k d_k + (1+\varepsilon )^2\lambda ^4 r_k^2 ] \\&\quad =1-\varepsilon \lambda ^2 r_k+(1+\varepsilon )^2\lambda ^4r_k^2 \,. \end{aligned}$$

An elementary calculation shows that \(\varepsilon \lambda ^2x-(1+\varepsilon )^2\lambda ^4x^2\ge 0\) for all \(x\in [0,1]\) and \(\lambda \in (0,\sqrt{\varepsilon }/(1+\varepsilon )]\). Using functional calculus of \(r_k\) again, we find

$$\begin{aligned} \varepsilon \lambda ^2 r_k-(1+\varepsilon )^2\lambda ^4r_k^2\ge 0 \end{aligned}$$

which gives the claim. Combining with (2.2), we obtain

$$\begin{aligned}&\tau \big (\exp [\lambda x_n+(1+\varepsilon )\lambda ^2D^2-(1+\varepsilon )\lambda ^2y_n]\big )\\&\quad \le \tau \big (\exp [\lambda x_{n-1}+(1+\varepsilon )\lambda ^2D^2-(1+\varepsilon )\lambda ^2y_{n-1}]\big ). \end{aligned}$$

Iteratively using (2.1) and the claim \(n-1\) times yields

$$\begin{aligned} \tau (e^{\lambda x_n}) \le \tau (\exp [(1+\varepsilon )\lambda ^2D^2])=\exp [(1+\varepsilon )\lambda ^2D^2] \end{aligned}$$

which completes the proof for \(M=1\). For arbitrary \(x_k\), considering \(x_k'=x_k/M\) leads to the conclusion. \(\square \)

We remark that the exponential inequality in this lemma is crucial for the proof of law of the iterated logarithms for noncommutative martingales by the second named author in [65].

Theorem 2.3

Let \((x_k)\) be a self-adjoint martingale sequence with respect to the filtration \(({\mathcal {N}}_k, E_k)\) and \(d_k:=dx_k=x_k-x_{k-1}\) be the associated martingale differences such that

(i):

\(\tau (x_k)=x_0=0\); (ii) \(\Vert d_k\Vert _\infty \le M\); (iii) \(\sum _{k=1}^n E_{k-1}(d_k^2) \le D^2 1\).

Then for \(t\ge 0\),

$$\begin{aligned} \mathrm{Prob }\left( x_n\ge t\right) \!\le \! \exp \left( \!-\!\frac{t^2}{4(1\!+\!\varepsilon )D^2\!+\!{2(1\!+\!\varepsilon )t M}/{\sqrt{\varepsilon }}}\!-\!\frac{\sqrt{\varepsilon }Mt^3}{2(1\!+\!\varepsilon )\left( 2\sqrt{\varepsilon }D^2+M t\right) ^2}\right) \,\end{aligned}$$

for all \(0<\varepsilon \le 1\).

Note that if \(\varepsilon =1\) the first term in our upper bound reduces to the same estimate as Oliveira’s. In fact, the first term is always dominating.

Proof

We assume \(M=1\) first. Let \(\varepsilon \in (0,1]\). By exponential Chebyshev’s inequality we have \(\tau \left( 1_{[t,\infty )} \left( x_n\right) \right) \le e^{-\lambda t}\tau \big (e^{\lambda x_n}\big )\) for \(t>0\). It follows from Lemma 2.2 that

$$\begin{aligned} \tau \left( 1_{[t,\infty )}\left( x_n\right) \right) \le \exp (-\lambda t+(1+\varepsilon )\lambda ^2D^2). \end{aligned}$$

Now we set

$$\begin{aligned} \lambda =\frac{t}{2(1+\varepsilon )D^2+{(1+\varepsilon )t}/{\sqrt{\varepsilon }}} \end{aligned}$$

which is less than \(\sqrt{\varepsilon }/(1+\varepsilon )\). Then,

$$\begin{aligned}&-\lambda t+(1+\varepsilon )\lambda ^2D^2=-t^2\cdot \frac{1+t/\left( \sqrt{\varepsilon } D^2\right) }{4(1+\varepsilon )D^2\left[ 1+t/\left( 2\sqrt{\varepsilon }D^2\right) \right] ^2}\\&\quad =-\frac{t^2}{4(1+\varepsilon )D^2\left[ 1+t/\left( 2\sqrt{\varepsilon }D^2\right) \right] } - \frac{\sqrt{\varepsilon }t^3}{2(1+\varepsilon )\left( 2\sqrt{\varepsilon }D^2+t\right) ^2}\,. \end{aligned}$$

Replacing \(t\) and \(D\) with \(t/M\) and \(D/M\) respectively yields the assertion. \(\square \)

Similar to the classical probability theory, we have for positive \(a\in {\mathcal {M}}\) and for all \(0<p<\infty \),

$$\begin{aligned} \Vert a\Vert _p^p=p\int _0^\infty t^{p-1}\mathrm{Prob}(a>t) dt\,\!. \end{aligned}$$
(2.3)

From here it is routine to estimate the \(p\)-th moment of \(x_n\) using Theorem 2.3.

Proposition 2.4

Under the assumption of Theorem 2.3, for \(2\le p<\infty \) we have

$$\begin{aligned} \Vert x_n\Vert _p\le 2^{3/2}(1+\varepsilon )^{1/2}\sqrt{p}\Bigg \Vert \sum _{i=1}^nE_{i-1}(dx_i^2)\Bigg \Vert _\infty ^{1/2} + 2^{5/2}\Big (\frac{1+\varepsilon }{\sqrt{\varepsilon }}\Big )p\sup _{i=1,\ldots ,n}\Vert dx_i\Vert _\infty \, \end{aligned}$$
(2.4)

for all \(0<\varepsilon \le 1\).

Proof

Our strategy is to integrate the first term in Theorem 2.3. The proof is similar to that of [36, Corollary 0.3]. Note that it follows from symmetry that

$$\begin{aligned} \mathrm{Prob}\left( |x_n| \ge t\right) \le 2\exp \left( -\frac{t^2}{4(1+\varepsilon )D^2+{2(1+\varepsilon )t M}/{\sqrt{\varepsilon }}}\right) \,. \end{aligned}$$

Using (2.3), we obtain

$$\begin{aligned} \frac{\Vert x_n\Vert _p^p}{2p}\!\le \! \int _0^{\frac{2\sqrt{\varepsilon }D^2}{M}} t^{p-1}\exp \left( -\frac{t^2}{8(1+\varepsilon )D^2}\right) dt\!+\!\int _\frac{2\sqrt{\varepsilon }D^2}{M}^\infty t^{p-1} \exp \left( -\frac{t\sqrt{\varepsilon }}{4(1+\varepsilon ) M}\right) dt. \end{aligned}$$

Let us estimate the first term on the right hand side. Using the fact that \(\Gamma (x)\le x^{x-1}\) for \(x\ge 1\), we have

$$\begin{aligned}&\int _0^{\frac{2\sqrt{\varepsilon }D^2}{M}}\! t^{p-1}\exp \left( -\frac{t^2}{8(1+\varepsilon )D^2}\right) dt \!=\! 2^{3p/2-1}(1\!+\!\varepsilon )^{p/2}D^p\int _0^{\frac{\varepsilon D^2}{2M^2(1+\varepsilon )}}r^{p/2-1}e^{-r} dr\\&\quad \le 2^{3p/2-1}(1+\varepsilon )^{p/2}D^p \int _0^\infty r^{p/2-1}e^{-r} dr \le 2^{3p/2-1}(1+\varepsilon )^{p/2}D^p ({p}/2)^{p/2-1}\\&\quad \le 2^{p}(1+\varepsilon )^{p/2}D^p p^{p/2-1}\,\!. \end{aligned}$$

For the second term on the right hand side,

$$\begin{aligned}&\int _\frac{2\sqrt{\varepsilon }D^2}{M}^\infty t^{p-1} \exp \left( -\frac{t\sqrt{\varepsilon }}{4(1+\varepsilon ) M}\right) dt \\&\quad \le 4^p\Big (\frac{1+\varepsilon }{\sqrt{\varepsilon }}\Big )^pM^p\int _0^\infty r^{p-1}e^{-r}dr \le 4^p\Big (\frac{1+\varepsilon }{\sqrt{\varepsilon }}\Big )^pM^p p^{p-1}\,\!. \end{aligned}$$

Hence, we find

$$\begin{aligned} \Vert x_n\Vert _p^p\le 2^{p+1}(1+\varepsilon )^{p/2}D^p p^{p/2}+2^{2p+1}\Big (\frac{1+\varepsilon }{\sqrt{\varepsilon }}\Big )^pM^p p^{p}. \end{aligned}$$

This yields

$$\begin{aligned} \Vert x_n\Vert _p&\le 2^{1+1/p}(1+\varepsilon )^{1/2}D \sqrt{p}+ 2^{2+1/p}\Big (\frac{1+\varepsilon }{\sqrt{\varepsilon }}\Big )Mp \\&\le 2^{3/2}(1+\varepsilon )^{1/2}D \sqrt{p} + 2^{5/2}\Big (\frac{1+\varepsilon }{\sqrt{\varepsilon }}\Big )Mp\,. \end{aligned}$$

Setting \(D^2=\big \Vert \sum _{i=1}^n E_{i-1}(dx_i^2)\big \Vert \) and \(M=\sup _{i=1,\ldots ,n}\Vert dx_i\Vert \) gives the assertion. \(\square \)

Another way to obtain (1.2) would be an improved Burkholder inequality for noncommutative martingales:

Problem 2.5

Is it true that for some function \(f(p)\) and constant \(C\),

$$\begin{aligned} \Bigg \Vert \sum _k dx_k\Bigg \Vert _p \le C\sqrt{p}\Vert \left( \sum _k dx_k^*dx_k + dx_kdx_k^*\right) ^{1/2}\Vert _p+f(p) \left( \sum _k \Vert dx_k\Vert _p^p\right) ^{1/p}\nonumber \\ \end{aligned}$$
(2.5)

holds for all noncommutative martingales.

For independent increments this has recently been proved in [36]. One would actually expect \(f(p)=p\). As will become clear in the following, the validity of (2.5) would improve our main results and imply a number of results in different contexts. At the time of this writing we are unable to decide whether (2.5) holds. However, the commutative case was known to be true due to the work of Pinelis [49], who attributed it to Hitczenko.

3 Noncommutative Burkholder–Davis–Gundy type inequalities

We refer the readers to [27, 29, 30] for further details about the facts mentioned in this section. Let \(x=(E_1x,\ldots ,E_nx)\) be a (finite) martingale sequence with martingale differences \(dx_k\). For \(1\le p\le \infty \), we define

$$\begin{aligned} \Vert x\Vert _{h_p^c} =\left\| \left( \sum _{k}E_{k-1}(dx_k^* dx_k)\right) ^{1/2}\right\| _p,\quad \Vert x\Vert _{h_p^r}=\Vert x^*\Vert _{h_p^c}\,, \end{aligned}$$

and

$$\begin{aligned} \Vert x\Vert _{h_p^d}=\left( \sum _{k}\Vert dx_k\Vert _p^p\right) ^{1/p}. \end{aligned}$$

We are going to use the continuous filtration \(({\mathcal {N}}_t)_{t\ge 0}\subset {\mathcal {N}}\) in the following. Recall that a martingale \(x\) is said to have almost uniform (or a.u. for short) continuous path if for every \(T>0\), every \(\varepsilon >0\) there exists a projection \(e\) with \(\tau (1-e)<\varepsilon \) such that the function \(f_e:[0,T]\rightarrow {\mathcal {N}}\) given by \(f_e(t)=x_te\in {\mathcal {N}}\) is norm continuous. Let \(\sigma =\{0=s_0,\ldots ,s_n=T\}\) be a partition of the interval \([0,T]\) and \(|\sigma |\) its cardinality. Put

$$\begin{aligned}&\Vert x\Vert _{h_p^c([0,T];\sigma )} =\Bigg \Vert \sum _{j=0}^{|\sigma |-1}E_{s_{j}}|E_{s_{j+1}}x-E_{s_j}x|^2 \Bigg \Vert ^{1/2}_{p/2}, \quad 2\le p\le \infty \,,\\&\Vert x\Vert _{h_p^d([0,T];\sigma )} =\left( \sum _{j=0}^{|\sigma |-1}\Vert E_{s_{j+1}}x-E_{s_j}x\Vert _p^p \right) ^{1/p}, \quad 2\le p<\infty \,, \end{aligned}$$

and \(\Vert x\Vert _{h_p^r([0,T];\sigma )}=\Vert x^*\Vert _{h_p^c([0,T];\sigma )}\). Let \({\mathcal {U}}\) be an ultrafilter refining the natural order given by inclusion on the set of all partitions of \([0,T]\). Let \(x\in L_p({\mathcal {N}})\). For \(2\le p< \infty \), we define

$$\begin{aligned} \langle x,x\rangle _T=\lim _{\sigma , {\mathcal {U}}}\sum _{i=0}^{|\sigma |-1}E_{s_i}|E_{s_{i+1}}x-E_{s_i}x|^2\,. \end{aligned}$$

Here the limit is taken in the weak* topology and it is shown in [27] that the convergence is also true in \(L_{p}\) norm \(\Vert \cdot \Vert _{p/2}\) for all \(2<p<\infty \). We define the continuous version of \(h_p\) norms for \(2\le p<\infty \),

$$\begin{aligned}&\Vert x\Vert _{h_p^c([0,T])}=\lim _{\sigma , {\mathcal {U}}}\Vert x\Vert _{h_p^c([0,T];\sigma )} \,\!,\\&\Vert x\Vert _{h_p^d([0,T])}=\lim _{\sigma , {\mathcal {U}}}\Vert x\Vert _{h_p^d([0,T];\sigma )} \,. \end{aligned}$$

and \(\Vert x\Vert _{h_p^r([0,T])}=\Vert x^*\Vert _{h_p^c([0,T])}\) for \(2\le p < \infty \). Then for all \(2<p<\infty \)

$$\begin{aligned} \Vert x\Vert _{h_p^c([0,T])}=\Vert \langle x,x\rangle _T\Vert ^{1/2}_{p/2}\,. \end{aligned}$$
(3.1)

A martingale \(x\) is said to be of vanishing variation if \(\Vert x\Vert _{h^d_p([0, T])}=0\) for all \(T>0\) and all \(2<p<\infty \). We also write

$$\begin{aligned} \mathrm{var }_p(x)=\Vert x\Vert _{h_p^d([0,T])},\quad \end{aligned}$$

and let \(V_p({\mathcal {N}})\) denote the \(L_2({\mathcal {N}})\) closure of \(\{x\in L_p({\mathcal {N}}): \mathrm{var }_p(x)=0\}\).

The following results are proved in [27]. For any \(y\in L_p({\mathcal {N}})\), we write \(d_j y= E_{s_{j}}y-E_{s_{j-1}}y\). Put \(\Vert x\Vert _{L_p(\mathrm{var })}=\sup _{\sigma } \Vert (d_jx)\Vert _{L_p(\ell _1)}\), where the supremum is taken over all finite partitions of \([0,T]\), and the norm \(\Vert \cdot \Vert _{L_p(\ell _1)}\) was defined in [37], which we will not use after the next result.

Theorem 3.1

Let \(2<p<\infty \) and \(x\in L_p({\mathcal {N}}_T)\). Then for all \(\delta >0\), there exists a decomposition \(x=y^\delta +z^\delta \) satisfying the following

(1):

\(\mathrm{var }_p(y^\delta )<\delta \), \(z^\delta \in L_p(\mathrm{var })\).

(2):

Let \(P(x)=w^*\hbox {-}\lim _\delta y^\delta \). Here \(w^*\hbox {-}\lim \) denotes the weak* limit. Then \(P: L_p({\mathcal {N}})\rightarrow V_p({\mathcal {N}})\) is an orthogonal projection.

(3):

\(P(x)=x\) for all \(x\) with vanishing variation.

One may take \(y^\delta =w^*\hbox {-}\lim _\sigma \sum _{j=1}^{|\sigma |} d_j(d_jx1_{[|d_jx|\le \delta ]})\) where \(1_B\) is the spectral projection of \(d_jx\) restricted to the Borel set \(B\).

Lemma 3.2

If \(x\) has a.u. continuous path, then it is of vanishing variation.

Now let us prove the main results of this section, which can be regarded as the noncommutative version of Burkholder–Davis–Gundy type inequalities.

Theorem 3.3

Let \(x\) be a mean 0 martingale with a.u. continuous path. Then for every \(T>0\), we have

(1):

For \(2\le p<\infty \), if \(x\) is self-adjoint, then

$$\begin{aligned} \Vert E_T x\Vert _p\le C\sqrt{p}\liminf _{\sigma ,{\mathcal {U}}}\Vert x\Vert _{h_\infty ^c([0,T];\sigma )}\,\!. \end{aligned}$$

If \(x\) is not necessarily self-adjoint, then

$$\begin{aligned} \Vert E_T x\Vert _p\le C\sqrt{p}\liminf _{\sigma ,{\mathcal {U}}}\big (\Vert x\Vert _{h_\infty ^c([0,T];\sigma )}+\Vert x\Vert _{h_\infty ^r([0,T];\sigma )}\big )\,, \end{aligned}$$

where we may take \(C=2\sqrt{2}\).

(2):

For all \(2\le p < \infty \),

$$\begin{aligned} \Vert E_T x\Vert _p\le C'{p}\max \big \{\Vert x\Vert _{h_p^c([0,T])},\,\Vert x\Vert _{h_p^r([0,T])}\big \}\,. \end{aligned}$$

Proof

(1) First assume that \(x\) is self-adjoint and that \(x\in {\mathcal {N}}_T\). We follow the strategy used in the proof of Theorem 3.1. Fix a partition \(\sigma \) of \([0,T]\). We write \(h_p(\sigma )\) for \(h_p([0,T];\sigma )\) in the following proof. Let \(\delta >0\). We have \(d_jx = d_jx 1_{[|d_jx|>\delta ]}+d_jx 1_{[|d_jx|\le \delta ]}\). Conditioning again, we obtain

$$\begin{aligned} d_jx=d_j(d_jx 1_{[|d_jx|>\delta ]})+d_j(d_jx 1_{[|d_jx|\le \delta ]})\,. \end{aligned}$$

Put \(z^\delta _\sigma = \sum _{j=1}^{|\sigma |}d_j(d_jx 1_{[|d_jx|>\delta ]}) \) and \(y^\delta _\sigma =\sum _{j=1}^{|\sigma |}d_j(d_jx 1_{[|d_jx|\le \delta ]})\). Then clearly

$$\begin{aligned} \sup _{j=1,\cdots , |\sigma |} \Vert d_j(d_jx 1_{[|d_jx|\le \delta ]})\Vert _\infty \le 2\delta \,. \end{aligned}$$

Using Proposition 2.4 for some fixed \(0<\varepsilon \le 1\), we find

$$\begin{aligned} \Vert y^\delta _\sigma -\tau (y^\delta _\sigma )\Vert _p\le 2^{3/2}(1+\varepsilon )^{1/2}\sqrt{p}\Vert y^\delta _\sigma \Vert _{h^c_\infty (\sigma )}+2^{7/2} \Big (\frac{1+\varepsilon }{\sqrt{\varepsilon }}\Big ) p\delta \,. \end{aligned}$$
(3.2)

Note that

$$\begin{aligned} 0&\le E_{s_{j-1}}|d_j(d_jx 1_{[|d_jx|<\delta ]})|^2=E_{s_{j-1}}[(d_jx 1_{[|d_jx|<\delta ]})^2]\!-\![E_{s_{j-1}}(d_jx 1_{[|d_jx|<\delta ]})]^2\\&\le E_{s_{j-1}}[(d_jx 1_{[|d_jx|<\delta ]})^2]\le E_{s_{j-1}}[(d_jx)^2]\,. \end{aligned}$$

Then we have

$$\begin{aligned} \Vert y^\delta _\sigma \Vert _{h^c_\infty (\sigma )}&= \Bigg \Vert \sum _{i=0}^{|\sigma |-1}E_{s_{j-1}}|d_j(d_jx 1_{[|d_jx|<\delta ]})|^2\Bigg \Vert _{\infty }^{1/2}\\&\le \Bigg \Vert \sum _{i=0}^{|\sigma |-1}E_{s_{j-1}} |d_jx|^2\Bigg \Vert ^{1/2}_\infty =\Vert x\Vert _{h^c_\infty (\sigma )}. \end{aligned}$$

According to Theorem 3.1 (in our context, \(y^\delta =w^*\hbox {-}\lim _\sigma y^\delta _\sigma \)) and Lemma 3.2, we have \(x=w^*\hbox {-}\lim _{\delta \rightarrow 0} y^\delta \). Hence for any \(\lambda _i\ge 0,\sum _{i=1}^k \lambda _i=1\), we have

$$\begin{aligned} x=w^*\hbox {-}\lim _{\begin{array}{c} \delta _i\rightarrow 0\\ i=1,\ldots ,k \end{array}}\sum _{i=1}^k \lambda _i y^{\delta _i}\,. \end{aligned}$$

Since in a Banach space the weak closure and the norm closure of a convex set are the same, by the reflexivity of \(L_p({\mathcal {N}})\) we can find a net \(x_\alpha \) in the convex hull of \(\{y^\delta \}\) such that \(x_\alpha \rightarrow x\) in \(L_p({\mathcal {N}})\). Therefore by sending \(\delta \rightarrow 0\), we deduce from (3.2) that

$$\begin{aligned} \Vert x\Vert _p\le 2\sqrt{2}(1+\varepsilon )^{1/2}\sqrt{p}\liminf _{\sigma ,{\mathcal {U}}}\Vert x\Vert _{h_\infty ^c(\sigma )}\,\!, \end{aligned}$$

for all \(0<\varepsilon \le 1\). Sending \(\varepsilon \rightarrow 0\) yields the first assertion. If \(x\) is not self-adjoint, we write \(x=\mathfrak {R}(x)+i\mathfrak {I}(x)\) where \(\mathfrak {R}(x)=\frac{x+x^*}{2}\) and \(\mathfrak {I}(x)=\frac{x-x^*}{2i}\). Then the second assertion follows from the self-adjoint case by triangle inequality.

(2) Since \(x\) is of vanishing variation, \(\Vert x\Vert _{h_p^d([0,T])}=0\) for all \(2< p<\infty \). Using (1.1), we have

$$\begin{aligned} \Vert x\Vert _p\le C'p (\Vert x\Vert _{h_p^c([0,T];\sigma )}+\Vert x\Vert _{h_p^r([0,T];\sigma )})+p\Vert x\Vert _{h_p^d([0,T];\sigma )}\,\!. \end{aligned}$$

Taking limits on the right hand side yields the assertion for \(2<p<\infty \). The case \(p=2\) is proved by sending \(p\downarrow 2\). \(\square \)

4 Poincaré type inequalities and applications

4.1 Poincaré type inequalities

Let \((T_t)_{t\ge 0}\) be a semigroup of operators acting on a finite von Neumann algebra \(({\mathcal {N}},\tau )\) where \(\tau (1)=1\). Following [30, 31] we say \((T_t)\) is a standard semigroup if it satisfies the following assumptions:

  1. (1)

    Every \(T_t\) is a normal completely positive map on \({\mathcal {N}}\) such that \(T_t(1) = 1\);

  2. (2)

    Every \(T_t\) is self-adjoint, i.e. \(\tau (T_t (x) y)=\tau (x T_t(y))\) for all \(x,y\in {\mathcal {N}}\).

  3. (3)

    The family \((T_t)\) is pointwise weak* continuous. Equivalently, \(\lim _{t\rightarrow t_0} T_t x = T_{t_0}x\) with respect to the strong operator topology in \({\mathcal {N}}\) for any \(x\in {\mathcal {N}}\); see [43].

It is well known that Assumption (3) is further equivalent to that \((T_t)\) is a strongly continuous semigroup on \(L_2({\mathcal {N}},\tau )\), where \(T_t\) extends to \(L_2({\mathcal {N}},\tau )\) by \(T_t \Lambda (x)=\Lambda (T_t x)\) for \(x\in {\mathcal {N}}\) and \(\Lambda : {\mathcal {N}}\rightarrow L_2({\mathcal {N}},\tau )\) is the natural embedding. By [16], \((T_t)\) extends to a strongly continuous contraction semigroup on \(L_p({\mathcal {N}})\) for every \(1\le p< \infty \) with generator \(A\), i.e. \(T_t=e^{-tA}\). Write for \(1\le p<\infty \)

$$\begin{aligned} \mathrm{Dom }_p(A)=\left\{ f\in L_p({\mathcal {N}}): \lim _{t\rightarrow 0} (f-T_tf)/t \text{ converges } \text{ in } L_p({\mathcal {N}})\right\} \!. \end{aligned}$$

Then the classical semigroup theory asserts that \(\mathrm{Dom }_p(A)\) is dense in \(L_p({\mathcal {N}})\) and that if \(x\in \mathrm{Dom }_p(A)\) then \(T_tx\in \mathrm{Dom }_p(A)\). We also denote \(\mathrm{Dom }(A)=\mathrm{Dom }_2(A)\). Note that \(A\) is a positive operator on \(L_2({\mathcal {N}},\tau )\). The standard assumptions also imply that \(\tau (T_tx)=\tau (x)\) and thus \(T_t\)’s are faithful. In addition, \(T_t\) is a contraction on \({\mathcal {N}}\). Indeed, for \(x\in {\mathcal {N}}\), we have

$$\begin{aligned} \Vert T_t x\Vert _\infty =\sup _{\Vert y\Vert _1\le 1} |\tau ((T_tx)y)|=\sup _{\Vert y\Vert _1\le 1} |\tau (x(T_ty))|\le \sup _{\Vert y\Vert _1\le 1} \Vert T_ty\Vert _1\Vert x\Vert _\infty \le \Vert x\Vert _\infty \,. \end{aligned}$$

Recall that \(T_t\) is said to admit a reversed Markov dilation if

  1. (H1)

    there exists a larger finite von Neumann algebra \({\mathcal {M}}\) and a family \(\pi _t: {\mathcal {N}}\rightarrow {\mathcal {M}}\) of trace preserving \(*\)-homomorphism;

  2. (H2)

    there is a decreasing filtration \(({\mathcal {M}}_{[s})_{0\le s<\infty }\) with \(\pi _r(x)\in {\mathcal {M}}_{[s}\) for all \(r>s\) such that \(E_{[s}(\pi _t(x))=\pi _s(T_{s-t}x)\) for all \(t<s\) and \(x\in {\mathcal {N}}\).

Here we have \({\mathcal {M}}_{[t}=E_{[t}({\mathcal {M}})\). For elements \(x,y\in \mathrm{Dom }(A)\) we may define the gradient form, which is called Meyer’s “carré du champ” in the commutative theory,

$$\begin{aligned} 2\Gamma (x,y)=A(x^*)y+x^*A(y)-A(x^*y) \end{aligned}$$

and for \(x,y \in \mathrm{Dom }(A^2)\) the second order gradient

$$\begin{aligned} 2\Gamma _2(x,y)=\Gamma (Ax,y)+\Gamma (x, Ay)-A\Gamma (x,y). \end{aligned}$$

Recall that \((T_t)\) is called a noncommutative diffusion (or nc-diffusion for short) semigroup if \(\Gamma (x,x)\in L_1({\mathcal {N}})\) for all \(x\in \mathrm{Dom }(A^{1/2})\). If \((T_t)\) is nc-diffusion, then \(\Gamma (x,x)\in L_1({\mathcal {N}})\) is well-defined for \(x\in \mathrm{Dom }(A^{1/2})\) by extension. By duality, \(\Gamma (x,x)\in L_p({\mathcal {N}})\) for \(1\le p< \infty \) if and only if there exists \(C>0\) such that \(|\tau (\Gamma (x,x)y)|\le C \Vert y\Vert _{p'}\) for all \(y\) and \(1/p+1/p'=1\).

We will use the following crucial results proved by Junge et al. in [32], which is a noncommutative version of the Stroock–Varadhan martingale problem.

Theorem 4.1

Suppose that \((T_t)_{t\ge 0}\) is a standard nc-diffusion semigroup. Then \(T_t\) admits a reversed Markov dilation \(({\pi }_t)\) with a.u. continuous path, i.e. in addition to (H1) and (H2), for all \(x\in \mathrm{Dom }(A)\) and all \(S>0\),

$$\begin{aligned} {m}_s(x):={\pi }_s(T_s(x)),\quad 0\le s \le S \end{aligned}$$

is a (reversed) martingale with a.u. continuous path.

Remark 4.2

Let \(2\le p <\infty \). For the purpose of our main result, we extend the theorem to \(x\in \mathrm{Dom }(A^{1/2})\). Indeed, since \(\mathrm{Dom }(A^{1/2})\cap {\mathcal {N}}\) is a \(*\)-subalgebra of \({\mathcal {N}}\) by [16] and \(\mathrm{Dom }(A)\) is dense in \(L_2({\mathcal {N}})\), there exists a sequence \((x_n)\in \mathrm{Dom }(A)\) such that \(\lim _{n\rightarrow \infty }\Vert x_n-x\Vert _2=0\). But \(E_{[r}({\pi }_s T_s(x_n))={\pi }_r T_r(x_n)\) for \(s<r\). Taking limits on both sides, we find \(E_{[r}({\pi }_s T_s(x))={\pi }_r T_r(x)\) in \(L_2({\mathcal {N}})\). According to [42], the set of a.u. continuous path martingales is closed in \(L_2({\mathcal {N}})\). Hence \({\pi }_s T_s(x)\) has a.u. continuous path. Similar argument applies to the forward martingales, but we only need the reversed martingales in this paper.

Put \(L_p^0({\mathcal {N}})=\{x\in L_p({\mathcal {N}}): \lim _{t\rightarrow \infty }T_tx=0 \}\) for \(1\le p\le \infty \). Here the limit is taken with respect to \(\Vert \cdot \Vert _{L_p({\mathcal {N}})}\) for \(1\le p<\infty \) and with respect to the weak* topology for \(p=\infty \). Let \(\mathrm{Fix}=\{x\in {\mathcal {N}}: T_t x=x \text{ for } \text{ all } t\ge 0\}\). Then it was shown in [35] that \(\mathrm{Fix}\) is a von Neumann subalgebra and \(\mathrm{Fix}^\perp =L^0_\infty ({\mathcal {N}})\). Denote by \(E_\mathrm{Fix}: {\mathcal {N}}\rightarrow \mathrm{Fix}\) the conditional expectation which extends to a contraction on \(L_p({\mathcal {N}})\). Then for all \(x\in L_p({\mathcal {N}})\) we have \(x-E_\mathrm{Fix} x\in L_p^0({\mathcal {N}})\) and \(L_p^0({\mathcal {N}})\) is a complemented subspace of \(L_p({\mathcal {N}})\).

Lemma 4.3

Let \(2\le p<\infty \) and \((T_t)_{t\ge 0}\) be a standard nc-diffusion semigroup. Then for all \(0\le s<t\le \infty \), and \(x\in \mathrm{Dom }(A^{1/2})\cap L_p^0({\mathcal {N}})\) with \(\Gamma (T_rx,T_rx)\) uniformly bounded for \(r\ge 0\) in \(L_p({\mathcal {N}})\) we have

$$\begin{aligned} \Vert {m}(x)\Vert _{h^c_p([s,t])}=\left\| 2\int _s^t{\pi }_r(\Gamma (T_{r}x, T_{r}x))dr\right\| ^{1/2}_{p/2}\,. \end{aligned}$$

Proof

By Theorem 4.1, Remark 4.2 and Lemma 3.2, \(\mathrm{var }_p(m)=0\) for all \(2<p<\infty \). (3.1) implies for \(2<p<\infty \),

$$\begin{aligned} \Vert m\Vert _{h_p^c([s,t])}=\Vert \langle m,m\rangle _t- \langle m, m \rangle _s\Vert ^{1/2}_{p/2}\,. \end{aligned}$$

It follows from [30, Lemma 2.4.1] and uniform boundedness that

$$\begin{aligned} \langle m, m\rangle _s-\langle m, m\rangle _t =2\int _s^t{\pi }_r(\Gamma (T_{r}x, T_{r}x))dr\,. \end{aligned}$$

Here the integral when \(t=\infty \) is well-defined for \(x\in L_p^0({\mathcal {N}})\) according to [30, Proposition 2.4.3]. This gives the assertion for \(2<p<\infty \). The case \(p=2\) follows by sending \(p\downarrow 2\). \(\square \)

We are now ready to state our main result of this section.

Theorem 4.4

Suppose \(2\le p<\infty \). Let \(T_t=e^{-tA}\) be a standard nc-diffusion semigroup and \(\Gamma \) the gradient form associated with \(A\). Assume \(x\in L_p({\mathcal {N}})\cap \mathrm{Dom }(A^{1/2})\) satisfies

$$\begin{aligned} \tau (y\Gamma (T_t x,T_t x))\le e^{-2\alpha t} \tau (y T_t\Gamma (x,x)), \quad y\in {\mathcal {N}}, y\ge 0\,, \end{aligned}$$
(4.1)

for some \(\alpha >0\). Then we have the following Poincaré type inequalities

$$\begin{aligned}&\Vert x-E_\mathrm{Fix} x\Vert _p \le C\sqrt{p/\alpha }\max \{\Vert \Gamma (x,x)^{1/2}\Vert _\infty , \Gamma (x^*,x^*)^{1/2}\Vert _\infty \},\end{aligned}$$
(4.2)
$$\begin{aligned}&\Vert x-E_\mathrm{Fix} x\Vert _p \le C'\alpha ^{-1/2}p\max \{\Vert \Gamma (x,x)^{1/2}\Vert _p,\Gamma (x^*,x^*)^{1/2}\Vert _p\}, \end{aligned}$$
(4.3)

where we can take \(C=4\sqrt{2}\) in general and \(C=2\sqrt{2}\) if \(x\) is self-adjoint.

Proof

First assume \(2<p<\infty \). Notice that \(E_\mathrm{Fix} x\) is in the multiplicative domain of \(T_t\). Then \(\Gamma (x,x)=\Gamma (x-E_\mathrm{Fix} x,x-E_\mathrm{Fix} x)\). Without loss of generality we may assume \(x\in L_p^0({\mathcal {N}})\), which implies \(\lim _{t\rightarrow \infty }T_tx=E_{\mathrm{Fix }}(x)=0\) in \(L_p\). Fix a constant \(0<M<\infty \) and consider the reversed martingale \({m}_t(x)\) in Theorem 4.1 for \(t\in [0,M]\). By Theorem 3.3 (applied to reversed martingales), noticing that \({m}_0(x)={\pi }_0(x)\), we have

$$\begin{aligned}&\Vert {m}_0-E_{[M}({m}_0)\Vert _p \\&\quad \le C\sqrt{p}\liminf _{\sigma ,{\mathcal {U}}}\big (\Vert {m}_0-E_{[M}({m}_0)\Vert _{h_\infty ^c([0,M];\sigma )} +\Vert {m}_0-E_{[M}({m}_0)\Vert _{h_\infty ^r([0,M];\sigma )}\big )\,. \end{aligned}$$

Using the reversed Markov dilation and [30, Lemma 2.4.1 (iii)], we find (see [30, (2.12)])

$$\begin{aligned}&\Vert {m}_0-E_{[M}({m}_0)\Vert _{h_\infty ^c([0,M];\sigma )}\\&\quad =\Bigg \Vert \sum _{j=0}^{|\sigma |-1} E_{[s_{j+1}}|{m}_{s_j}(x)-{m}_{s_{j+1}}(x)|^2\Bigg \Vert ^{1/2}_{\infty }\\&\quad =\Bigg \Vert \sum _{j=0}^{|\sigma |-1} E_{[s_{j+1}}(\pi _{s_j}(|T_{s_j}x|^2)) -\pi _{s_{j+1}}(|T_{s_{j+1}}x|^2)\Bigg \Vert ^{1/2}_{\infty }\\&\quad =\Bigg \Vert \sum _{j=0}^{|\sigma |-1} \pi _{s_{j+1}}(T_{s_{j+1}-s_j}|T_{s_j}x|^2 -|T_{s_{j+1}}x|^2)\Bigg \Vert ^{1/2}_{\infty }\\&\quad =\Bigg \Vert 2\sum _{j=0}^{|\sigma |-1}\pi _{s_{j+1}}\left( \int _{0}^{s_{j+1}-s_j} T_{s_{j+1}-s_j-r} (\Gamma (T_{r+s_j} x,T_{r+s_j} x)) dr\right) \Bigg \Vert ^{1/2}_{\infty }\\&\quad =\Bigg \Vert 2\sum _{j=0}^{|\sigma |-1}\int _{s_j}^{s_{j+1}}E_{[s_{j+1}} {\pi }_r(\Gamma (T_rx,T_rx)) dr \Bigg \Vert ^{1/2}_{\infty }\,. \end{aligned}$$

Since \(E_{[s_{j+1}}\) and \({\pi }_r\) are contractions, we deduce from (4.1) that

$$\begin{aligned}&\Vert {m}_0-E_[M({m}_0)\Vert _{h_\infty ^c([0,M];\sigma )} \le \sqrt{2}\left( \sum _{j=0}^{|\sigma |-1}\int _{s_j}^{s_{j+1}} \Vert \Gamma (T_rx,T_rx)\Vert _\infty dr\right) ^{1/2} \\&\quad =\sqrt{2}\left( \sum _{j=0}^{|\sigma |-1}\int _{s_j}^{s_{j+1}}\sup _{y\ge 0,y\in {\mathcal {N}}, \Vert y\Vert _1\le 1}\tau (y\Gamma (T_rx,T_rx))dr\right) ^{1/2}\\&\quad \le \sqrt{2}\left( \sum _{j=0}^{|\sigma |-1}\int _{s_j}^{s_{j+1}}e^{-2\alpha r}\sup _{y\ge 0,y\in {\mathcal {N}}, \Vert y\Vert _1\le 1}\tau (yT_r\Gamma (x,x))dr\right) ^{1/2}\\&\quad =\sqrt{2}\left( \sum _{j=0}^{|\sigma |-1}\int _{s_j}^{s_{j+1}} e^{-2\alpha r} \Vert T_r\Gamma (x,x)\Vert _\infty dr \right) ^{1/2} \le \sqrt{2}\left( \int _0^M e^{-2\alpha r} \Vert \Gamma (x,x) \Vert _\infty dr \right) ^{1/2}\\&\quad \le \alpha ^{-1/2}\Vert \Gamma (x,x)\Vert _\infty ^{1/2}\,. \end{aligned}$$

Similarly,

$$\begin{aligned} \Vert {m}_0-E_{[M}({m}_0)\Vert _{h_\infty ^r([0,M];\sigma )} \le \alpha ^{-1/2}\Vert \Gamma (x^*,x^*)\Vert _\infty ^{1/2}\,. \end{aligned}$$

Hence we have

$$\begin{aligned} \Vert {m}_0-E_{[M}({m}_0)\Vert _p \le C\Big (\frac{p}{\alpha }\Big )^{1/2} (\Vert \Gamma (x,x)\Vert _\infty ^{1/2}+\Vert \Gamma (x^*,x^*)\Vert _\infty ^{1/2})\,. \end{aligned}$$

By the reversed Markov dilation,

$$\begin{aligned} \Vert E_{[M}({m}_0)\Vert _p=\Vert E_{[M}({\pi }_0(x))\Vert _p=\Vert \pi _M T_M x\Vert _p\le \Vert T_M x\Vert _p\,. \end{aligned}$$

Note that \(\lim _{M\rightarrow \infty }\Vert T_M x\Vert _p=0\) and that \(\Vert x\Vert _p=\Vert {m}_0\Vert _p\le \Vert {m}_0-E_{[M}({m}_0)\Vert _p+\Vert E_{[M}({m}_0)\Vert _p\). Sending \(M\rightarrow \infty \) gives the first assertion for \(2<p<\infty \). Sending \(p\downarrow 2\) gives the case \(p=2\).

For (4.3), note that (4.1) implies \(\Gamma (T_t x,T_t x)\) is uniformly bounded in \(L_p({\mathcal {N}})\). Then Theorem 3.3 and Lemma 4.3 imply that for \(M>0\), \(2<p<\infty \), and \(x\in \mathrm{Dom }(A^{1/2})\), we have

$$\begin{aligned}&\Vert {m}_0-E_{[M}({m}_0)\Vert _p\\&\quad \le \sqrt{2}C'{p}\max \left\{ \left\| \int _0^M {\pi }_r(\Gamma (T_{r}x, T_{r}x))dr \right\| _{p/2}^{1/2},\,\left\| \int _0^M {\pi }_r(\Gamma (T_{r}x^*, T_{r}x^*))dr \right\| _{p/2}^{1/2}\right\} \,. \end{aligned}$$

Similar to the above argument, (4.1) yields

$$\begin{aligned} \left\| \int _0^M {\pi }_r(\Gamma (T_{r}x, T_{r}x))dr \right\| _{p/2}^{1/2} \le \frac{1}{\sqrt{2\alpha }}\Vert \Gamma (x,x)\Vert _{p/2}^{1/2}\,. \end{aligned}$$

The rest of proof is the same as that of the first assertion. \(\square \)

If \(A\) has a spectral gap in \(L_p\), we can deduce the second inequality (4.3) from the main result of [30] on the noncommutative Riesz transform. However, we have explicit order \(p\) here. So far as we know, no previous method has achieved the order \(\sqrt{p}\) in the first inequality in the noncommutative setting.

Remark 4.5

In fact, if (4.1) holds for all \(x\in \mathrm{Dom }(A^{1/2})\), then \(T_t\) is a nc-diffusion semigroup. Indeed, it was proved in [32] that \(T_t\Gamma (x,x)\in L_1({\mathcal {N}})\) for \(t > 0\). Then (4.1) implies that \(\Gamma (T_tx, T_tx)\in L_1({\mathcal {N}})\). Taking limit gives \(\Gamma (x,x)\in L_1({\mathcal {N}})\).

Condition (4.1) is not convenient to check. In practice, we may pose stronger assumptions which are easy to verify. The following lemma is of course well known in the commutative case.

Lemma 4.6

Let \(T_t=e^{-tA}\) be a standard nc-diffusion semigroup. Let \(x\in {\mathcal {N}}\) be such that \(\Gamma _2(x,x)\) is well-defined. Then \(\Gamma _2(x,x)\ge \alpha \Gamma (x,x)\) implies (4.1).

Proof

Since \(T_t\) is positive, \(\alpha T_t\Gamma (x,x)\le T_t\Gamma _2(x,x)\). Let \(\widetilde{T}_t=e^{2\alpha t}T_t\). Consider the function

$$\begin{aligned} f(s)=\widetilde{T}_{t-s}\Gamma (\widetilde{T}_sx,\widetilde{T}_sx)=e^{2\alpha (t+s)}T_{t-s}\Gamma (T_sx,T_sx). \end{aligned}$$

Due to the assumption, \(f(s)\) is differentiable. Then

$$\begin{aligned} f'(s)&= 2\alpha e^{2\alpha (t+s)}T_{t-s}\Gamma (T_sx,T_sx)+e^{2\alpha (t+s)}T_{t-s}A\Gamma (T_sx,T_sx)\\&\quad -\, e^{2\alpha (t+s)}T_{t-s}[\Gamma (AT_sx,T_sx) +\Gamma (T_sx,AT_sx)]\\&= 2\alpha e^{2\alpha (t+s)}T_{t-s}\Gamma (T_sx,T_sx) - 2e^{2\alpha (t+s)}T_{t-s}\Gamma _2(T_sx,T_sx)\le 0 \end{aligned}$$

for all \(0< s< t\). We have by continuity \(\Gamma (\widetilde{T}_t x,\widetilde{T}_tx)=f(t)\le f(0)=\widetilde{T}_t\Gamma (x,x)\), or \(\Gamma (T_t x,T_t x)\le e^{-2\alpha t} T_t\Gamma (x,x)\) which implies (4.1). \(\square \)

For the purpose of future development, let us recall the definition of positive forms. Suppose \(\Theta : {\mathcal {N}}\times {\mathcal {N}}\rightarrow L_1({\mathcal {N}},\tau )\) is a sesquilinear form whose domain is a weakly dense \(*\)-subalgebra \(\mathrm{Dom }(\Theta )\) such that \(1\in \mathrm{Dom }(\Theta )\). In this paper, we follow the convention that a sesquilinear form is conjugate linear in the first component. \(\Theta \) is said to be positive if for all \(n\in {\mathbb {N}}\), \(x_1,\ldots ,x_n\in \mathrm{Dom }(\Theta )\), \((\Theta (x_i,x_j))_{i,j=1}^n\) is positive in \(M_n(L_1({\mathcal {N}}))\cong L_1(M_n\bar{\otimes }{\mathcal {N}})\). Given another sesquilinear form \(\Phi \), \(\Theta \ge \Phi \) if \(\Theta -\Phi \ge 0\). We refer the readers to [48, 56] for more details. For any \(n\in {\mathbb {N}}\), and any \(a_1,\ldots ,a_n\in \mathrm{Dom }(A)\) the \(n\times n\) matrix \((\Gamma (a_i,a_j))_{i,j=1}^n\) with entries in \({\mathcal {N}}\) is positive in \(M_n({\mathcal {N}})\). The following useful fact was due to Peterson [48]; see also [56] for the implication “\(\Rightarrow \)”.

Theorem 4.7

Let \((T_t)\) be a strongly continuous semigroup on \(L_2({\mathcal {N}})\). Then \((T_t)\) is a completely positive semigroup if and only if \(\Gamma \) is a positive form.

As in the commutative case, the domain of \(\Gamma \) and \(\Gamma _2\) is a delicate issue. Theorem 4.4 avoided this difficulty by considering individual element. In many cases we are interested in the Poincaré type inequalities for the whole space. Our next result is meant for this purpose.

Corollary 4.8

Let \(T_t=e^{-tA}\) be a standard nc-diffusion semigroup. Suppose that there exists a weakly dense self-adjoint subalgebra \({\mathcal {A}}\subset {\mathcal {N}}\) such that

(i) \(A({\mathcal {A}})\subset {\mathcal {A}}\); (ii) \(T_t({\mathcal {A}})\subset {\mathcal {A}}\); (iii) \({\mathcal {A}}\) is dense in \(\mathrm{Dom }(A^{1/2})\) in the graph norm of \(A^{1/2}\).

Assume \(\Gamma _2(x,x)\ge \alpha \Gamma (x,x)\) for some \(\alpha >0\) and for all \(x\in {\mathcal {A}}\). Then (4.1) holds for all \(x\in \mathrm{Dom }(A^{1/2})\). Moreover, all \(x\in L_p({\mathcal {N}})\) satisfies (4.2) and (4.3).

Proof

For \(x\in \mathrm{Dom }(A^{1/2})\) we deduce from assumption iii) that there exist \((x_n)\subset {\mathcal {A}}\) with \(\Gamma _2(x_n,x_n)\ge \alpha \Gamma (x_n,x_n)\) such that \(\Vert x_n-x\Vert _2\rightarrow 0\) and \(\Vert A^{1/2}x_n-A^{1/2}x\Vert _2\rightarrow 0\) as \(n\rightarrow \infty \). By [12, Section 9], we have \(\Vert \Gamma (x,x)\Vert _1=\langle A^{1/2} x, A^{1/2}x\rangle _{L_2({\mathcal {N}})}\). Since \(\Gamma \) is a complete positive form, we have for \(x,y\in \mathrm{Dom }(A^{1/2})\),

$$\begin{aligned} \left( \begin{array}{c@{\quad }c} \Gamma (x,x)&{} \Gamma (x,y)\\ \Gamma (y,x) &{} \Gamma (y,y) \end{array} \right) \ge 0\,. \end{aligned}$$

Note that \(\Gamma (x,y)^*=\Gamma (y,x)\). Then \(\Vert (\Gamma (x,x)+\varepsilon 1 )^{-1/2}\Gamma (x,y)(\Gamma (y,y)+\varepsilon 1)^{-1/2}\Vert _\infty \le 1\) for any \(\varepsilon >0\). Hence

$$\begin{aligned}&\Vert \Gamma (x,y)\Vert _1\\&\quad \le \Vert (\Gamma (x,x)+\varepsilon 1)^{1/2}\Vert _2 \Vert (\Gamma (x,x)+\varepsilon 1)^{-1/2}\Gamma (x,y)(\Gamma (y,y)+\varepsilon 1)^{-1/2}\Vert _\infty \\&\qquad \times \Vert (\Gamma (y,y)+\varepsilon 1)^{1/2}\Vert _2\\&\quad \le \Vert \Gamma (x,x)+\varepsilon 1\Vert _1^{1/2}\Vert \Gamma (y,y)+\varepsilon \Vert _1^{1/2}\,. \end{aligned}$$

Sending \(\varepsilon \rightarrow 0\), we have \(\Vert \Gamma (x,y)\Vert _1\le \Vert \Gamma (x,x)\Vert _1^{1/2}\Vert \Gamma (y,y)\Vert _1^{1/2}\). It follows that

$$\begin{aligned}&\Vert \Gamma (x_n,x_n)-\Gamma (x,x)\Vert _1\le \Vert \Gamma (x_n-x,x_n-x)\Vert _1+2\Vert \Gamma (x_n-x,x)\Vert _1\\&\quad \le \langle A^{1/2} (x_n-x), A^{1/2}(x_n-x)\rangle _{L_2({\mathcal {N}},\tau )}\\&\qquad +\,2 \langle A^{1/2} (x_n-x), A^{1/2}(x_n-x)\rangle ^{1/2}_{L_2({\mathcal {N}},\tau )} \Vert \Gamma (x,x)\Vert _1^{1/2} \,. \end{aligned}$$

Hence \(\lim _{n\rightarrow \infty } \Gamma (x_n,x_n)=\Gamma (x,x)\) in \(L_1({\mathcal {N}})\). Notice that \(T_t\) and \(A^{1/2}\) commute. Then for all \(t\ge 0\) and \(x\in \mathrm{Dom }(A^{1/2})\), \(T_t x\in \mathrm{Dom }(A^{1/2})\) and a similar argument as above gives that \(\lim _{n\rightarrow \infty } \Gamma (T_t x_n,T_t x_n)=\Gamma (T_t x,T_t x)\) in \(L_1({\mathcal {N}})\). Since Lemma 4.6 implies

$$\begin{aligned} \tau (y\Gamma (T_t x_n,T_t x_n))\le e^{-2\alpha t} \tau (y T_t\Gamma (x_n,x_n)) \end{aligned}$$

for all \(y\in {\mathcal {N}},y\ge 0\), sending \(n\rightarrow \infty \) on both sides yields the first assertion. For the “moreover” part, note that we only need to prove (4.2) and (4.3) for

$$\begin{aligned} \max \{\Vert \Gamma (x,x)^{1/2}\Vert _p, \Vert \Gamma (x^*,x^*)^{1/2}\Vert _p\}<\infty . \end{aligned}$$

Recall that \(\Gamma (x,x)\) is understood as the weak* limit of \(\Gamma ^{A_\varepsilon }(x,x)\) where \(A_\varepsilon = (I+\varepsilon A)^{-1} A\) (see [12, (3.2)]). If this limit exists in \(L_{p/2}\) for \(p> 2\), then \(\tau (\Gamma (x,x))\) is finite and hence \(x\in \mathrm{Dom }(A^{1/2})\). The individual result Theorem 4.4 then comes into play and completes the proof. \(\square \)

The condition \(\Gamma _2(x,x)\ge \alpha \Gamma (x,x)\) we posed here is usually called the curvature condition or \(\Gamma _2\)-criterion. The expression \(\max \{\Vert \Gamma (x,x)^{1/2}\Vert _\infty ,~\Vert \Gamma (x^*,x^*)^{1/2}\Vert _\infty \}\) is the so-called Lipschitz norm in the commutative theory. In the classical diffusion setting, Bakry and Emery [5] showed that the \(\Gamma _2\)-criterion implies logarithmic Sobolev inequality, which in turn yields the \(L_p\) Poincaré inequalities with constant \(C\sqrt{p}\) due to Aida and Stroock [2]; see also [1] for another proof. In the general non-diffusion setting, we will show that the first implication is not true. At the time of this writing, we do not know whether the \(L_p\) Poincaré inequalities follow from LSI in full generality. However, adapting our theory to the classical diffusion setting will result in a shortcut. Namely, we can directly show that \(\Gamma _2\)-criterion implies the \(L_p\) Poincaré inequalities with constants \(C\sqrt{p}\). Let \(T_t=e^{-tL}\) be a symmetric classical diffusion semigroup with infinitesimal generator \(L\) acting on a probability space \(({\mathbb {R}}^d, \mu )\). By the well known diffusion theory (see e.g. [52, Section VII.2]), under certain regularity conditions on \(L\), one can always construct a diffusion process \(X_t\) corresponding to \(T_t\) by solving a martingale problem.

Theorem 4.9

Assume that \(X_t\) is a classical diffusion process defined on \((\Omega , {\mathbb {P}})\) for the semigroup \(T_t=e^{-tL}\). Suppose \(\Gamma _2(f,f)\ge \alpha \Gamma (f,f)\) for the real-valued function \(f\). Then for \(2\le p<\infty \),

$$\begin{aligned} \Vert f-E_{\mathrm{Fix }}f \Vert _p\le C\sqrt{p}\Vert \Gamma (f,f)^{1/2}\Vert _p. \end{aligned}$$

Proof

The proof follows the same strategy as for (4.3). Assume \(2<p<\infty \) and \(\lim _{t\rightarrow \infty }T_tf=0\) in \(L_p\). By approximation, we may assume that \(f\) is compactly supported. Recall that for the diffusion process \(X_t\), we have

$$\begin{aligned} {\mathbb {E}}\,_{x}[f(X_{t})]=f(x)+{\mathbb {E}}\,_{x}\left[ \,\int _0^{t}-Lf(X_{s})ds\right] , \end{aligned}$$

where \({\mathbb {E}}\,_x\) is the expectation conditioning on \(\{X_0=x\}\). We obtain a martingale

$$\begin{aligned} M_t^f=f(X_t)-f(X_0)+\int _0^t Lf(X_s)ds \end{aligned}$$

adapted to \({\mathcal {F}}_t:=\sigma (X_s: s\le t)\). Fix a large constant \(K>0\). For \(0\le t\le K\), define \(\widetilde{X}_t =X_{K-t}\), \(\widetilde{\mathcal {F}}_{[t}={\mathcal {F}}_{K-t}\), \(N_t^f=(T_t f)(X_{K-t})\). Note that in this setting, the reversed Markov dilation is given by \(\pi _t f= f(\widetilde{X}_t)\). Then \((N_t^f)_{0\le t\le K}\) is a reversed martingale with respect to the filtration \(\widetilde{\mathcal {F}}_{[t}\). Indeed, by the Markov property, for \(s<t\)

$$\begin{aligned} E[ N_s^f|\widetilde{\mathcal {F}}_{[t}]&= E[(T_sf) (X_{K-s})|{\mathcal {F}}_{K-t}]\\&= T_{(K-s)-(K-t)}T_sf(X_{K-t})=T_tf(X_{K-t})=N_t^f. \end{aligned}$$

By Lemma 4.3 with \(\pi _r f= f(X_{K-r})\), we have

$$\begin{aligned} \langle N^f,N^f\rangle _K -\langle N^f,N^f\rangle _0=2\int _0^K \Gamma (T_rf,T_rf)(X_{K-r})dr. \end{aligned}$$

Applying the BDG inequality (1.2) (see [6, Proposition 4.2]) to \(N_t^f\) on \([0,K]\), we have

$$\begin{aligned} \Vert T_Kf(X_0)-f(X_K)\Vert _p\le C\sqrt{p} \Vert \langle N^f,N^f\rangle _K -\langle N^f,N^f\rangle _0\Vert _{p/2}^{1/2}. \end{aligned}$$

Here we used the continuity of the sample paths of the diffusion process \(X_t\) so that the conditional square function and the unconditional square functions coincide in continuous time (see e.g. [17, Remark VII.43(b)] or [54, VI.34]). Since \(\mu \) is the invariant measure, we have for any \(0\le t\le K\),

$$\begin{aligned} \Vert f(X_t)\Vert _p^p=\int |f(X_t)|^p d{\mathbb {P}}=\int {\mathbb {E}}\,_x|f(X_t)|^p \mu (dx)=\int T_t|f|^p(x)\mu (dx)=\int |f|^pd\mu \,. \end{aligned}$$

It follows that \(\Vert (T_Kf)(X_0)\Vert _p=\Vert T_K f\Vert _p \rightarrow 0\) as \(K\rightarrow \infty \). By the triangle inequality,

$$\begin{aligned} \Vert f\Vert _p=\Vert f(X_K)\Vert _p\le \Vert f(X_K)-T_Kf(X_0)\Vert _p+\Vert T_K f(X_0)\Vert _p. \end{aligned}$$

The rest of the proof is the same as that of (4.3). \(\square \)

Our first example is very simple, but it clarifies that \(\Gamma _2\)-criterion no longer implies LSI in the general non-diffusion setting. Let us recall the following generalized Schwartz inequality, which is called Choi’s inequality; see [11, Corollary 2.8].

Lemma 4.10

Let \(\phi : {\mathcal {M}}\rightarrow {\mathcal {N}}\) be a contractive completely positive map between von Neumann algebras. Then \([\phi (x_i^*x_j)]\ge [\phi (x_i^*)\phi (x_j)]\) for any \(n\in {\mathbb {N}}\) and any \(x_1,\ldots ,x_n\in {\mathcal {M}}\).

Proof

Since \(\phi \) is complete positive, \(\phi \otimes I_n\) is 2-positive, here \(I_n\) is the identity matrix. Let

$$\begin{aligned} X=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} x_1 &{}x_2&{}\cdots &{}x_n\\ 0&{} 0&{}\cdots &{}0\\ \vdots &{} &{}\ddots &{} \\ 0&{}0&{}\cdots &{}0 \end{array}\right) \!. \end{aligned}$$

By [47, Exercise 3.4], \([\phi \otimes I_n (X)]^*[\phi \otimes I_n (X)]\le \phi \otimes I_n (X^*X)\). The proof is complete. \(\square \)

Example 4.11

(Conditional expectation) Let \(E:{\mathcal {M}}\rightarrow {\mathcal {N}}\) be the conditional expectation and \(A=I-E\). For \(x,y\in {\mathcal {M}}\), a calculation gives

$$\begin{aligned} 2\Gamma (x,y)=x^* y-E(x^*)y-x^*E(y)+E(x^*y)\,. \end{aligned}$$

By Lemma 4.10, \(2[\Gamma (x_i,x_j)]\ge [(x_i-Ex_i)^*(x_j-Ex_j)]\ge 0\) for \(x_1,\ldots ,x_n\in {\mathcal {M}}\). We deduce from Theorem 4.7 that \(A\) generates a completely positive semigroup \(T_t=e^{-tA}\) acting on \({\mathcal {M}}\). It is easy to check \(T_t\) is a standard nc-diffusion semigroup. Let \(\Gamma ,\Gamma _2\) be the gradient forms associated to \(T_t\).

Proposition 4.12

\(\Gamma _2\ge \frac{1}{2}\Gamma \) in \({\mathcal {M}}\).

Proof

Note that \(AE=EA=0\). We find

$$\begin{aligned} 4\Gamma _2(x,y)=x^*y-E(x^*)y-x^*E(y)-2E(x^*)E(y)+3E(x^*y)\,. \end{aligned}$$

Hence \((\Gamma _2-\frac{1}{2}\Gamma )(x,y)=\frac{1}{2}(E(x^*y)-E(x^*)E(y))\). Since \(E\) is contractive completely positive, it follows from Lemma 4.10 that \(\Gamma _2-\frac{1}{2}\Gamma \) is a positive form. \(\square \)

The logarithmic Sobolev inequality fails, however. Indeed, the LSI reads as follows in this case: for \(x\ge 0\),

$$\begin{aligned} \tau (x^2\ln (|x|/\Vert x\Vert _2))\le C\Big (\Vert x\Vert _2^2-\frac{1}{2}[\tau (E(x^*)x)+\tau (x^*E(x))]\Big )\,. \end{aligned}$$

It is easy to see this is not true. Indeed, let us consider the Lebesgue probability space \(([0,1],dt)\) with \(E\) being the expectation, i.e. \(E(x)=\int _{[0,1]}x(t)dt=\tau (x)\). Set \(x_n(t)=\sqrt{n}1_{[0,1/n]}(t)\). Then \(\Vert x_n\Vert _2=1\) and the left-hand side is \(\tau (x_n^2\ln x_n)=\frac{1}{2}\ln n\). However, the right-hand side is less than a constant \(C>0\), which is impossible for large \(n\).

4.2 Deviation and transportation inequalities

As is well-known, the Poincaré inequality with constant \(\sqrt{p}\) implies the concentration phenomenon. We are going to prove a noncommutative version of exponential integrability due to Bobkov and Götze [8] in the commutative case. The following variant was due to Efraim and Lust-Piquard in the case of Walsh system.

Corollary 4.13

Under the assumptions of Theorem 4.4, we have

$$\begin{aligned} \tau (e^{|x-E_\mathrm{Fix} x|})\le 2\exp \Big (\frac{C}{\alpha }\max \{\Vert \Gamma (x,x)\Vert _\infty , \Vert \Gamma (x^*,x^*)\Vert _\infty \}\Big )\,, \end{aligned}$$
(4.4)

and for \(t>0\)

$$\begin{aligned} \mathrm{Prob }(|x-E_\mathrm{Fix} x|\ge t) \le 2 \exp \Big (-\frac{\alpha t^2}{4C\max \{\Vert \Gamma (x,x)\Vert _\infty ,\Vert \Gamma (x^*,x^*)\Vert _\infty \}}\Big )\,.\qquad \quad \end{aligned}$$
(4.5)

We may take \(C=32e\) in general and \(C=8e\) for \(x\) self-adjoint.

Proof

We follow the proof in the commutative case; see [19, Corollary 4.1 and 4.2]. Since \(\Gamma (x,x)=\Gamma (x-E_\mathrm{Fix} x, x- E_\mathrm{Fix} x)\), we may assume \(E_\mathrm{Fix}(x)=0\). Put \(M=\max \{\Vert \Gamma (x,x)^{1/2}\Vert _\infty , \Gamma (x^*,x^*)^{1/2}\Vert _\infty \}\). Note that \(\frac{k^k}{(2k-1)!!}\le \big (\frac{e}{2} \big )^k\) for all \(k\in {\mathbb {N}}\). By functional calculus and (4.2),

$$\begin{aligned} \frac{1}{2} \tau (e^{|x|})&\le \tau (\cosh x) = 1+\sum _{k=1}^\infty \frac{1}{(2k)!}\Vert x\Vert _{2k}^{2k}\\&\le 1+\sum _{k=1}^\infty \frac{C^{2k} (2k)^k}{\alpha ^k(2k)!}M^{2k} \le 1+\sum _{k=1}^\infty \frac{k^k (CM)^{2k}}{\alpha ^k k!(2k-1)!!}\\&\le 1+\sum _{k=1}^\infty \frac{(e/2)^k (CM)^{2k}}{\alpha ^k k!} =\exp \Big (\frac{eC^2M^2}{2\alpha }\Big )\,. \end{aligned}$$

We have proved the first assertion for \(C=32e\) and we can take \(C=8e\) if \(x\) is self-adjoint. For the second inequality, we deduce from Chebyshev inequality that

$$\begin{aligned} \tau (1_{[t,\infty )}(|x|)) \le e^{-\lambda t} \tau (e^{\lambda |x|})\le 2 e^{-\lambda t + C\lambda ^2M^2/\alpha }\,. \end{aligned}$$

Then the assertion follows from minimizing the right hand side with respect to \(\lambda \). \(\square \)

The improvement in the situation of commutative diffusion in Theorem 4.9 also gives an intermediate term in (4.4) for self-adjoint element \(x\), i.e.

$$\begin{aligned} \tau (e^{|x-E_\mathrm{Fix} x|})&\le 2\tau \exp \Big (\frac{C'}{\alpha }\Gamma (x,x)\Big ) \le 2\exp \Big (\frac{C'}{\alpha }\Vert \Gamma (x,x)\Vert _\infty \Big ). \end{aligned}$$

We do not have such an intermediate term in the fully noncommutative generality without the help of (2.5). However, it seems the Lipschitz norm is the right choice in application to the concentration inequality. In this sense, we did not lose much even if we use a larger norm. The following result is simply a one side version of Corollary 4.13. We record it here for future references.

Proposition 4.14

Under the hypotheses of Theorem 4.4, assume further that \(x\) is self-adjoint. Then for \(t\in {\mathbb {R}}\)

$$\begin{aligned} \tau (e^{t(x-E_\mathrm{Fix} x)})\le e^{c \Vert \Gamma (x,x)\Vert _\infty t^2}\,, \end{aligned}$$
(4.6)

and for \(t>0\),

$$\begin{aligned} \mathrm{Prob }(x-E_\mathrm{Fix} x \ge t) \le \exp \Big (-\frac{t^2}{4c\Vert \Gamma (x,x)\Vert _\infty }\Big ) \end{aligned}$$
(4.7)

where the constant \(c\) only depends on \(\alpha \).

Proof

Again it suffices to consider (4.6) for \(x\) with \(E_\mathrm{Fix}(x)=0\) since \(\Gamma (x,x)=\Gamma (x-E_\mathrm{Fix} x, x-E_\mathrm{Fix} x)\). From the proof of (4.4), we know there exists \(C>0\) such that for \(t\in {\mathbb {R}}\)

$$\begin{aligned} \tau (e^{tx})\le \tau (e^{tx})+\tau (e^{-tx})\le 2 e^{C \Vert \Gamma (x,x)\Vert _\infty t^2/\alpha }\,. \end{aligned}$$

Then for \(t^2\Vert \Gamma (x,x)\Vert _\infty \,{\ge }\, 1\), we have \(\tau (e^{tx})\,{\le }\, e^{(\ln 2+C/\alpha )\Vert \Gamma (x,x)\Vert _\infty t^2}\). For \(t^2\Vert \Gamma (x,x)\Vert _\infty < 1\),

$$\begin{aligned} \tau (e^{tx})&= 1+\sum _{k=2}^\infty \frac{t^k\tau (x^k)}{k!}\le 1+\sum _{k=2}^\infty \frac{t^kC^k k^{k/2}\Vert \Gamma (x,x)\Vert _\infty ^{k/2}}{\alpha ^{k/2}k!}\\&\le 1+ c \Vert \Gamma (x,x)\Vert _\infty t^2\le e^{c\Vert \Gamma (x,x)\Vert _\infty t^2} \end{aligned}$$

for some constant \(c=c(\alpha )\) since \(\tau (x)=0\) and the series \(\sum _{k=2}^\infty \frac{k^{k/2}}{k!}\) converges. The second assertion follows in the same way as (4.5). \(\square \)

The exponential integrability result (4.6) was proved by Bobkov and Götze [8] in the commutative case by using a variant of LSI. They also deduced a transportation inequality from (4.6). We will follow their approach to obtain a noncommutative version of transportation inequality. Since LSI is not available in our noncommutative theory, our Poincaré inequalities might be a more universal approach to the transportation inequality. Let us first define Wasserstein distance and entropy in the noncommutative setting.

Definition 4.15

Let \(\rho \) and \(\sigma \) be positive \(\tau \)-measurable operators (e.g. density matrices) affiliated with \(({\mathcal {M}},\tau )\). The noncommutative entropy of \(\rho \in L_1({\mathcal {M}},\tau )\) is given by

$$\begin{aligned} \mathrm{Ent }(\rho )=\tau (\rho \ln (\rho /\tau (\rho ))\,. \end{aligned}$$

Let \(\phi \) and \(\psi \) be states on \({\mathcal {M}}\). The \(L_1\)-Wasserstein distance between \(\phi \) and \(\psi \) is defined by

$$\begin{aligned} W_1^A(\phi ,\psi ) =\sup \{ |\phi (x)-\psi (x)|: x \, \mathrm{self}\hbox {-}\mathrm{adjoint },\,\Vert \Gamma (x,x)\Vert _\infty \le 1\}\,. \end{aligned}$$

The \(L_1\)-Wasserstein distance between \(\rho \) and \(\sigma \) is \(W_1^A(\rho ,\sigma )=W_1^A(\phi _\rho ,\phi _\sigma )\) for \(\phi _\rho (\cdot )=\tau (\cdot \rho )/\tau (\rho )\) and \(\phi _\sigma (\cdot )=\tau (\cdot \sigma )/\tau (\sigma )\).

Here the superscript \(A\) in \(W_1^A\) is to emphasize the dependence on the generator of the semigroup \(T_t\). We may ignore the superscript \(A\) for simplicity in the following. It is easy to check that \(W_1^A\) is a pseudometric but may not be a metric in general. Our definition of Wasserstein distance coincides with the classical definition in the commutative case due to the Kantorovich–Rubinstein theorem; see e.g. [60, Theorem 5.10]. It is also closely related to the quantum metric in the sense of Rieffel [53]. Now we state a general fact on the relationship between conditional expectation and entropy.

Lemma 4.16

Let \(\rho \in L_1({\mathcal {M}},\tau )\) with \(\rho \ge 0\) and \(\tau (\rho )=1\) and \(E: {\mathcal {M}}\rightarrow {\mathcal {N}}\) the conditional expectation onto subalgebra \({\mathcal {N}}\). Then

$$\begin{aligned} \tau (E\rho \ln E\rho )\le \tau (\rho \ln \rho )\,. \end{aligned}$$

Proof

Let \(\rho _n=\rho 1_{[0,n]}(\rho )\). Then \(\rho _n\in L_p({\mathcal {M}},\tau )\) for all \(p\ge 1\). It is easy to see that \(\rho _n\rightarrow \rho \) in the measure topology. Notice that \(\tau [(\rho _n/\tau (\rho _n))\ln (\rho _n/\tau (\rho _n))] = \lim _{p\downarrow 1}\frac{\Vert \rho _n/\tau (\rho _n)\Vert _p^p -1}{p-1}\). This yields

$$\begin{aligned}&\tau \left( \frac{E\rho _n}{\tau (\rho _n)} \ln \frac{E\rho _n}{\tau (\rho _n)}\right) =\lim _{p\downarrow 1}\frac{\Vert E\rho _n/\tau (\rho _n)\Vert ^p_p -1}{p-1}\\&\quad \le \lim _{p\downarrow 1}\frac{\Vert \rho _n/\tau (\rho _n)\Vert ^p_p -1}{p-1} =\tau \left( \frac{\rho _n}{\tau (\rho _n)}\ln \frac{\rho _n}{\tau (\rho _n)}\right) \,. \end{aligned}$$

Let \(\mu \) be the distribution of \(\rho \). Then

$$\begin{aligned} \tau \left( \frac{\rho _n}{\tau (\rho _n)}\ln \frac{\rho _n}{\tau (\rho _n)}\right) =\frac{1}{\tau (\rho _n)}\int _0^n x\ln x \mu (dx)-\ln \tau (\rho _n)\rightarrow \tau (\rho \ln \rho ). \end{aligned}$$

Following [20], we denote the generalized singular number of \(\rho \) by \(\mu _t(\rho )\). Note that \(\Vert E\rho _n-E\rho \Vert _1\le \Vert \rho _n-\rho \Vert _1\rightarrow 0\) as \(n\rightarrow \infty \). We have for every \(t>0\),

$$\begin{aligned} \mu _t(E\rho _n-E\rho )\le t^{-1}\int _0^t \mu _s(E\rho _n- E\rho ) ds\le t^{-1}\Vert E\rho _n-E\rho \Vert _1\,. \end{aligned}$$

Then \(\lim _{n\rightarrow \infty }\mu _t(E\rho _n-E\rho )=0\). By [20, Lemma 3.1], \((E\rho _n)\) converges to \(E\rho \) in the measure topology. Since \(0\le E\rho _n\le E\rho \), by [20, Lemma 2.5], \(\mu _t(E\rho _n)\le \mu _t(E\rho )\). We deduce from [20, Lemma 3.4] that \(\lim _{n\rightarrow \infty }\mu _t(E\rho _n)=\mu _t(E\rho )\) for \(t>0\). Now consider \(g(x)=x\ln x = x\ln x 1_{[1,\infty )(x)}- (-x\ln x 1_{(0,1)})\). Both functions in the decomposition are nonnegative Borel functions vanishing at the origin. It follows from [20, (3)] that \(\tau (E\rho \ln E\rho )=\int _0^1 \mu _t(E\rho )\ln \mu _t(E\rho ) dt\). Since for fixed \(\varepsilon >0\), \([\mu _t(E\rho _n)\ln \mu _t(E\rho _n)]_n\) is uniformly bounded on \(t\in [\varepsilon , 1]\), we have

$$\begin{aligned}&\tau (E\rho \ln E\rho )=\sup _{\varepsilon >0} \int _{\varepsilon }^1 \mu _t(E\rho )\ln \mu _t(E\rho )dt\\&\quad =\sup _{\varepsilon >0}\lim _{n\rightarrow \infty }\int _{\varepsilon }^1\mu _t(E\rho _n)\ln \mu _t(E\rho _n)dt\le \limsup _{n\rightarrow \infty }\tau (E\rho _n\ln E\rho _n)\\&\quad \le \limsup _{n\rightarrow \infty }\tau (\rho _n\ln \rho _n)\,. \end{aligned}$$

This completes the proof. \(\square \)

The next result in the commutative setting is well known; see e.g. [18, Section 6.2].

Lemma 4.17

Let \(\sigma \) be a self-adjoint \(\tau \)-measurable operator. Then,

$$\begin{aligned} \ln \tau (e^{\sigma })=\sup \{\tau (\rho \sigma )-\tau (\rho \ln \rho ):\rho \ge 0, \tau (\rho )=1\}\,. \end{aligned}$$
(4.8)

Therefore, for all positive \(\rho \in L_1({\mathcal {M}},\tau )\)

$$\begin{aligned} \mathrm{Ent }(\rho )=\sup \{\tau (\sigma \rho ): \sigma \,\mathrm{self-adjoint},\, \tau (e^\sigma )\le 1\}\,. \end{aligned}$$
(4.9)

Proof

Let \(\sigma \) be a self-adjoint operator \(\tau \)-measurable operator. Consider the von Neumann subalgebra \({\mathcal {N}}\) generated by \(\{f(\sigma ):f: {\mathbb {C}}\rightarrow {\mathbb {C}}\,\text { bounded measurable}\}\). Then there exists a conditional expectation \(E:{\mathcal {M}}\rightarrow {\mathcal {N}}\) which can extend to a contraction \(L_p({\mathcal {M}},\tau )\rightarrow L_p({\mathcal {N}},\tau )\) for all \(1\le p\le \infty \). Assume \(\tau (\rho )=1\). Then \(E(\rho )\in L_1({\mathcal {N}},\tau )\). But \({\mathcal {N}}\) is commutative and \(\tau (E(\rho ))=\tau (\rho )=1\). After identifying \(\tau \) with a probability measure denoted still by \(\tau \), we use Jensen’s inequality for the measure \(E(\rho ) d\tau \) to deduce that

$$\begin{aligned} \tau (\sigma E(\rho ))-\tau (E(\rho )\ln E(\rho ))=\tau (\ln (e^\sigma E(\rho )^{-1}) E(\rho )))\le \ln \tau (e^\sigma )\,. \end{aligned}$$

Using Lemma 4.16 and noticing that \(\tau (\sigma \rho )=\tau (\sigma E(\rho ))\), we find

$$\begin{aligned} \tau (\sigma \rho )- \tau (\rho \ln \rho )\le \ln \tau (e^\sigma )\,. \end{aligned}$$

For the reverse inequality, put \(\sigma _n=\sigma 1_{(-\infty ,n]}(\sigma )\) for \(n\in {\mathbb {N}}\) where \(1_{(-\infty ,n]}(\sigma )\) is the spectral projection of \(\sigma \). Plugging \(\rho _n=e^{\sigma _n}/\tau (e^{\sigma _n})\) into the right hand side of (4.8), we have

$$\begin{aligned} \tau (\sigma \rho _n)- \tau (\rho _n\ln \rho _n)=\frac{\tau ((\sigma -\sigma _n)e^{\sigma _n})}{\tau (e^{\sigma _n})}+ \ln \tau (e^{\sigma _n}) \,. \end{aligned}$$

By the spectral decomposition theorem of \(\sigma \), \(\tau ((\sigma -\sigma _n)e^{\sigma _n})\ge 0\). Then for all \(n\) we have

$$\begin{aligned} \sup \{\tau (\rho \sigma )-\tau (\rho \ln \rho ):\rho \ge 0, \tau (\rho )=1\} \ge \ln \tau (e^{\sigma _n})\,. \end{aligned}$$

By Fatou’s lemma [20, Theorem 3.5] \(\liminf _{n\rightarrow \infty }\ln \tau (e^{\sigma _n})\ge \ln \tau (e^\sigma )\). This proves (4.8). For (4.9), note that, by (4.8), \(\tau (e^\sigma )\le 1\) implies \(\tau (\sigma \rho )\le \tau (\rho \ln \rho )\) for all positive \(\rho \in L_1({\mathcal {M}},\tau )\) with \(\tau (\rho )=1\). If \(\tau (\rho )\ne 1\), we consider \(\rho '=\rho /\tau (\rho )\) and find \(\tau (\sigma \rho )\le \tau (\rho \ln (\rho /\tau (\rho )))\). The equality is achieved by \(\sigma =\ln \rho -\ln \tau (\rho )\). This proves the second assertion. \(\square \)

Theorem 4.18

Let \(({\mathcal {M}},\tau )\) be a noncommutative probability space. Then

$$\begin{aligned} W_1(\rho ,1)\le \sqrt{2c \mathrm{Ent }(\rho )}, \end{aligned}$$
(4.10)

for all \(\rho \ge 0\) with \(\tau (\rho )=1\) if and only if for every self-adjoint \(\tau \)-measurable operator \(x\) affiliated with \({\mathcal {M}}\) such that \(\Vert \Gamma (x,x)\Vert _\infty \le 1\) and \(\tau (x)=0\),

$$\begin{aligned} \tau (e^{tx})\le e^{ct^2/2}, \,\mathrm{for all } \,t\in \mathbb {R}_+\,. \end{aligned}$$
(4.11)

Proof

Thanks to the preceding two lemmas, the proof is the same as that in the commutative case in [8]. We provide it here for completeness. Setting \(\sigma =tx-ct^2/2\) in (4.9) and assuming (4.11) we find

$$\begin{aligned} \tau ((tx-ct^2/2)\rho )\le \mathrm{Ent }(\rho )\,. \end{aligned}$$
(4.12)

Since \(\tau (x)=0\) and \(\tau (\rho )=1\), it follows that \(\tau (x\rho -x)\le \frac{ct}{2}+\frac{1}{t} \mathrm{Ent }(\rho ).\) Minimizing right hand side gives

$$\begin{aligned} \tau (x\rho -x)\le \sqrt{2c\mathrm{Ent }(\rho )}\,. \end{aligned}$$
(4.13)

Note that \(\tau (x\rho -x)=\tau (\mathring{x}\rho -\mathring{x})\) for all \(x\) where \(x=\mathring{x}+\tau (x)\). Taking \(\sup \) over all self-adjoint \(x\) with \(\Vert \Gamma (x,x)\Vert _\infty \le 1\) on the left hand side of (4.13) gives (4.10). For the other direction, note that (4.12) is equivalent to (4.10) by reversing the above argument. Then (4.11) follows from (4.8) by setting \(\sigma =tx-ct^2/2\). \(\square \)

If \(\mathrm{Fix} = {\mathbb {C}}1\) (i.e. the system \(({\mathcal {M}},T_t)\) is ergodic), then combining the above theorem with (4.6), we find the transportation inequality (4.10) under the assumptions of Theorem 4.4. In fact, we even have a non-ergodic version of transportation inequality.

Corollary 4.19

Suppose \(\tau (e^{t(x-E_\mathrm{Fix} x)})\le e^{ct^2}\) for any \(\tau \)-measurable self-adjoint operator \(x\) affiliated to \({\mathcal {M}}\) such that \(\Vert \Gamma (x,x)\Vert _\infty \le 1\). Then

$$\begin{aligned} W_1(\rho , E_\mathrm{Fix} \rho )\le \sqrt{2c \mathrm{Ent }(\rho )}\,. \end{aligned}$$
(4.14)

for all \(\rho \ge 0\) with \(\tau (\rho )=1\). In particular, (4.10) holds under the additional assumption \(E_\mathrm{Fix} \rho =1\).

Proof

The proof modifies a little that of Theorem 4.18. Since \(\tau (\rho )=1\), we have \(\tau ([t(x-E_\mathrm{Fix} x)-ct^2/2]\rho )\le \mathrm{Ent }(\rho )\). Then we deduce that \(\tau (\rho x -\rho E_\mathrm{Fix}(x))\le \sqrt{2c\mathrm{Ent }(\rho )}\). Since \(\tau (\rho E_\mathrm{Fix}(x))=\tau (E_\mathrm{Fix}(\rho )x)\), we have

$$\begin{aligned} \tau (\rho x-E_\mathrm{Fix}(\rho )x)\le \sqrt{2c\mathrm{Ent }(\rho )}\,. \end{aligned}$$

Taking \(\sup \) over all self-adjoint \(x\) with \(\Vert \Gamma (x,x)\Vert _\infty \le 1\) gives the assertion. \(\square \)

It is easy to see that the assumptions are fulfilled by the hypotheses of Theorem 4.4. The point here is that even though the fixed point algebra \(\mathrm{Fix}\) is not trivial we still have a transportation inequality although in certain situation the inequality does fail.

Remark 4.20

Let \(\rho \) be a positive operator with \(\tau (\rho )=1\). For \(\rho \in \mathrm{Fix}\), define \(B(\rho )=\{f\in L_1({\mathcal {N}}):E_{\mathrm{Fix }}(f)=\rho \}\). Then for \(f_1,f_2\in B(\rho )\), we have \(W_1(f_1,f_2)\le W_1(f_1,\rho )+ W_1(f_2,\rho )<\infty \). However, if \(f_1\in B(\rho _1), f_2\in B(\rho _2)\) and \(\rho _1\ne \rho _2\), then

$$\begin{aligned} W_1(\rho _1,\rho _2)\ge \sup \{|\tau (\rho _1 x-\rho _2 x)|:{x\in \mathrm{Fix}, \Vert \Gamma (x,x)\Vert _\infty \le 1}\}=\infty \,. \end{aligned}$$

It follows that \(W_1(f_1,f_2)\ge |W_1(\rho _1,\rho _2)-W_1(f_1,\rho _1)-W_1(f_2,\rho _2)|=\infty \). This yields an interesting geometric picture: operators in the same “fiber” \(B(\rho )\) have finite distance between one another while operators belonging to different “fibers” have infinite distance.

The following simple result provides another way (under the assumption of finite diameter) to obtain the transportation inequality.

Corollary 4.21

Suppose for self-adjoint \(x\in {\mathcal {N}}\), \(E_\mathrm{Fix}(x)=0\) and \(\Vert \Gamma (x,x)\Vert _\infty \le 1\) imply \(\Vert x\Vert _\infty \le K\). Then, (4.14) holds with \(c=K^2\) for all \(\rho \ge 0\) such that \(\tau (\rho )=1\).

Proof

A calculation gives \(e^x-x\le e^{x^2}\). Assume \(E_\mathrm{Fix}(x)=0\) and \(\Vert \Gamma (x,x)\Vert _\infty \le 1\). Then for \(t>0\), \(\tau (e^{tx})=\tau (e^{tx}-tx)\le \tau (e^{t K}-t K)\le e^{K^2t^2}\). The claim now follows from Corollary 4.19. \(\square \)

Suppose in Theorem 4.4 we only have \(\Gamma _2\ge 0\) but not the \(\Gamma _2\)-condition. Junge and Mei proved in [30] as the main result

$$\begin{aligned} \Vert A^{1/2} x\Vert _p\le c(p)\max \{\Vert \Gamma (x,x)^{1/2}\Vert _p,\Vert \Gamma (x^*,x^*)^{1/2}\Vert _p\} \end{aligned}$$

in this setting. Using the proof of Theorem 1.1.7 in the same paper [30], it can be shown that if

$$\begin{aligned} \Vert T_t: L_1^0({\mathcal {N}})\rightarrow L_\infty ({\mathcal {N}})\Vert \le Ct^{-n/2}, \end{aligned}$$
(4.15)

then \(\Vert A^{-1/2}:L_p^0({\mathcal {N}})\rightarrow L_\infty ({\mathcal {N}})\Vert \le C(n)\) for \(p>n\). Indeed, we consider the composition of operators

where \(s,q,p,\alpha ,\beta \) are chosen so that

$$\begin{aligned} 1<s<q, n<p<q<\infty , \alpha +\beta =\frac{1}{2}, \alpha =\frac{n}{2}\left( \frac{1}{p}-\frac{1}{q}\right) \!, \beta =\frac{n}{2s}. \end{aligned}$$

For example, \(p=2n,q=4n,s=\frac{4n}{3}\) satisfy these conditions. By [30, Corollary 1.1.4], \(\Vert A^{-\alpha }:L_p^0({\mathcal {N}})\rightarrow L_q^0({\mathcal {N}})\Vert \le C(n)\). Using [30, Lemma 1.1.3], we have \(\Vert A^{-\beta }: L_{s,1}^0({\mathcal {N}})\rightarrow L_\infty ({\mathcal {N}}) \Vert \le C(n)\). The embedding \(L_q^0({\mathcal {N}})\hookrightarrow L_{s,1}^0({\mathcal {N}})\) follows from interpolation theory. Hence, \(\Vert A^{-1/2}:L_p^0({\mathcal {N}})\rightarrow L_\infty ({\mathcal {N}})\Vert \le C(n)\). This gives \(\Vert x\Vert _\infty \le C(n)\Vert A^{1/2}x\Vert _p\) for large \(p\) and \(E_\mathrm{Fix}(x)=0\). Assuming \(\Gamma (x,x)\le 1\) for self-adjoint \(x\), it follows that \(\Vert x\Vert _\infty \le C(n,p)\). By choosing e.g. \(p=2n\), the constant \(C(n,p)\) actually only depends on \(n\). In light of Corollary  4.21, we obtain the following result.

Corollary 4.22

Let \(T_t\) be a standard nc-diffusion semigroup acting on \({\mathcal {N}}\) with \(\Gamma _2\ge 0\). Then (4.15) with finite dimension \(n\) implies the transportation inequality (4.14) for all \(\rho \ge 0\) such that \(\tau (\rho )=1\).

In the commutative theory, the transportation inequality (4.10) implies isoperimetric type inequality by Marton’s argument in [8]. So far it is not clear what isoperimetric inequality means in noncommutative probability. We hope to give a noncommutative analog of isoperimetric inequality.

Definition 4.23

Let \(e,f\in ({\mathcal {M}},\tau )\) be projections. The distance between \(e\) and \(f\) is

$$\begin{aligned} d(e,f)=\inf \{W_1(\phi ,\psi ): \phi \,\mathrm{and }\, \psi \,\mathrm{are\,states, }\,s(\phi )=e,s(\psi )=f\}, \end{aligned}$$

where \(s(\phi )\) is the support of \(\phi \).

Here our definition generalizes directly the distance of sets in the commutative theory. Thus in general \(d\) is not a metric, as in the commutative setting. Then the following result follows from the same proof as in the commutative setting given in [8].

Proposition 4.24

Let \(e,f\in ({\mathcal {M}},\tau )\) be projections. Then under the assumptions of Theorem 4.4 and assuming \(\mathrm{Fix}={\mathbb {C}}1\),

$$\begin{aligned} d(e,f)\le \sqrt{-2c\ln \tau (e)}+\sqrt{-2c\ln \tau (f)}\,. \end{aligned}$$

Equivalently, for every \(h>\sqrt{-2c\ln \tau (e)}\) and every projection \(p\) such that \(d(p,e)>h\),

$$\begin{aligned} \tau (p)\le \exp \Big (-\frac{1}{2c}\Big (h-\sqrt{-2c\ln \tau (e)}\Big )^2\Big )\,. \end{aligned}$$

Proof

Put \(\phi _e(\cdot )=\tau (e\cdot )/\tau (e)\) and \(\phi _f=\tau (f\cdot )/\tau (f)\). It is easy to see that \(d(e,f)\le W_1(\phi _e,\phi _f)\). Then triangle inequality and (4.10) yield \(d(e,f)\le \sqrt{2c\mathrm{Ent }(e)}+\sqrt{2c\mathrm{Ent }(f)}\). By spectral decomposition theorem of the identity, \(\mathrm{Ent }(e)= \int _0^1\frac{1_{A}}{\tau (e)}\ln \frac{1_{A}}{\tau (e)} d\mu \) where \(A\) is a Borel set such that \(1_A(Id)=e\). Hence we find \(\mathrm{Ent }(e)=-\ln \tau (e)\), which gives the first assertion. The equivalent formulation is a simple calculation. \(\square \)

To conclude this section, we remark that the best possible \(\alpha \) in \(\Gamma _2\ge \alpha \Gamma \) sometimes characterizes the dynamical system \(({\mathcal {M}},T_t)\); see the example of hyperfinite \(II_1\) factor below.

5 Application to the group von Neumann algebras

Starting from this section, we will investigate a variety of examples which satisfy the assumptions of Theorem 4.4. All the Poincaré type, deviation and transportation inequalities we derived previously will hold in these examples. The key point is to check \(\Gamma _2\ge \alpha \Gamma \). It may be of independent interest because it means a strictly positive Ricci curvature from the geometric point of view. We consider the \(\Gamma _2\)-criterion for group von Neumann algebras in this section.

Let \(G\) be a countable discrete group. In this paper we say that \(\psi :\psi \rightarrow {\mathbb {R}}_+\) is a conditionally negative definite length (cn-length) function if it vanishes at the identity \(e\), \(\psi (g)=\psi (g^{-1})\) and is conditionally negative which means that \(\sum _{g}\xi _g=0\) implies \(\sum _{g,h}\bar{\xi }_g\xi _h\psi (g^{-1}h)\le 0\).

Let \(\lambda : G\rightarrow B(\ell _2(G))\) be the left regular representation given by \(\lambda (g)\delta _h=\delta _{gh}\) where \(\delta _g\)’s form the unit vector basis of \(\ell _2(G)\). Let \({\mathcal {L}}(G) = \lambda (G)''\), the von Neumann algebra generated by \(\{\lambda (g): g\in G\}\). Any \(f\in {\mathcal {L}}(G)\) can be written as

$$\begin{aligned} f=\sum _{g\in G}\hat{f}(g)\lambda (g)\,. \end{aligned}$$

It is well known that \(\tau (f)=\langle \delta _e, f \delta _e \rangle \) defines a faithful normal tracial state and \(\tau (f)=\hat{f}(e)\) where \(e\) is the identity element. In what follows, we define the semigroup associated to the cn-length function \(\psi \) by \(T_t(\lambda (g))=T_t^\psi (\lambda (g))=\phi _t(g)\lambda (g)\) for \(g\in G\), where \(\phi _t(g)=e^{-t\psi (g)}\). The infinitesimal generator of \(T_t\) is given by \(A\lambda (g)=\psi (g)\lambda (g)\). Recall that the Gromov form is defined as

$$\begin{aligned} K(g,h)=K_{g,h} =\frac{1}{2}(\psi (g)+\psi (h)-\psi (g^{-1}h)), \quad g, h\in G\,. \end{aligned}$$

Given \(f=\sum _{x\in G}\hat{f}(x)\lambda (x)\) and \(g=\sum _{y\in G}\hat{g}(y)\lambda (y)\), a straightforward calculation gives

$$\begin{aligned}&\Gamma (f,g)=\sum _{x,y\in G}\bar{\hat{f}}(x)\hat{g}(y)K(x,y)\lambda (x^{-1}y)\,,\\&\Gamma _2(f,g)=\sum _{x,y\in G}\bar{\hat{f}}(x)\hat{g}(y)K(x,y)^2\lambda (x^{-1}y)\,. \end{aligned}$$

Let \({\mathcal {A}}\) be the subalgebra of \({\mathcal {L}}(G)\) which consists of elements that can be written as finite combination of \(\lambda _g, g\in G\). Then \({\mathcal {A}}\) is weakly dense in \({\mathcal {L}}(G)\) such that \(A{\mathcal {A}}\subset {\mathcal {A}}\) and \(T_t{\mathcal {A}}\subset {\mathcal {A}}\).

Lemma 5.1

\({\mathcal {A}}\) is dense in \(\mathrm{Dom }(A^{1/2})\) in the graph norm of \(A^{1/2}\) and \(T_t\) is a standard nc-diffusion semigroup acting on \({\mathcal {L}}(G)\).

Proof

For \(f=\sum _{g_i\in G}\hat{f}(g_i)\lambda (g_i)\in \mathrm{Dom }(A^{1/2})\), put \(f_n=\sum _{i=1}^n \hat{f}(g_i)\lambda (g_i)\). Note that \(f\in L_2({\mathcal {L}}(G))\). Then

$$\begin{aligned} \Vert f_n-f\Vert _{L_2({\mathcal {L}}(G))}=\tau ((f_n^*-f^*)(f_n-f)) =\sum _{i=n+1}^\infty |\hat{f}(g_i)|^2\rightarrow 0, \,\mathrm{as } \,n\rightarrow \infty \,. \end{aligned}$$

Since \(\langle Af,f\rangle =\sum _{i=1}^\infty \psi (g_i)|\hat{f}(g_i)|^2 <\infty \), we have

$$\begin{aligned} \langle A(f_n-f), f_n-f\rangle _{L_2({\mathcal {L}}(G),\tau )} =\sum _{i=n+1}^\infty \psi (g_i) |\hat{f}(g_i)|^2\rightarrow 0, \,\mathrm{as } \,n\rightarrow \infty \,. \end{aligned}$$

Therefore \({\mathcal {A}}\) is dense in the graph norm. Since \(\psi \) is conditionally negative, Schoenberg’s theorem implies that \(T_t\) is completely positive; see e.g. [10, Appendix D]. It can be directly checked that \(T_t\) is normal and unital. Since \(\psi (g)=\psi (g^{-1})\),

$$\begin{aligned} \tau (T_t(x)y)&= \left\langle \delta _e,\sum _{g}e^{-t\psi (g)}\hat{x}(g)\lambda _g\sum _h \hat{y}(h)\lambda _h\delta _e\right\rangle \\&= \sum _{g} e^{-t\psi (g)}\hat{x}(g)\hat{y}(g^{-1}) =\tau (xT_ty)\,. \end{aligned}$$

Hence \(T_t\) is self-adjoint. To check that \((T_t)\) is weak* continuous on \({\mathcal {L}}(G)\), it suffices to verify that \((T_t)\) is a strongly continuous semigroup on \(L_2({\mathcal {L}}(G))\). For \(f\in L_2({\mathcal {L}}(G))\), we have

$$\begin{aligned} \Vert T_t f-f\Vert ^2_{L_2({\mathcal {L}}(G))}=\sum _{g\in G} (e^{-t\psi (g)}-1)^2|\hat{f}(g)|^2 \rightarrow 0 \text{ as } t\rightarrow 0\,. \end{aligned}$$

We have proved that \((T_t)\) is a standard semigroup. Let \(f\in {\mathcal {A}}\). Then

$$\begin{aligned} \Vert \Gamma (f,f)\Vert _1 =\left\langle \delta _e,\sum _{g\in G}|\hat{f}(g)|^2K_{g,g} \delta _e \right\rangle =\sum _{g\in G}\psi (g)|\hat{f}(g)|^2 =\Vert A^{1/2}f\Vert _2<\infty \,. \end{aligned}$$

Since \({\mathcal {A}}\) is dense in \(\mathrm{Dom }(A^{1/2})\) in the graph norm, by an approximation argument (see the proof of Lemma 4.8), the above equality holds for all \(f\in \mathrm{Dom }(A^{1/2})\) and thus \((T_t)\) is a nc-diffusion semigroup. \(\square \)

By virtue of Lemma 5.1 and Corollary 4.8, our Poincaré inequalities will follow if the \(\Gamma _2\)-criterion holds. Put \(\mathrm{Fix}=\{f\in {\mathcal {L}}(G):\psi (f)=0\}\).

Corollary 5.2

Let \(2\le p<\infty \) and assume \(\Gamma _2(f,f)\ge \alpha \Gamma (f,f)\) for \(f\in {\mathcal {A}}\). Then there exists a constant \(C\) such that for all self-adjoint \(f\in L_p({\mathcal {L}}(G),\tau )\),

$$\begin{aligned} \Vert f-E_\mathrm{Fix}(f)\Vert _p\le C\alpha ^{-1/2}\min \{\sqrt{p} \Vert \Gamma (f,f)^{1/2}\Vert _\infty ,\,{p} \Vert \Gamma (f,f)^{1/2}\Vert _p\}\,. \end{aligned}$$

If \(\psi (g) = 0\) only if \(g\) is the identity element, then \(E_\mathrm{Fix}(f)=\tau (f)\) for \(f\in {\mathcal {L}}(G)\).

Among the examples we will consider below, the free group on \(n\) generators \({\mathbb {F}}_n\) satisfies \(E_\mathrm{Fix}(f)=\tau (f)\) but the finite cyclic group \({\mathbb {Z}}_n\) has nontrivial \(\mathrm{Fix}\). With the help of Lemma 5.1 and Corollary 4.8, we only need to check the \(\Gamma _2\)-criterion on the finitely supported elements in order to fulfill the hypotheses of our main theorem. We call

$$\begin{aligned}{}[\Gamma _2(f^i, f^j)]\ge \alpha [\Gamma (f^i,f^j)] \text{ for } \text{ any } n\in {\mathbb {N}} \text{ and } f^1,\ldots ,f^n\in {\mathcal {A}}\end{aligned}$$

the algebraic \(\Gamma _2\)-condition (or \(\Gamma _2\)-criterion) and abbreviate it to “\(\Gamma _2\ge \alpha \Gamma \) in \({\mathcal {L}}(G)\)”. This is the theme of two sections from now on. This condition is seemingly stronger than needed. However, \(\Gamma _2(f,f)\ge \alpha \Gamma (f,f)\) for all \(f=\sum _{g \mathrm{finite}} \hat{f}_g\lambda (g)\in {\mathcal {L}}(G)\) amounts to check \([\Gamma _2(\lambda (g_i), \lambda (g_j))]\ge \alpha [\Gamma (\lambda (g_i), \lambda (g_j))]\) for \(g_i\in G\). This algebraic condition is also easier to check because it can be reduced to check the positivity of certain matrices as will be shown below. The following technical lemmas will be used repeatedly.

Lemma 5.3

Suppose \(K=(K_{g,h})_{g,h\in G}\) is a matrix indexed by \(G\) with entries in \({\mathbb {C}}\) and define a sesquilinear form \(\Theta :{\mathcal {A}}\times {\mathcal {A}}\rightarrow {\mathcal {A}}\), \(\Theta (f^i,f^j)=\sum _{g,h\in G} \bar{\hat{f}}^i(g) \hat{f^j}(h)K_{g,h} \lambda (g^{-1}h)\). Then \(K\) is nonnegative definite if and only if \(\Theta \) is positive.

Proof

Our proof is based on Lance [38, Proposition 2.1]. Assume \(K\) is nonnegative definite. Write \(K=X^*X\) for \(X=(x_{g,h}), x_{g,h}\in {\mathbb {C}}\). Then \(K=\sum _{l} (\sum _{g,h}x_{lg}^* x_{lh}\otimes e_{g,h})\). Given \(f^1,\ldots ,f^n\in {\mathcal {A}}\), we have

$$\begin{aligned} \Theta (f^i,f^j)&= \sum _{l\in G}\sum _{g,h\in G} \bar{\hat{f^i}}(g)\hat{f^j}(h)x_{lg}^* x_{lh}\lambda (g^{-1}h) \\&= \sum _{l\in G} \left( \sum _g {\hat{f^i}}(g)x_{lg}\lambda _g\right) ^*\left( \sum _h {\hat{f^j}}(h)x_{lh}\lambda _h\right) \,. \end{aligned}$$

Here we understand all indices are finite. Put \(\theta ^i_l = \sum _g {\hat{f^i}}(g)x_{lg}\lambda _g\). We then have

$$\begin{aligned} \Theta =\sum _{l\in G} \sum _{i,j=1}^n{\theta ^i_l}^* \theta ^j_l\otimes e_{i,j} =\sum _{l\in G} \left( \,\sum _{i=1}^n \theta ^i_l\otimes e_{1,i}\right) ^*\left( \,\sum _{j=1}^n \theta ^j_l\otimes e_{1,j}\right) \,, \end{aligned}$$

which is positive in \(M_n({\mathcal {L}}(G))\). Conversely, let \((x_i)=\sum _i x_i\delta _{k_i}\in \ell _2(G)\) and write \(g_i= {k_i}^{-1}\in G\). Then \((x_i)=\sum _{i\in {\mathbb {N}}} x_i \delta _{g_i^{-1}}\). Let \(f^i=\lambda (k_i)\) so that \((\Theta (f^i,f^j))=(K_{i,j}\lambda (k_i^{-1}k_j))\) is positive in \(M_n({\mathcal {L}}(G))\) for all \(n\in {\mathbb {N}}\). Then for \(h_i=x_i\delta _{g_i}\in \ell _2(G)\), we have for all \(n\in {\mathbb {N}}\)

$$\begin{aligned} 0&\le \langle [\Theta (f^i,f^j)] (h_1,\ldots ,h_n), (h_1,\ldots ,h_n)\rangle _{\ell _2^n(\ell _2(G))}\\&= \sum _{i,j=1}^n \langle K_{i,j}\lambda (k_i^{-1}k_j)x_j\delta _{g_i}, x_i\delta _{g_i}\rangle _{\ell _2(G)} =\sum _{i,j=1}^n K_{i,j}x_j \bar{x_i} \,, \end{aligned}$$

which implies that \(K\) is nonnegative definite. \(\square \)

The next lemma is useful when we deal with the product of groups. Note that

$$\begin{aligned} {\mathcal {L}}\left( \,\,\prod _{i=1}^m G_i\right) \cong \bar{\otimes }_{i=1}^m{\mathcal {L}}(G_i). \end{aligned}$$

The identification is given by \(\lambda (g_1,\ldots ,g_m)\mapsto \lambda (g_1)\otimes \cdots \otimes \lambda (g_m)\) for \(g_i\in G_i\). We associated the form \(\Gamma _2\) to a matrix \(K\) as follows: \(\Gamma _2^K(f,g)=\sum _{x,y\in G} \bar{\hat{f}}(x)\hat{g}(y)K^2_{x,y}\lambda (x^{-1}y)\). In what follows the matrix \(K\) will be the Gromov form. If \(K=K_1\otimes K_2\), then it is easy to check \(\Gamma ^K(f^i\otimes g^i, f^j\otimes g^j)=\Gamma ^{K_1}(f^i,f^j)\otimes \Gamma ^{K_2}(g^i,g^j)\) for \(f_i\in {\mathcal {L}}(G_1), g_i\in {\mathcal {L}}(G_2)\).

Lemma 5.4

Let \((K_i)_{i=1}^m\) be nonnegative definite matrices and \(\Gamma ^{K_i}\) the associated gradient forms in the sense of Lemma 5.3. Suppose \(\Gamma ^{K_i}_2\ge \alpha \Gamma ^{K_i}\). Then

$$\begin{aligned} \Gamma _2^{K}\ge \alpha \Gamma ^{K}, \end{aligned}$$

where \(K=\sum _{i=1}^m 1\!\!1\otimes \cdots \otimes K_i\otimes \cdots \otimes 1\!\!1\) with \(K_i\) in the \(i\)th position and in what follows \(1\!\!1\) always denotes the matrix with every entry equal to 1.

Proof

In light of Lemma 5.3 it suffices to verify \(K\bullet K\ge \alpha K\). Here and in the following \(A\bullet B\) denotes the Schur product of matrix. Note that trivially \(1\!\!1\ge 0\). Since \(K_i\ge 0\), all the “cross terms” of the form

$$\begin{aligned} 1\!\!1\otimes \cdots \otimes K_{i_1}\otimes \cdots \otimes K_{i_2} \otimes \cdots \otimes 1\!\!1 \end{aligned}$$

are nonnegative matrices for all \(1\le i_1 < i_2 \le m\). It follows that

$$\begin{aligned} K\bullet K\ge \sum _{i=1}^m 1\!\!1\otimes \cdots \otimes (K_i\bullet K_i) \otimes \cdots \otimes 1\!\!1 \ge \alpha K\,. \end{aligned}$$

\(\square \)

5.1 The free groups

Let \({\mathbb {F}}_n\) denote the free group on \(n\) generators with length function \(\psi =|\cdot |\), where for \(g\in {\mathbb {F}}_n\), \(|g|\) is the length of (the freely reduced form of) \(g\). Note that the Gromov form \(K(g,h) = |\min (g,h)|:= \max \{|w|: g=wg',h=wh'\}\) where \(\min (g,h)\) is the longest common prefix subword of \(g\) and \(h\). It is well known that \(\psi \) is conditionally negative due to Haagerup [24].

Proposition 5.5

\( \Gamma _2 \ge \Gamma \) holds in \({\mathcal {L}}({\mathbb {F}}_n)\) for the semigroup \((e^{-t\psi })\) where \(\psi \) is defined as above.

Proof

For a freely reduced word \(x\in {\mathbb {F}}_n\), write \(g_i\prec g\) for the prefix subword of \(g\) with length \(i\). Following Haagerup’s construction, we define a map

$$\begin{aligned} V: {\mathbb {F}}_n \rightarrow \ell _2({\mathbb {F}}_n), \quad g \mapsto V(g)=\sum _{g_i\prec g}\sqrt{2(i-1)}\delta _{g_i}\,. \end{aligned}$$

Then we have

$$\begin{aligned} \widetilde{K}_{g,h}:=K^2_{g,h}-K_{g,h} =\langle V(g), V(h)\rangle _{\ell _2({\mathbb {F}}_n)}=V(g)^*V(h)\,, \end{aligned}$$

where \(V(g)^*\) is a row vector and \(V(h)\) a column vector. It follows that \(\widetilde{K}=(\widetilde{K}_{g,h})_{g,h}\) is a nonnegative definite matrix. We deduce from Lemma 5.3 that \(\Gamma _2\ge \Gamma \). \(\square \)

The particular case \(n=1\) gives some interesting results in classical Fourier analysis. Indeed, \({\mathcal {L}}({\mathbb {F}}_1)={\mathcal {L}}({\mathbb {Z}})=L_\infty ({\mathbb {T}})\) and \(L_p({\mathcal {L}}({\mathbb {F}}_1))=L_p({\mathbb {T}})\) after identifying \(\lambda (k)(x)=e^{2\pi i kx}\). In this case

$$\begin{aligned} K(j,k)=\left\{ \begin{array}{l@{\quad }l} \min (|j|,|k|) , &{} jk>0,\\ 0, &{} \mathrm{otherwise}. \end{array}\right. \end{aligned}$$

Corollary 5.6

Let \(2\le p<\infty \). Then there exists constants \(C\) and \(C'\) such that for all \(f\in L_p({\mathbb {T}})\), we have

$$\begin{aligned}&\Vert f-\hat{f}(0)\Vert _p\le C\sqrt{p}\left\| \sum _{j,k\in {\mathbb {Z}},jk>0}\bar{\hat{f}}(j)\hat{f}(k)\min (|j|,|k|)e^{2\pi i(k-j)\cdot }\right\| ^{1/2}_\infty ,\\&\Vert f-\hat{f}(0)\Vert _p\le C'{p}\left\| \sum _{j,k\in {\mathbb {Z}},jk>0}\bar{\hat{f}}(j)\hat{f}(k)\min (|j|,|k|)e^{2\pi i(k-j)\cdot }\right\| ^{1/2}_{p/2}. \end{aligned}$$

Remark 5.7

Observe that this example is purely commutative. However, commutative probability theory seems insufficient to establish these inequalities. Intuitively, the multiplier \(|j|\) corresponds to \(\Delta ^{1/2}\). The Markov process generated by \(\Delta ^{1/2}\) is the Cauchy process with discontinuous path. The classical diffusion theory does not apply here. But it is still nc-diffusion so that our noncommutative theory is essential in this regard. In general, whenever the process has discontinuous path but its semigroup still satisfies our assumptions, the noncommutative theory seems to be a natural choice due to the existence of Markov dilation with a.u. continuous path as stated in Theorem 3.2. We will have more examples of this kind in the following.

Remark 5.8

It was shown in [30, Remark 1.3.2] that

$$\begin{aligned} \Vert T_t: L_1^0({\mathcal {L}}({\mathbb {F}}_n))\rightarrow L_\infty ({\mathcal {L}}({\mathbb {F}}_n))\Vert \le C t^{-3}. \end{aligned}$$

Therefore, Theorem 4.18 and Corollary 4.22 give two different ways to prove the transportation inequality (4.10) for \({\mathcal {L}}({\mathbb {F}}_n)\).

5.2 Application to the noncommutative tori \({\mathcal {R}}_\Theta \)

We recall the definition following [28]. Let \(\Theta \) be a \(d\times d\) antisymmetric matrix with entries \(0\le \theta _{ij}<1\). The noncommutative torus (or the rotation algebra) with \(d\) generators associated to \(\Theta \) is the von Neumann algebra \({\mathcal {R}}_\Theta \) generated by \(d\) unitaries \(u_1,\ldots ,u_d\) satisfying \(u_j u_k= e^{2\pi i\theta _{jk}}u_k u_j\). Every element of \({\mathcal {R}}_{\Theta }\) is in the closure of the span of words of the form \(w_k=u_1^{k_1},\ldots ,u_d^{k_d}\) for \(k=(k_1,\ldots ,k_d)\in {\mathbb {Z}}^d\). \({\mathcal {R}}_\Theta \) admits a unique normal faithful trace \(\tau \) given by \(\tau (x)=\hat{x}(0)\) where \(x=\sum _{k\in {\mathbb {Z}}^d} \hat{x}(k) u_1^{k_1} ,\ldots , u_d^{k_d}\in {\mathbb {R}}_\Theta \). Our goal is to show that \({\mathcal {R}}_\Theta \) admits a standard nc-diffusion semigroup with the \(\Gamma _2\)-criterion. We start with the von Neumann algebra of \({\mathbb {Z}}^d\). It is well known that \({\mathcal {L}}({\mathbb {Z}}^d)\cong L_\infty ({\mathbb {T}}^d)\). We define \(\psi (k)=\Vert k\Vert _1=\sum _{i=1}^d |k_i|\) for \(k=(k_1,\ldots ,k_d)\in {\mathbb {Z}}^d\). Clearly, \(\psi \) is a cn-length function and thus generate a standard nc-diffusion semigroup \(P_t\) by Lemma 5.1. In fact \(P_t=\widetilde{P}_t^{\otimes d}\) where \(\widetilde{P}_t\) is the Poisson semigroup on \(L_\infty ({\mathbb {T}})\).

Proposition 5.9

Let \(\Gamma \) be the gradient form associated to \(P_t\). Then \(\Gamma _2\ge \Gamma \) in \({\mathcal {L}}({\mathbb {Z}}^d)\).

Proof

Let \(K^d\) be the Gromov form associated with \(\psi \). A calculation shows that \(K^d(j,k)=K(j_1,k_1)+ \cdots +K(j_d,k_d)\) for \(j=(j_1,\ldots ,j_d),k=(k_1,\ldots ,k_d)\in {\mathbb {Z}}^d\), where \(K\) is the Gromov form of \({\mathbb {Z}}={\mathbb {F}}_1\) considered in the proceeding subsection. Alternatively, we may write \(K^d=\sum _{i=1}^d 1\!\!1\otimes \cdots \otimes K\otimes \cdots \otimes 1\!\!1\) where \(K\) is in the \(i\)th position. But we know from Proposition 5.5 that \(\Gamma _2\ge \Gamma \) in \({\mathcal {L}}({\mathbb {Z}})\). The assertion follows from Lemma 5.3. \(\square \)

Proposition 5.10

\({\mathcal {R}}_\Theta \) admits a standard nc-diffusion semigroup with \(\Gamma _2\ge \Gamma \).

Proof

Let \(k\in {\mathbb {Z}}^d\). Consider an action \(\alpha : {\mathbb {T}}^d\rightarrow \mathrm {Aut}({\mathcal {R}}_\Theta )\) given by: for \(s\in {\mathbb {T}}^d\),

$$\begin{aligned} \alpha _s(u_1^{k_1} ,\ldots , u_d^{k_d}) = e^{2\pi i \sum _{j=1}^d k_j s_j}u_1^{k_1} ,\ldots , u_d^{k_d}. \end{aligned}$$

It is easy to check that \(\alpha _s\) is a trace preserving automorphism. Define a map

$$\begin{aligned}&\pi : {\mathcal {R}}_\Theta \rightarrow L_\infty ({\mathbb {T}}^d)\bar{\otimes } {\mathcal {R}}_\Theta ,\\&\quad w_k=u_1^{k_1} ,\ldots , u_d^{k_d}\mapsto \pi (w_k)(s)=\alpha _s(w_k)=e^{2\pi i \langle k,s\rangle } u_1^{k_1} ,\ldots , u_d^{k_d}\,. \end{aligned}$$

Then \(\pi \) is an injective \(*\)-homomorphism. Define \(T_t:{\mathcal {R}}_\Theta \rightarrow {\mathcal {R}}_\Theta , T_t(w_k)=e^{-t\Vert k\Vert _1}w_k\). We claim that \((T_t)_{t\ge 0}\) is the desired semigroup. Indeed, by Lemma 6.1 and Proposition 5.9, \(P_t\otimes Id\) acting on \(L_\infty ({\mathbb {T}}^d)\bar{\otimes }{\mathcal {R}}_\Theta \) is a standard nc-diffusion semigroup and satisfies \(\Gamma _2\ge \Gamma \). Then since \(\pi \) is injective and

$$\begin{aligned} P_t\otimes Id (\pi (w_k))=e^{-t\Vert k\Vert _1} e^{2\pi i \langle k,\cdot \rangle } \otimes u_1^{k_1} ,\ldots , u_d^{k_d}=\pi (T_t(w_k)) \end{aligned}$$

leaves \(\pi ({\mathcal {R}}_\Theta )\) invariant, we deduce that \(T_t\) is a standard nc-diffusion semigroup acting on \({\mathcal {R}}_\Theta \) with \(\Gamma _2\ge \Gamma \). \(\square \)

5.3 The finite cyclic group \({\mathbb {Z}}_n\)

We consider the group von Neumann algebra \({\mathcal {L}}({\mathbb {Z}}_n)\) in this subsection. Let \((e_j)_{j=1}^n\) be the standard basis of \({\mathbb {C}}^n\). Each \(e_j\) can be regarded as a vector in \({\mathbb {R}}^{2n}\) by canonical identification. Given \(k\in {\mathbb {Z}}_n\), define the \(2n\times 2n\) diagonal matrix \(\alpha _k = (e^{2\pi i kj/n})_{j=0}^{n-1}\) where each \(e^{2\pi i kj/n}\) is on diagonal and is identified with the \(2\times 2\) rotation matrix

$$\begin{aligned} \left( \begin{array}{c@{\quad }c} \cos (2\pi kj/n) &{} -\sin (2\pi kj/n)\\ \sin (2\pi kj/n)&{}\cos (2\pi kj/n) \end{array}\right) \!. \end{aligned}$$

Consider the finite cyclic group \({\mathbb {Z}}_n\) with 1-cocycle structure \((b,\alpha , {\mathbb {R}}^{2n})\), where

$$\begin{aligned} b(k)=\frac{1}{\sqrt{n}}\left( \,\sum _{j=1}^n\alpha _k(e_j)-e_j\right) =\frac{1}{\sqrt{n}}\sum _{j=1}^n \left( \begin{array}{l}\cos (2\pi k(j-1)/n)-1\\ \sin (2\pi k(j-1)/n)\end{array}\right) \otimes e_j\,. \end{aligned}$$

Then the length function \(\psi \) given by \(\psi (g)=\Vert b(g)\Vert _2^2\) is conditionally negative; see e.g. [10, Appendix D].

Lemma 5.11

Let \(K(k,h)\) be the Gromov form. Then \(K(k,h)=\langle b(k),b(h)\rangle \).

Proof

Since the length function \(\psi (k)=\Vert b(k)\Vert ^2\), by the cocycle property,

$$\begin{aligned} K(k,h)&= \frac{1}{2}(\Vert b(k)\Vert ^2+\Vert b(h)\Vert ^2-\Vert b(h-k)\Vert ^2)\\&= \frac{1}{2}(\Vert b(-k)\Vert ^2+\Vert \alpha _{-k}(b(h))\Vert ^2-\Vert \alpha _{-k}(b(h))+b(-k)\Vert ^2)\\&= -\langle b(-k), \alpha _{-k}(b(j))\rangle = -\langle \alpha _k(b(-k)), b(h)\rangle \\&= \langle b(k),b(h)\rangle \,. \end{aligned}$$

\(\square \)

Clearly, \(K(k,h)=0\) if \(k=0\) or \(h=0\). For \(k,h\ne 0\), a computation gives

$$\begin{aligned} K(k,h)&= \frac{1}{n}\sum _{j=0}^{n-1} \big [(1-\cos (2\pi kj/n))(1-\cos (2\pi hj/n)) + \sin (2\pi kj/n)\sin (2\pi hj/n)\big ]\\&= 1+\frac{1}{n}\sum _{j=0}^{n-1}\cos \Big (\frac{2\pi (k-h)j}{n}\Big )=1+\delta _{k,h}\,, \end{aligned}$$

where \(\delta _{k,h}\) is the Kronecker delta function. It follows that \(\psi (k)=2(1-\delta _{k,0})\). For reasons that will become clear later, we normalize \(\psi \) and still denote it by \(\psi \) so that \(\psi (k)=1-\delta _{k,0}\) for \(k\in {\mathbb {Z}}_n\). Then the associated Gromov form satisfies \(K_{k,h}=\frac{1}{2}(1+\delta _{k,h})\) for \(k,h\ne 0\) and \((K_{k,h}^2-\frac{1}{2} K_{k,h}) \ge 0\). It is an immediate consequence of Lemma 5.3 that \(\Gamma _2\ge \frac{1}{2}\Gamma \) in \({\mathcal {L}}({\mathbb {Z}}_n)\). In fact, we can do better.

Proposition 5.12

For all \(0<\alpha \le \frac{n+2}{2n}\), we have \(\Gamma _2\ge \alpha \Gamma \) in \({\mathcal {L}}({\mathbb {Z}}_n)\). Moreover, \(\alpha _n= \frac{n+2}{2n}\) is the largest possible \(\alpha \) with the \(\Gamma _2\)-criterion.

Proof

Note that the \(n\times n\) matrix \(K\) can be written as a block matrix

$$\begin{aligned} K=\left( \begin{array}{c@{\quad }c} 0&{} 0\\ 0&{} \frac{1}{2}(I_{n-1}+1\!\!1_{n-1}) \end{array}\right) \end{aligned}$$

where \(I_{n-1}\) is the \(n-1\) dimensional identity matrix and every entry of \(1\!\!1_{n-1}\) is 1. Write \(\widehat{K}=\frac{1}{2}(I_{n-1}+1\!\!1_{n-1})\). Since \(1\!\!1_{n-1}\le (n-1) I_{n-1}\), for \(0<\alpha \le \frac{n+2}{2n}\) we have

$$\begin{aligned}&4\widehat{K}{}\bullet \widehat{K}-4\alpha \widehat{K}=(3-2\alpha )I_{n-1}- (2\alpha -1)1\!\!1_{n-1}\\&\quad \ge (2+n-2\alpha n)I_{n-1}\ge 0 \,. \end{aligned}$$

Plugging \(\varvec{x}=(\frac{1}{\sqrt{n-1}},\ldots ,\frac{1}{\sqrt{n-1}})\) into \(\varvec{x}'(\widehat{K}\bullet \widehat{K}-\alpha \widehat{K})\varvec{x}\ge 0\) reveals that \(\alpha _n = \frac{n+2}{2n}\) is sharp. Then Lemma 5.3 leads to the \(\Gamma _2\)-criterion. \(\square \)

5.4 The discrete Heisenberg group \(H_3({\mathbb {Z}}_n)\)

We consider the Heisenberg group \(H_3({\mathbb {Z}}_n)={\mathbb {Z}}_n\times {\mathbb {Z}}_n\times {\mathbb {Z}}_n\) over \({\mathbb {Z}}_n\) as a subalgebra of \(M_3({\mathbb {Z}}_n)\) as follows:

$$\begin{aligned} \left( \begin{array}{c@{\quad }c@{\quad }c} 1&{}b&{}a\\ 0&{}1&{}c\\ 0&{}0&{}1 \end{array}\right) \left( \begin{array}{c@{\quad }c@{\quad }c} 1&{}b'&{}a'\\ 0&{}1&{}c'\\ 0&{}0&{}1 \end{array}\right) = \left( \begin{array}{c@{\quad }c@{\quad }c} 1&{}b+b'&{}a+a'+bc'\\ 0&{}1&{}c+c'\\ 0&{}0&{}1 \end{array}\right) ; \end{aligned}$$

see e.g. [15, Section VII.5] for more details. We will write \(H\) for \(H_3({\mathbb {Z}}_n)\) as long as there is no confusion. The multiplication here is given by

$$\begin{aligned} (a,b,c)(a',b',c')=(a+a'+bc', b+b',c+c'),\quad (a,b,c),(a',b',c')\in H\,. \end{aligned}$$

Other multiplications have been considered in the literature.

Proposition 5.13

Let \(\psi (a,b,c)=2-\delta _{b,0}-\delta _{c,0}\). Then

(1):

\(\psi \) is conditionally negative and thus the semigroup \((T_t)\) determined by \(\psi \) is a standard nc-diffusion semigroup.

(2):

Let \(\Gamma \) be the gradient form associated to \(\psi \). Then \(\Gamma _2\ge \frac{n+2}{2n}\Gamma \) in \({\mathcal {L}}(H)\).

Proof

  1. (1)

    The length function of \({\mathbb {Z}}_n\) considered in Sect. 5.3 is given by \(\widetilde{\psi }(k)=(1-\delta _{k,0})\), which extends to \({\mathbb {Z}}_n\times {\mathbb {Z}}_n\) as \(\widetilde{\psi }(k,l)=(1-\delta _{k,0})+(1-\delta _{l,0})\). Define a group homomorphism

    $$\begin{aligned} \beta : H\rightarrow {\mathbb {Z}}_n\times {\mathbb {Z}}_n, \quad (a,b,c)\mapsto (b,c). \end{aligned}$$

    Since \(\widetilde{\psi }\) is conditionally negative, it follows from the definition that \(\psi = \widetilde{\psi }\circ \beta \) is also conditionally negative. Lemma 5.1 yields that \((T_t)\) is a standard nc-diffusion semigroup.

  2. (2)

    Let \(K\) and \(\widetilde{K}\) be the Gromov form of \((H,\psi )\) and \(({\mathbb {Z}}_n,\widetilde{\psi })\) respectively. A calculation shows that for indices \((a,b,c), (a',b',c')\in H\),

    $$\begin{aligned} K((a,b,c),(a',b',c'))&= \widetilde{K}(b,b')+\widetilde{K}(c,c')\\&= (\widetilde{K}\otimes 1\!\!1 + 1\!\!1\otimes \widetilde{K})((b,b')\otimes (c,c'))\,. \end{aligned}$$

By Proposition 5.12 and Lemma 5.4 with \(m=2\), we have \(\Gamma _2^{K}\ge \frac{n+2}{2n}\Gamma ^{K}\) in \({\mathcal {L}}(H)\), as desired. \(\square \)

Let \(e_{i,j}\) be the standard basis of the matrix algebra \(M_n({\mathbb {C}})\) and \(\delta _j\) the standard basis of \(\ell _2({\mathbb {Z}}_n)\). Define the diagonal matrix \(u_k=\sum _{j=1}^{n}e^{2\pi i k(j-1)/n}\otimes e_{j,j}\) and the shift operator \(v_l(\delta _j)=\delta _{j+l}\) which is nothing but the left regular representation of \({\mathbb {Z}}_n\) on \(\ell _2({\mathbb {Z}}_n)\). It is easy to see that \(u_k, v_l\in M_n=B(\ell _2({\mathbb {Z}}_n))\) and they satisfy \(u_k v_l=e^{2\pi ikl/n}v_l u_k\).

Proposition 5.14

Let \({\mathcal {L}}(H)\) be the group von Neumann algebra of \(H\). Then

$$\begin{aligned} {\mathcal {L}}(H)\cong L_\infty ({\mathbb {Z}}_n^2)\oplus M_n\oplus {\mathcal {M}}_2\oplus \cdots \oplus {\mathcal {M}}_{n-1}\,, \end{aligned}$$

where \({\mathcal {M}}_x, x=2,\ldots , n-1\) are von Neumann algebras acting on \(\ell _2({\mathbb {Z}}_n^2)\). Moreover, if \(T_t\) is the semigroup associated to \(\psi (a,b,c)=2-\delta _{b,0}-\delta _{c,0}\), then \(T_t\) leaves each component invariant and \(T_t|_{{\mathcal {M}}_x}\)is a standard nc-diffusion semigroup.

Proof

Let us first determine the center of \({\mathcal {L}}(H)\) denoted by \({\mathcal {Z}}\). The identity

$$\begin{aligned} \lambda (a,b,c)\lambda (a',b',c')=\lambda (a',b',c')\lambda (a,b,c) \end{aligned}$$

for all \((a',b',c')\in H\) holds if and only if \(b=c=0\). Thus \({\mathcal {L}}({\mathbb {Z}}_n,0,0)\subset {\mathcal {Z}}\). Let \({\mathcal {F}}\) denote the discrete Fourier transform of the first component on \(\ell _2(H)\). For \(\delta _{(x,0,0)}\in \ell _2({\mathbb {Z}}_n,0,0)\), we have

$$\begin{aligned} {\mathcal {F}}(\delta _{(x,0,0)})=\frac{1}{\sqrt{n}}\sum _{k=0}^{n-1}e^{-\frac{2\pi i k x}{n}}\delta _{(k,0,0)}\,. \end{aligned}$$

A calculation gives

$$\begin{aligned} {\mathcal {F}}\lambda (a,0,0){\mathcal {F}}^{-1} \delta _{(x,0,0)}=e^{-\frac{2\pi i a x}{n}}\delta _{(x,0,0)}\,. \end{aligned}$$

This shows that \({\mathcal {F}}{\mathcal {L}}({\mathbb {Z}}_n,0,0){\mathcal {F}}^{-1}=\{e^{-\frac{2\pi i a \cdot }{n}}: a\in {\mathbb {Z}}_n\}'' = L_\infty ({\mathbb {Z}}_n)\). Since for fixed \(x\in {\mathbb {Z}}_n\), \(\delta _{(x,\cdot ,\cdot )}\in \ell _2({\mathbb {Z}}_n^2)\), we have the Hilbert space decomposition \(\ell _2(H)=\bigoplus _{x\in {\mathbb {Z}}_n} \ell _2^x({\mathbb {Z}}_n^2)\), where the superscript \(x\) is used to distinguish different copies. We may drop \(x\) if there is no ambiguity. Then by the decomposition theorem of von Neumann algebras for subalgebras of the center (see e.g. [58, Theorem IV.8.21]),

$$\begin{aligned} {\mathcal {L}}(H)\cong \bigoplus _{x\in {\mathbb {Z}}_n} {\mathcal {M}}_x\,\!, \end{aligned}$$

where \({\mathcal {M}}_x\) is determined by the unitary \({\mathcal {F}}\), the discrete Fourier transform on the first component. Let \(q_x\in \ell _\infty ({\mathbb {Z}}_n,0,0)\) be the central projection given by \(q_x: \ell _2(H)\rightarrow \mathrm{span}\{\delta _{(x,y,z)}: y,z\in {\mathbb {Z}}_n\}\). Put \(p_x={\mathcal {F}}^{-1}q_x{\mathcal {F}}\in {\mathcal {L}}({\mathbb {Z}}_n,0,0)\). Then \(p_x\) is a central projection, \(\sum _{x=0}^{n-1}p_x=1\) and \(p_x {\mathcal {L}}(H)={\mathcal {F}}^{-1}{\mathcal {M}}_x{\mathcal {F}}\). We observe that \(T_t\lambda (a,0,0)=\lambda (a,0,0)\). Hence \({\mathcal {L}}({\mathbb {Z}}_n,0,0)\) is contained in the multiplicative domain of \(T_t\). Then by the property of multiplicative domain (see e.g. [47, Theorem 3.18]) for all \(x\in {\mathbb {Z}}_n,\xi \in {\mathcal {L}}(H)\), \(T_t(p_x \xi )=p_xT_t(\xi )\in p_x{\mathcal {L}}(H)\). Let \(\eta \in {\mathcal {M}}_x\) with \(p_x\xi ={\mathcal {F}}^{-1}\eta {\mathcal {F}}\). We define \(\widetilde{T}_t\eta ={\mathcal {F}}T_t(p_x\xi ){\mathcal {F}}^{-1}\). Since \({\mathcal {F}}\) is a unitary, \(\widetilde{T}_t\) restricted to \({\mathcal {M}}_x\) is a standard nc-diffusion semigroup. By abuse of notation, we will write \(T_t\) for \(\widetilde{T}_t\) on \({\mathcal {M}}_x\) in the future.

To get more precise description of \({\mathcal {M}}_x\), we define a family of maps for \(x\in {\mathbb {Z}}_n\)

$$\begin{aligned} \pi _x: {\mathcal {L}}(H)\rightarrow B(\ell _2^x({\mathbb {Z}}_n^2)),\quad \lambda (a,b,c)\mapsto \pi _x(\lambda (a,b,c)), \end{aligned}$$

where \(\pi _x(\lambda (a,b,c))\) acts on \(\delta _{(k,l)}\) by

$$\begin{aligned} {\mathcal {F}}\lambda (a,b,c) {\mathcal {F}}^{-1}\delta _{(x,k,l)}=e^{-\frac{2\pi i x (a+bl)}{n}}\lambda _{(b,c)}\delta _{(k,l)}\,. \end{aligned}$$

Here \(\lambda (b,c)\) is the shift operator on \(\ell _2({\mathbb {Z}}_n^2)\) given by \(\lambda {(b,c)}\delta _{(k,l)}=\delta _{(k+b,l+c)}\). Then

$$\begin{aligned} {\mathcal {M}}_x&= \{\pi _x(\lambda (a,b,c)): (a,b,c)\in H\}''\\&= \left\{ e^{-\frac{2\pi i ax}{n}}v_b\otimes (v_c u_{-xb}):(a,b,c)\in H\right\} ''\,. \end{aligned}$$

Here we have used the convention \(\lambda (b,c)=v_b\otimes v_c\). If \(x=0\), we have

$$\begin{aligned} {\mathcal {M}}_0=\{\lambda (b,c):(b,c)\in {\mathbb {Z}}_n^2\}''={\mathcal {L}}({\mathbb {Z}}_n^2)=L_\infty ({\mathbb {Z}}_n^2). \end{aligned}$$

If \(x=1\), it can be checked that \(\{v_c u_{-b}: (b,c)\in {\mathbb {Z}}_n^2\}''=M_n\); see e.g. [15, Theorem VII.5.1]. Define for \((b,c)\in {\mathbb {Z}}_n^2\)

$$\begin{aligned} \rho (v_b\otimes (v_c u_{-b})) =v_c u_{-b}\,. \end{aligned}$$

Then \(\rho \) is a \(*\)-isomorphism and thus \({\mathcal {M}}_1=M_n\). \(\square \)

Consider the semigroup \(T_t\) acting on \(M_n({\mathbb {C}})\) defined by \(T_t|_{M_n({\mathbb {C}})}\) in the preceding proposition. Explicitly, \(T_t\) is determined by \(T_t (v_c u_b) = e^{-t\psi (b,c)}(v_c u_b)\) where \(\psi (b,c)=2-\delta _{b,0}-\delta _{c,0}\). Then the \(\Gamma _2\)-criterion for \(M_n\) follows from Proposition 5.13. We record this fact below.

Proposition 5.15

\(M_n\) admits a standard nc-diffusion semigroup \((T_t)_{t\ge 0}\). Let \(\Gamma ^{M_n}\) be the gradient form associated to \(T_t\). Then \(\Gamma ^{M_n}_2\ge \frac{n+2}{2n}\Gamma ^{M_n}\) in \(M_n\).

5.5 Application to the generalized Walsh system

Let us recall some basic facts about the Walsh system following [19]. Let \(\Omega _n^m=\{1,e^{2\pi i/n}, e^{2\pi i 2/n},\ldots ,e^{2\pi i(n-1)/n}\}^m\) be the \(m\)-dim discrete cube equipped with uniform probability measure \(P\). Let \(\omega _j,j=1, \ldots ,m\) denote the \(j\)th coordinate function on \(\Omega _n^m\). For a nonempty subset \(B\subset \{1,\ldots ,m\}\) and \(\varvec{x}=(x_1,\ldots ,x_m)\in {\mathbb {Z}}_n^m\), define

$$\begin{aligned} \omega _B(\varvec{x})=\prod _{j\in B}\omega _j^{x_j}, \end{aligned}$$

and \(\omega _\emptyset = 1\). Put \(G=\{\omega _B(\varvec{x}):B\subset \{1,\ldots ,m\}, \varvec{x}\in {\mathbb {Z}}_n^m\}\). Then \(G\) is clearly a group and \(L_\infty (\Omega _n^m)\) is spanned by the elements of \(G\).

We consider the abelian group \({\mathbb {Z}}_n^m\). Define

$$\begin{aligned} \psi (x_1,\ldots ,x_m)=(m-\delta _{x_1}- \cdots -\delta _{x_m}),\quad (x_1,\ldots ,x_m)\in {\mathbb {Z}}_n^m \,, \end{aligned}$$

where \(\delta _x = \delta _{x,0}\). Given \(\varvec{x} =(x_1,\ldots ,x_m)\), put \(B_{\varvec{x}}=\{i: x_i\ne 0, i=1,\ldots ,m\}\). Clearly \(\psi (x_1,\ldots ,x_m)=|B_{\varvec{x}}|\), where \(|B|\) is the cardinality of \(B\). An argument similar to the proof of Proposition 5.13 shows that \(\psi \) is a cn-length function and the associated \(\Gamma \) form satisfies

$$\begin{aligned} \Gamma _2\ge \frac{n+2}{2n}\Gamma \,\mathrm{in }\, {\mathcal {L}}({\mathbb {Z}}_n^m)\,. \end{aligned}$$
(5.1)

Here the constant \(\frac{n+2}{2n}\) is given by Proposition 5.12. Define a map

$$\begin{aligned} \beta : {\mathbb {Z}}_n^m\rightarrow G,\quad \varvec{x} =(x_1,\ldots ,x_m)\mapsto \prod _{j\in B_{\varvec{x}}}\omega _j^{x_j}\,. \end{aligned}$$

It is easy to check that \(\beta \) is a group isomorphism from \({\mathbb {Z}}_n^m\) to \(G\). The idea here is to convert addition to multiplication. Under the identification \(\beta \), \({\mathcal {L}}({\mathbb {Z}}_n^m)=L_\infty (\Omega _n^m)\) and thus every \(f\in {\mathcal {L}}({\mathbb {Z}}_n^m)\) can be written as

$$\begin{aligned} f ={\mathbb {E}}\,(f) + \sum _{\varvec{x}\in {\mathbb {Z}}_n^m, \varvec{x}\ne \varvec{0}} \hat{f}_{\varvec{x}} \prod _{j\in B_{\varvec{x}}}\omega _j^{x_j}\,, \end{aligned}$$

where \({\mathbb {E}}\,(f)=\tau (f)\) is the expectation associated to the uniform probability. By abuse of notation, we still denote by \(\psi \) the cn-length function induced by \(\beta \) on \(\{\omega _B\}\), i.e.

$$\begin{aligned} \psi (\omega _{B_{\varvec{x}}})=\psi (\beta (\varvec{x})) := \psi (\varvec{x}),\quad \varvec{x}\in {\mathbb {Z}}_n^m\,. \end{aligned}$$

Then we have

$$\begin{aligned} \psi (\omega _{B_{\varvec{x}}})=\psi (x_1,\ldots ,x_m)=|B_{\varvec{x}}|\,. \end{aligned}$$

Therefore the infinitesimal generator \(A\) of the heat semigroup \(T_t\) in this case is the number operator for the generalized Walsh system which counts non-zero elements

$$\begin{aligned} A\,\omega _B=|B|\omega _B\,. \end{aligned}$$

Moreover, it follows from (5.1) that \(\Gamma _2\ge \frac{n+2}{2n}\Gamma \) in \(L_\infty (\Omega _n^m)\). The case \(n=2\) is of particular interest. Indeed, we may write \(f\in {\mathcal {L}}({\mathbb {Z}}_2^m)\) as

$$\begin{aligned} f ={\mathbb {E}}\,(f) + \sum _{B\subset \{1,\ldots ,m\},B\ne \emptyset } \hat{f}_B \omega _B\,. \end{aligned}$$

Note that in this case \(\omega _B^{-1}\omega _C=\omega _{B \triangle C}\). Then the Gromov form of \(\{\omega _B\}\) is given by

$$\begin{aligned} K(\omega _B, \omega _C) =\frac{1}{2}(|B|+|C|-|B \triangle C|)=|B\cap C|\,. \end{aligned}$$

Hence, we find the gradient form

$$\begin{aligned} \Gamma (f,f)=\sum _{B,C\subset \{1,\ldots ,m\}}\bar{\hat{f}}_B\hat{f}_C|B\cap C|\omega _{B\triangle C}\,. \end{aligned}$$

Let \(e_j=(1,\ldots ,-1,\ldots ,1)\) where \(-1\) is only at the \(j\)th position. For \(x\in \Omega _n^m\), put \((\partial _j f) (x)=\frac{1}{2}(f(x)-f(xe_j))\) and define the discrete gradient \(\nabla f = (\partial _j f)_{j=1}^m\). Then a calculation gives \(\Gamma (f,f)=|\nabla f|^2\), where \(|\cdot |\) is the Euclidean norm of a vector in \({\mathbb {C}}^n\). If we simply write \(|\nabla f|=\Gamma (f,f)^{1/2}\) for any \(n=2,3,\ldots \), our Poincaré inequalities for the generalized Walsh system is a dimension \(m\) free estimate.

Corollary 5.16

Let \(2\le p<\infty \). Then for all \(f\in L_p(\Omega _n^m,P)\),

$$\begin{aligned} \Vert f-{\mathbb {E}}\,(f)\Vert _p\le C\sqrt{\frac{2n}{n+2}}\min \{\sqrt{p}\Vert \,|\nabla f|\,\Vert _\infty ,\,{p}\Vert \,|\nabla f|\,\Vert _p\}\,. \end{aligned}$$

One of Efraim and Lust-Piquard’s main results in [19] asserts that for \(2<p<\infty \) and \(f\in L_p(\Omega _2^m,P)\),

$$\begin{aligned} \Vert f-{\mathbb {E}}\,(f)\Vert _p\le C\sqrt{p}\Vert \,|\nabla f|\,\Vert _p\,. \end{aligned}$$

Our version for the case \(n=2\) is weaker. But our approach works from general \(n\) while their result is only for \(n=2\). For arbitrary \(n>2\), it is unclear to us whether it is possible to obtain similar results based on their method. Moreover, their concentration inequality [19, Corollary 4.2] due to Bobkov and Götze is now a special case of our (4.4) with the same order. Efraim and Lust-Piquard’s inequality would follow from our general theory if (2.5) were true.

5.6 The \(q\hbox {-}\)Gaussian algebras

We first recall some definitions and basic facts following [9]. Throughout this section \(-1\le q\le 1\). Let \({\mathcal {H}}\) be a separable real Hilbert space with complexification \({\mathcal {H}}_{\mathbb {C}}\). Let \((F_q({\mathcal {H}}),\langle \cdot ,\cdot \rangle _q)\) be the \(q\)-Fock space with vacuum vector \(\Omega \) and \(\Gamma _q({\mathcal {H}})\) the \(q\)-Gaussian algebra which is the von Neumann algebra generated by \(s(f)=l(f)+l^*(f)\) for \(f\in {\mathcal {H}}\) where

$$\begin{aligned} l^*(f)f_1\otimes \cdots \otimes f_n=f\otimes f_1\otimes \cdots \otimes f_n \end{aligned}$$

and

$$\begin{aligned} l(f)f_1\otimes \cdots \otimes f_n=\sum _{j=1}^n q^{j-1}\langle f_j,f\rangle f_1\otimes \cdots \otimes f_{j-1}\otimes f_{j+1}\otimes \cdots \otimes f_n\,\end{aligned}$$

are the creation and annihilation operators respectively. The vacuum vector gives rise to a canonical tracial state \(\tau _q(X)=\langle X\Omega ,\Omega \rangle _q\) for \(X\in \Gamma _q({\mathcal {H}})\). The \(q\)-Ornstein–Uhlenbeck semigroup \(T_t^q=\Gamma _q(e^{-t}I_{\mathcal {H}})\) is a standard semigroup and extends to a semigroup of contractions on \(L_p\) spaces. The generator on \(L_2\) is the number operator \(N^q\) which acts on the Wick product by

$$\begin{aligned} N^q W(f_1\otimes \cdots \otimes f_n)= n W(f_1\otimes \cdots \otimes f_n), \quad f_1,\ldots ,f_n\in {\mathcal {H}}_{\mathbb {C}}\,, \end{aligned}$$

where \(W\) is the Wick operator. It is easy to check that \((T_t)\) is a nc-diffusion semigroup.

Let \(\ell _2^n\) be the real Hilbert space with dimension \(n\) and \(\{e_1,\ldots ,e_n\}\) an orthonormal basis. For \(j=1,\ldots ,n\), consider the embedding

$$\begin{aligned} \iota _j: {\mathcal {H}}\rightarrow {\mathcal {H}}\otimes \ell _2^n,\quad h\mapsto h\otimes e_j\,. \end{aligned}$$

According to [9, Theorem 2.11], there exists a unique map \(\Gamma _q(\iota ):\Gamma _q({\mathcal {H}})\rightarrow \Gamma _q({\mathcal {H}}\otimes \ell _2^n)\) such that \(\Gamma _q(\iota _j)(s(h))=s(h\otimes e_j)\). The map \(\Gamma _q(\iota )\) is linear, bounded, unital completely positive and preserves the canonical trace. Define \(s_j^q(h)=s(h\otimes e_j)\). If \(q=1\), we write \(g_j(h)=s(h\otimes e_j)\) and it is well-known \(g_j(h)\) is a standard Gaussian random variable if \(\Vert h\Vert =1\). For \(h\in {\mathcal {H}}\), put

$$\begin{aligned} u_n(h) =\frac{1}{\sqrt{n}}\sum _{j=1}^n s_j^q(h)\otimes g_j(h)\,. \end{aligned}$$

Write \({\mathcal {M}}_n^q = \Gamma _q({\mathcal {H}}\otimes \ell _2^n)\). We consider the von Neumann algebra ultraproduct \({\mathcal {M}}= \prod _{\mathcal {U}}{\mathcal {M}}_n^q\bar{\otimes }{\mathcal {M}}_n^1\). Any element of \({\mathcal {M}}\) can be written as \(u_\omega (h)=(u_n(h))^\bullet \). We need the following fact proved in [4].

Lemma 5.17

The map \(s(h)\mapsto u_\omega (h)\) extends to an injective trace preserving *-homomorphism \(\pi : \Gamma _q({\mathcal {H}})\rightarrow {\mathcal {M}}\). Moreover, for \(x\in \Gamma _q({\mathcal {H}})\),

$$\begin{aligned} \pi (T_t^q(x))=(Id\otimes T_t^1)^\bullet \pi (x)\,. \end{aligned}$$

Proposition 5.18

For \(-1\le q\le 1\), \(\Gamma _2\ge \Gamma \) in \(\Gamma _q({\mathcal {H}}).\)

Proof

Since \(\pi \) is injective, it suffices to prove \(\pi \Gamma _2^{N^q}\ge \pi \Gamma ^{N^q}\) in \({\mathcal {M}}\). By Lemma 5.17, we have \(\pi N^q (x)=(Id\otimes N^1)^\bullet \pi (x)\). It follows that

$$\begin{aligned} \pi (\Gamma ^{N^q}(x,y))=\Gamma ^{(Id\otimes N^1)^\bullet }(\pi (x),\pi (y)), \end{aligned}$$

and similar identity is true for \(\Gamma _2^{N^q}\). It is proved in [4] by using the central limit theorem of Speicher [57] that \(\Gamma _2\ge \Gamma \) in \(\Gamma _1({\mathcal {H}})\). Here \(\Gamma _1({\mathcal {H}})\) is the von Neumann algebra acting on the symmetric Fock space. It follows that for all \(n\in {\mathbb {N}}\) and in \({\mathcal {M}}_n^q\otimes {\mathcal {M}}_n^1\), \(\Gamma _2^{Id\otimes N^1}\ge \Gamma ^{Id\otimes N^1}\). Hence, we find

$$\begin{aligned} \Gamma _2^{(Id\otimes N^1)^\bullet }(\pi (x_i),\pi (x_j))\ge \Gamma ^{(Id\otimes N^1)^\bullet }(\pi (x_i),\pi (x_j)) \end{aligned}$$

where for any \(m\in {\mathbb {N}}\) and \(x_i\in \Gamma _q({\mathcal {H}}), i=1,\ldots ,m\), as desired. \(\square \)

5.7 The hyperfinite \(II_1\) factor

Our goal in this subsection is to show that the hyperfinite \(II_1\) factor \(R\) admits different standard nc-diffusion semigroups with \(\Gamma _2\)-criterion and that the best possible \(\alpha \) characterizes the corresponding dynamical system. It is well known that \(R\) can be approximated by matrix algebras \(\{M_{n^k}:k\in {\mathbb {N}}\}\). We will embed \(M_{n^{m/2}}\) into the group von Neumann algebra of the generalized discrete Heisenberg group \(H_n^{m+1}={\mathbb {Z}}_n/2\times {\mathbb {Z}}_n^{m}\).

Let \(\Theta =(\theta _{jk})\) be an antisymmetric \(m\times m\) matrix with \(\theta _{jk}=\frac{1}{2}\) if \(j<k\). The multiplication in \(H_n^{m+1}={\mathbb {Z}}_n/2\times {\mathbb {Z}}_n^{m}\) is given by

$$\begin{aligned} (x,\xi )(y,\eta )=(x+y+B(\xi ,\eta ), \xi +\eta )\,, \end{aligned}$$

where \(B:{\mathbb {Z}}_n\times {\mathbb {Z}}_n\rightarrow {\mathbb {Z}}_n/2\) is a bilinear form given by \(B(\xi ,\eta )=\sum _{j,k=1}^m\theta _{jk}\xi _j\eta _k=\langle \xi ,\Theta \eta \rangle \). For \((r,\xi )\in H_n^{m+1}\), put \(\psi (r,\xi )=\sum _{j=1}^m 1-\delta _{\xi _j,0}=\#\{\xi _j\ne 0\}\). Define a semigroup acting on \({\mathcal {L}}(H_n^{m+1})\) by \(T_t \lambda (r,\xi )=e^{-t\psi }\lambda (r,\xi )\) for \(\lambda (r,\xi )\in {\mathcal {L}}(H_n^{m+1})\) and \(t\ge 0\). Using Lemma 5.4, an argument similar to Proposition 5.13 shows that \((T_t)_{t\ge 0}\) is a standard nc-diffusion semigroup and that the associated gradient form satisfies \(\Gamma _2\ge \frac{n+2}{2n}\Gamma \) in \({\mathcal {L}}(H_n^{m+1})\).

Lemma 5.19

Let \(m\ge 2\) be an even integer and \(n\ge 3\). We have

$$\begin{aligned} {\mathcal {L}}(H_n^{m+1})\cong \bigoplus _{x\in {\mathbb {Z}}_n/2} {\mathcal {M}}_x\,, \end{aligned}$$

where \({\mathcal {M}}_2=M_{n^{m/2}}\). Furthermore, \(T_t\) leaves each \({\mathcal {M}}_x\) invariant.

Proof

Most of the argument utilizes the proof of Proposition 5.14 and [28, Lemma 5.3]. Note that \(\lambda ({\mathbb {Z}}_n/2,0)\) lives in the center of \({\mathcal {L}}(H_n^{m+1})\). By the decomposition of von Neumann algebras for the subalgebras of the center we obtain the first assertion. Write \(e_j,j=1,\ldots ,m\) for the canonical basis of \({\mathbb {Z}}_n^m\) and put \(u_r^j=\lambda (0,re_j)\) for \(r\in {\mathbb {Z}}_n\). Then these \(u_r^j\)’s generate \(\lambda (0,{\mathbb {Z}}_n^m)\) and \(u_r^ju_s^k(\delta _{(x,\cdot )})=e^{{2\pi i rs\theta _{jk}x}/n} u_s^ku_r^j(\delta _{(x,\cdot )})\). Acting on \(H_2:=\mathrm{span}\{\delta _{(2,\cdot )}\}\), \(u_r^j\)’s satisfy \(u_r^ju_s^k=e^{{2\pi i rs}/n} u_s^ku_r^j\) for \(j<k\) and \(u_r^ju_s^k=e^{-{2\pi i rs}/n} u_s^ku_r^j\) for \(j>k\). It is clear that \(y(r_1,\ldots ,r_m)=u_{r_1}^1,\ldots ,u_{r_m}^m\) is a basis for \({\mathcal {M}}_2\) which satisfies the equation

$$\begin{aligned} u_r^jy(r_1,\ldots ,r_m){u_r^j}^*=C(r,j,r_1,\ldots ,r_m)y(r_1,\ldots ,r_m)\,, \end{aligned}$$

where \(C(r,j,r_1,\ldots ,r_m)=\exp (2\pi i r(r_1+ \cdots +r_{j-1}-r_{j+1}- \cdots -r_m)/n)\). In order to determine the center of \({\mathcal {M}}_2\), we consider the equation \(C(r,j,r_1, \ldots ,r_m)=1\) for all \(r\in {\mathbb {Z}}_n, j=1,\ldots ,m\). This leads to a linear system over \({\mathbb {Z}}_n\)

$$\begin{aligned} -r_2- \cdots -r_m&=0,\\ r_1-r_3- \cdots -r_m&=0,\\ \vdots \quad&\quad \vdots \\ r_1+ \cdots +r_{m-1}&=0. \end{aligned}$$

Solving this system, we find \(r_1= \cdots =r_m=0\). Here we used the crucial assumption that \(m\) is even. Hence \({\mathcal {M}}_2\) has trivial center. Since it has dimension \(n^m\), it follows that \({\mathcal {M}}_2=M_{n^{m/2}}\), as desired. By restricting \(T_t\) to \(M_{n^{m/2}}\) and repeating the argument of Proposition 5.14, we can prove the last assertion. \(\square \)

It follows from the lemma that \(M_{n^{m/2}}\) admits a standard nc-diffusion semigroup \(T_t\) with \(\Gamma _2\ge \frac{n+2}{2n}\Gamma \) in \(M_{n^{m/2}}\) for all \(m\in 2{\mathbb {N}}\). Since the hyperfinite \(II_1\) factor \(R\) is the weak closure of \(\cup _{k=1}^\infty M_{n^{k}}\), we have proved the following result.

Proposition 5.20

For any integer \(n\ge 2\), there exists a standard nc-diffusion semigroup \(T_t^n\) acting on \(R\) such that the associated gradient form \(\Gamma ^n\) satisfies \(\Gamma _2^n\ge \frac{n+2}{2n}\Gamma ^n\) in \(R\). The constant \(\alpha _n=\frac{n+2}{2n}\) is best possible.

The last conclusion follows from Proposition 5.12. We want to show that the semigroups \(T^n_t\) for different \(n\) are different. Let us now recall a definition from dynamical systems; see e.g. [62, Definition 2.4]. Let \((X,{\mathcal {B}}_X,\mu , T)\) be a measure-preserving dynamic system (MPDS) where \((X,{\mathcal {B}}_X,\mu )\) is a probability space and \(T\) is a measure-preserving transformation. A MPDS \((Y,{\mathcal {B}}_Y, \nu ,S)\) is said to be isomorphic to \((X,{\mathcal {B}}_X,\mu , T)\) if there exist (i) full measure sets \(X_1\subset X\) and \(Y_1\subset Y\) such that \(T(X_1)\subset X_1\) and \(S(Y_1)\subset Y_1\); and (ii) an invertible measure-preserving measurable map \(\phi : X\rightarrow Y\) such that \(\phi (T x)=S(\phi x)\) for all \(x\in X_1\). This motivates our following definition.

Definition 5.21

Let \(S_t\) and \(T_t\) be standard semigroups acting on noncommutative probability spaces \(({\mathcal {N}},\tau )\) and \(({\mathcal {M}},\tau ')\) respectively. We say \(({\mathcal {M}},T_t)\) and \(({\mathcal {N}}, S_t)\) are isomorphic if there exist \(\lambda >0\) and a trace-preserving \(*\)-isomorphism \(\phi :{\mathcal {M}}\rightarrow {\mathcal {N}}\) such that \(S_{\lambda t}(\phi x)=\phi (T_t x)\) for all \(x\in {\mathcal {M}}\).

The following result shows that \(R\) admits infinitely many non-isomorphic standard nc-diffusion semigroups.

Proposition 5.22

Let \(T_t^n\) be the semigroup considered in Proposition 5.20. If \((R, T_t^n)\) and \((R, T_{t}^{n'})\) are isomorphic, then \(\alpha _n=\alpha _{n'}\).

Proof

There exists a trace-preserving \(*\)-isomorphism \(\phi : R\rightarrow R\) such that

$$\begin{aligned} \phi (T_t^n x)=T_{\lambda t}^{n'}(\phi x) \end{aligned}$$
(5.2)

for \(x\in {\mathcal {M}}\). Let \(A^n\) be the generator of \(T_t^n\). \(\Gamma _2^{A^{n'}}\ge \frac{n'+2}{2n'}\Gamma ^{A^{n'}}\) implies \(\Gamma _2^{\lambda A^{n'}}\ge \lambda \frac{n'+2}{2n'}\Gamma ^{\lambda A^{n'}}\). This together with (5.2) gives \(\Gamma _2^{A^n}\ge \lambda \frac{n'+2}{2n'}\Gamma ^{A^n}\). But the best \(\alpha \) is \(\alpha _n=\frac{n+2}{2n}\). Hence we have \(\frac{n+2}{2n}\ge \lambda \frac{n'+2}{2n'}\). It is clear that \(\mathrm{sp}(A^n)={\mathbb {N}}\) and \(\mathrm{sp}(\lambda A^n)=\lambda {\mathbb {N}}\). Here \(\mathrm{sp}(A^n)\) denotes the spectrum of \(A^n\). (5.2) implies \(\mathrm{sp}(\lambda A^n)=\mathrm{sp}(A^n)\) and thus \(\lambda = 1\). Hence \(n'\ge n\). Repeating the argument by starting from \(\Gamma _2^{n}\ge \frac{n+2}{2n}\Gamma ^{n}\) gives \(n\ge n'\). \(\square \)

6 Tensor products and free products

In this section we will construct further examples with the \(\Gamma _2\)-criterion based on the examples considered in the previous section. This is done via the powerful algebraic tools—tensor products and free products. It is not difficult to see that the property “standard nc-diffusion” is stable under tensor products and free products. Due to the reason explained in the previous section, it suffices to consider the algebraic \(\Gamma _2\)-condition. That is, we always work with a dense subalgebra contained in the domain of the form under consideration.

6.1 Tensor products

The following result is our starting point to understand tensor products.

Lemma 6.1

Let \(\Theta :{\mathcal {A}}\times {\mathcal {A}}\rightarrow {\mathcal {M}}\) and \(\Phi :{\mathcal {B}}\times {\mathcal {B}}\rightarrow {\mathcal {N}}\) be positive sesquilinear forms, where \({\mathcal {A}}\subset {\mathcal {M}}\) and \({\mathcal {B}}\subset {\mathcal {N}}\) are dense subalgebras so that \(\Theta \) and \(\Phi \) are well-defined. Then \(\Theta \otimes \Phi : {\mathcal {A}}\otimes {\mathcal {B}}\times {\mathcal {A}}\otimes {\mathcal {B}}\rightarrow {\mathcal {M}}\otimes {\mathcal {N}}\) is positive where for \(\xi ^i=\sum _{k=1}^{n_i} x_k^i\otimes y_k^i\in {\mathcal {A}}\otimes {\mathcal {B}}\),

$$\begin{aligned} \Theta \otimes \Phi \Big ( \xi ^i, \xi ^j\Big ):=\sum _{k=1}^{n_i}\sum _{l=1}^{n_j}\Theta (x_k^i,x_l^j)\otimes \Phi (y_k^i,y_l^j)\,. \end{aligned}$$

Proof

For \(r\in {\mathbb {N}}\), let \((x_k^i)\subset {\mathcal {A}}, (y_k^i)\subset {\mathcal {B}}\) where \(k=1,\ldots ,n_i, i=1,\ldots ,r\). Put \(m=\sum _{i=1}^r n_i\). Without loss of generality we may assume \(n_i=n\) for \(i=1,\ldots ,r\). Suppose \({\mathcal {M}}\) and \({\mathcal {N}}\) act on Hilbert spaces \(H\) and \(K\) respectively. Then \((\Theta (x_k^i,x_l^j))_{k,l,i,j}=\sum _{i,j,k,l} \Theta (x_k^i,x_l^j)\otimes e_{(k,i),(l,j)}\ge 0\) as an operator on \(\ell _2^m(H)\) where \(1\le k,l\le n\). Similarly, \((\Phi (y_k^i,y_l^j))_{k,l,i,j}\ge 0\) on \(\ell _2^m(K)\). It follows that

$$\begin{aligned}&(\Theta (x_k^i,x_l^j))\otimes (\Phi (y_{k'}^{i'},y_{l'}^{j'}))\\&\quad =\sum _{i,j,k,l,i',j',k',l'}\Theta (x_k^i,x_l^j)\otimes \Phi (y_{k'}^{i'},y_{l'}^{j'})\otimes e_{(k,i),(l,j)}\otimes e_{(k',i'),(l',j')}\ge 0 \end{aligned}$$

on \(\ell _2^m(H)\otimes \ell _2^m(K)\). Define

$$\begin{aligned}&v:\ell _2^m(H\otimes K)\rightarrow \ell _2^m(H)\otimes \ell _2^m(K),\\&\quad \sum _{s}(\xi _t^s\otimes \eta _t^s)\otimes e_t\mapsto \sum _{s}(\xi _t^s\otimes e_t)\otimes (\eta _t^s\otimes e_t)\,. \end{aligned}$$

Here \(\xi _t^s\in H\), \(\eta _t^s\in K\), and \((e_t)\) is the canonical basis of \(\ell _2^m\) for \(t=1,\ldots ,m\). Then

$$\begin{aligned} v^*\Big [\sum _s(\xi _t^s\otimes e_t)\otimes (\eta _t^s\otimes e_t)\Big ]=\sum _{s}(\xi _t^s\otimes \eta _t^s)\otimes e_t\,. \end{aligned}$$

It is clear that \(v^*[(\Theta (x_k^i,x_l^j))\otimes (\Phi (y_{k'}^{i'},y_{l'}^{j'}))] v\ge 0\). But

$$\begin{aligned} v^*[(\Theta (x_k^i,x_l^j))\otimes (\Phi (y_{k'}^{i'},y_{l'}^{j'}))] v&= \sum _{i,j,k,l}\Theta (x_k^i,x_l^j)\otimes \Phi (y_{k}^{i},y_{l}^{j})\otimes e_{(k,i),(l,j)}\\&= [\Theta (x_k^i,x_l^j)\otimes \Phi (y_{k}^{i},y_{l}^{j})]_{(k,i),(l,j)}\,. \end{aligned}$$

Let \(\varvec{1}_n\) be a \(n\times 1\) column vector with each entry equal to \(1_{\mathcal {M}}\otimes 1_{\mathcal {N}}\) and \(I_r\) the \(r\times r\) identity matrix. Define an operator \(w: \ell _2^r(H\otimes K)\rightarrow \ell _2^m(H\otimes K)\) by

$$\begin{aligned} w=\varvec{1}_n\otimes I_r=\left( \begin{array} {c@{\quad }c@{\quad }c@{\quad }c} \varvec{1}_n&{} &{}&{} \\ &{}\varvec{1}_n &{} &{}\\ &{}&{} \ddots &{}\\ &{}&{}&{} \varvec{1}_n \end{array} \right) \,. \end{aligned}$$

\(w\) is an \(m\times r\) matrix. Note that \([\Theta (x_k^i,x_l^j)\otimes \Phi (y_{k}^{i},y_{l}^{j})]\) is an \(m\times m\) matrix. Then

$$\begin{aligned} 0&\le w^* \left[ \sum _{i,j,k,l}\Theta (x_k^i,x_l^j)\otimes \Phi (y_{k}^{i},y_{l}^{j})\otimes e_{(k,i),(l,j)}\right] w\\&= \sum _{i,j=1}^r \sum _{k=1}^{n_i}\sum _{l=1}^{n_j}\Theta (x_k^i,x_l^j)\otimes \Phi (y_k^i,y_l^j)\otimes e_{i,j}\\&= \sum _{i,j=1}^r \Theta \otimes \Phi (\xi ^i,\xi ^j)\otimes e_{i,j}\,, \end{aligned}$$

which completes the proof. \(\square \)

Lemma 6.2

Let \((T_t)_{t\ge 0}\) and \((S_t)_{t\ge 0}\) be standard semigroups with generator \(A\) and \(B\) acting on finite von Neumann algebras \({\mathcal {M}}\) and \({\mathcal {N}}\) respectively such that \(\Gamma ^A_2\ge \alpha \Gamma ^A\) in \({\mathcal {M}}\) and \(\Gamma ^B_2\ge \alpha \Gamma ^B\) in \({\mathcal {N}}\). Then \(\Gamma _2^{A\otimes I}\ge \alpha \Gamma ^{A\otimes I}\), \(\Gamma _2^{I\otimes B}\ge \alpha \Gamma ^{I\otimes B}\) and \(\Gamma _2^{A\otimes I+I\otimes B}\ge \alpha \Gamma ^{A\otimes I+I\otimes B}\) all in \({\mathcal {M}}\otimes {\mathcal {N}}\).

Proof

The first inequality follows from Lemma 6.1 with \(\Theta =\Gamma ^A\) and \(\Phi (y_1,y_2)=y_1^*y_2\). The second inequality can be shown similarly. For the last one, note that

$$\begin{aligned}&\Gamma _2^{A\otimes I+I\otimes B}-\alpha \Gamma ^{A\otimes I+I\otimes B}\\&\quad =(\Gamma ^{A\otimes I}_2-\alpha \Gamma ^{A\otimes I})+(\Gamma ^{I\otimes B}_2-\alpha \Gamma ^{I\otimes B})+2\Gamma ^{A}\otimes \Gamma ^{ B}\,. \end{aligned}$$

Then the first two inequalities and Lemma 6.1 with \(\Theta =\Gamma ^{A}\) and \(\Phi =\Gamma ^{B}\) yield the assertion. \(\square \)

Proposition 6.3

Let \(A_j\) be self-adjoint generators of standard nc-diffusion semigroups \((T_t^{A_j})\) acting on \({\mathcal {N}}_j\) and \(\Gamma ^{A_j}_2\ge \alpha \Gamma ^{A_j}\) respectively for \(j=1,\ldots ,n\) with the same constant \(\alpha >0\). Then the tensor product generator \(\otimes A_j(x_1\otimes \cdots \otimes x_n)=\sum _{j} x_1\otimes \cdots \otimes x_{j-1}\otimes A_j(x_j)\otimes x_{j+1}\otimes \cdots \otimes x_n\) generates a standard nc-diffusion semigroup \((T_t^{\otimes A_j})\) with

$$\begin{aligned} \Gamma _2^{\otimes A_j}\ge \alpha \Gamma ^{\otimes A_j} \,. \end{aligned}$$

Proof

Note that \(T_t^{\otimes A_j}= T_t^{A_1}\otimes \cdots \otimes T_t^{A_n}\). Since \((T_t^{A_j})\) is a standard nc-diffusion semigroup for \(j=1,\ldots ,n\), so is \(T_t^{\otimes A_j}\). We prove the \(\Gamma _2\)-condition by induction. The case \(n=2\) follows from Lemma 6.2. The general case follows by induction and repeatedly invoking Lemma 6.1 to deal with “cross terms” like \(\Gamma ^{I\otimes \cdots A_i \cdots \otimes I}\otimes \Gamma ^{I\otimes \cdots A_j \cdots \otimes I}\). \(\square \)

Example 6.4

(Tensor product of matrix algebras) Let \(A\) be the generator of the semigroup \(T_t\) acting on \(M_n\) considered in Proposition 5.15. Let \(\Gamma \) be the gradient form associated to \(\sum _{i=1}^m I\otimes \cdots \otimes A\otimes \cdots \otimes I\) where \(A\) is in the \(i\)th position. Then it follows from Proposition 6.3 that \(\Gamma _2\ge \frac{2+n}{2n} \Gamma \) in \(\otimes _{i=1}^m M_n\).

Example 6.5

(Random matrices) Let \((\Omega , {\mathbb {P}})\) be a probability space. Consider \(I\otimes T_t\) acting on \(L_\infty (\Omega ,{\mathbb {P}})\otimes M_n\) where \(T_t\) is the semigroup considered in Proposition 5.15. By Lemma 6.2, \(I\otimes T_t\) is a standard nc-diffusion semigroup and satisfies \(\Gamma _2\ge \frac{2+n}{2n} \Gamma \) in \(L_\infty (\Omega ,{\mathbb {P}})\otimes M_n\). Hence our results apply for random matrices.

Example 6.6

(Product measure) Here we consider \(A_j=I-E_j\) for \(E_j\) a conditional expectation on \({\mathcal {N}}_j\) for \(j=1,\ldots ,n\). By example 4.11, \(A_j\) generates a standard nc-diffusion semigroup and \(\Gamma _2^{A_j}\ge \frac{1}{2}\Gamma ^{A_j}\). Then we deduce from Proposition 6.3 that \(\Gamma _2^A\ge \frac{1}{2}\Gamma ^A\) for the tensor product generator \(A =\otimes A_j\). For \(x=x_1\otimes \cdots \otimes x_n\), put \(\Gamma _j(x,x)=x_1^*x_1\otimes \cdots \otimes \Gamma ^{A_j}(x_j,x_j)\otimes \cdots \otimes x_n^*x_n\). Then we have

$$\begin{aligned} \Gamma (x,x)=\sum _{j=1}^n \Gamma _j(x,x)\,. \end{aligned}$$

We want to investigate an easy consequence of our general theory for the product measure space. Let \((\Omega _i, {\mathbb {P}}_i)\), \(i=1,\ldots ,n\) be a family of probability spaces and denote by \((\Omega , {\mathbb {P}})\) the product probability space. Then \(L_\infty (\Omega , {\mathbb {P}})=\otimes _{i=1}^n L_\infty (\Omega _i, {\mathbb {P}}_i)\). Define \(E_i(f)=\int f d{\mathbb {P}}_i\) for \(f\in L_\infty (\Omega , {\mathbb {P}})\) and put \(A_i=I-E_i\). Then

$$\begin{aligned} \Gamma _i(f,f)&= \frac{1}{2} \left( |f|^2 - f\int \bar{f} d{\mathbb {P}}_i- \bar{f}\int f d{\mathbb {P}}_i +\int |f|^2 d{\mathbb {P}}_i\right) \\&= \frac{1}{2}\left( |f-\int f d{\mathbb {P}}_i|^2+ \int \left( |f|^2-\Big |\int f d{\mathbb {P}}_i\Big |^2\right) d{\mathbb {P}}_i\right) \,. \end{aligned}$$

It is straightforward to check that the fixed point subalgebra of the semigroup \(e^{-t(\otimes A_i)}\) is \({\mathbb {C}}1\). Hence \(E_\mathrm{Fix} f = {\mathbb {E}}\,f\) for \(f\in L_\infty (\Omega , {\mathbb {P}})\) where \({\mathbb {E}}\,\) is the expectation operator of \({\mathbb {P}}\). Then (4.7) yields

$$\begin{aligned} {\mathbb {P}}(f-{\mathbb {E}}\,(f)\ge t)&\le \exp \left( -\frac{ct^2}{\Vert \sum _{i=1}^n \Gamma _i(f,f)\Vert _\infty }\right) \nonumber \\&\le \exp \left( -\frac{2ct^2}{\sum _{i=1}^n\Vert f-\int fd{\mathbb {P}}_i\Vert _\infty ^2+\Vert \int (|f|^2-|\int f d{\mathbb {P}}_i |^2 ) d{\mathbb {P}}_i\Vert _\infty }\right) \!.\nonumber \\ \end{aligned}$$
(6.1)

Note that we do not impose any concrete condition on the probability spaces. This shows that the sub-Gaussian tail behavior is always true for product measures. We do not know whether such results were known before.

6.2 Free products with amalgamation

Here we want to prove that the condition \(\Gamma _2\ge \alpha \Gamma \) is stable under free products. Our general reference is [61]. We need some preliminary facts about free product of semigroups \(T_t= *_{k}T_t^{ A_k}\) acting on \({\mathcal {N}}:=*_{{\mathcal {D}}, k} {\mathcal {N}}_k\) with generators \(A_k\) acting on von Neumann algebra \({\mathcal {N}}_k\supset {\mathcal {D}}\). Here \({\mathcal {D}}\) is a von Neumann subalgebra of all \({\mathcal {N}}_k\). Similar to the tensor products considered before, if \((T_t^{A_k})\) is a standard nc-diffusion semigroup for \(k=1,\ldots ,n\), so is \(*_{k}T_t^{ A_k}\). We assume that \(A_k\) commutes with the conditional expectation \(E: {\mathcal {N}}_k\rightarrow {\mathcal {D}}\) for which we amalgamate and even

$$\begin{aligned} A_kE =EA_k =0 \,. \end{aligned}$$

Our first task is to calculate the gradient \(\Gamma \). For simplicity of notation, we always assume the elements we consider are chosen so that \(\Gamma \) and \(\Gamma _2\) are well-defined. Let us now consider elementary words \(x=a_1,\ldots , a_m\) and \(y=b_1,\ldots , b_n\) of mean \(0\) elements \(a_k\in {\mathcal {N}}_{i_k}\), \(b_k\in {\mathcal {N}}_{j_k}\). Recall that the free product generator is given by

$$\begin{aligned} A(b_1 ,\ldots , b_n)=\sum _{l=1}^n b_1 ,\ldots , b_{l-1}A_{j_l}(b_l)b_{l+1} ,\ldots , b_n \,. \end{aligned}$$

In the future we will ignore the index for \(A\). If we want to apply the free product generator \(A\) on the product \(x^*y\), we have to know the mean \(0\) decomposition

$$\begin{aligned} x^*y&= a_m^* ,\ldots , a_1^*b_1 ,\ldots , b_n \\&= \sum _{k=1}^{\min (n,m)} a_m^* ,\ldots , a_{k+1}^*\mathop {\overbrace{a_k^*E(a_{k-1}^* ,\ldots , a_1^*b_1 ,\ldots , b_{k-1})b_k}}\limits ^{\circ { }}b_{k+1} ,\ldots , b_n \\&\quad +\,{\left\{ \begin{array}{ll} E(a_m^*,\ldots , a_1^* b_1 ,\ldots , b_m)b_{m+1} ,\ldots , b_n, \,\text{ if } \, m\le n,\\ a_m^* ,\ldots , a_{n+1}^*E(a_n^* ,\ldots , a_1^* b_1 ,\ldots , b_m),\, \text{ if } \,m> n.\\ \end{array}\right. } \end{aligned}$$

Here \( \mathring{x}=x-E(x)\). Let \(k_0=\inf \{i\in {\mathbb {N}}: k_i\ne j_i\}\). The equality \(T_t(E(x))=E(x)\) implies that

$$\begin{aligned} A(E(x)y) =\lim _{t \rightarrow 0} \frac{T_t(E(x)y)-E(x)y}{t} =E(x)A(y) \,. \end{aligned}$$
(6.2)

It is easy to see that all terms containing \(A(a_i^*)\), \(A(b_i)\) for \(i\ge k_0\) will cancel out in \(\Gamma (x,y)\) and thus

$$\begin{aligned} 2\Gamma (x,y)&= \sum _{i=1}^{k_0-1} a_m^* ,\ldots , a_{i+1}^*A(a_i^*)a_{i-1} ,\ldots , a_1^*b_1 ,\ldots , b_n\\&\quad + \sum _{i=1}^{k_0-1} a_m^* ,\ldots , a_1^*b_1 ,\ldots , b_{i-1}A(b_i)b_{i+1} ,\ldots , b_n\\&\quad -\, a_m^* ,\ldots , a_{k_0}^*A(a_{k_0-1}^* ,\ldots , a_1^*b_1 ,\ldots , b_{k_0-1})b_{k_0} ,\ldots , b_n\\&= 2a_m^* ,\ldots , a_{k_0}^* \Gamma (a_1 ,\ldots , a_{k_0-1}, b_1 ,\ldots , b_{k_0-1}) b_{k_0} ,\ldots , b_n\,. \end{aligned}$$

Lemma 6.7

Let \(a_i,b_i\in {\mathcal {N}}_{k_i}\) be mean \(0\) elements for \(i=1,\ldots ,r\). Then

$$\begin{aligned} \Gamma (a_1 ,\ldots , a_r, b_1 ,\ldots , b_r)&= a_{r}^*\Gamma (a_1 ,\ldots , a_{r-1}, b_1 ,\ldots , b_{r-1})b_r\\&\quad +\, \Gamma (a_r,E(a_{r-1}^* ,\ldots , a_1^*b_1 ,\ldots , b_{r-1})b_r)\,. \end{aligned}$$

Proof

Using the mean 0 decomposition, we have

$$\begin{aligned}&2\Gamma (a_1 ,\ldots , a_r, b_1 ,\ldots , b_r)=A(a_r^*)a_{r-1}^* ,\ldots , b_{r-1}b_r + a_r^*A(a_{r-1}^* ,\ldots , a_1^*)b_1 ,\ldots , b_r \\&\qquad +\,a_r^* ,\ldots , a_1^*A(b_1 ,\ldots , b_{r-1})b_r+a_r^* ,\ldots , b_{r-1}A(b_r)\\&\qquad -\, A(a_r^*)\left( \sum _{i=1}^{r-1} a_{r-1}^* ,\ldots , a_{i+1}^*\mathop {\overbrace{a_i^* E(a_{i-1}^* ,\ldots , a_1^*b_1 ,\ldots , b_{i-1})b_i}}\limits ^{\circ { }}b_{i+1} ,\ldots , b_{r-1} \right) b_r\\&\qquad -\, a_r^*\left( \sum _{i=1}^{r-1} a_{r-1}^* ,\ldots , a_{i+1}^*\mathop {\overbrace{a_i^* E(a_{i-1}^* ,\ldots , a_1^*b_1 ,\ldots , b_{i-1})b_i}}\limits ^{\circ { }}b_{i+1} ,\ldots , b_{r-1} \right) A(b_r)\\&\qquad -\,a_r^*A(a_{r-1}^* ,\ldots , a_1^*b_1 ,\ldots , b_{r-1})b_r-A(a_r^*E(a_{r-1}^* ,\ldots , b_{r-1})b_r)\\&\quad =2a_r^*\Gamma (a_1 ,\ldots , a_{r-1}, b_1 ,\ldots , b_{r-1})b_r + A(a_r^*)E(a_{r-1}^* ,\ldots , b_{r-1})b_r\\&\qquad +\,a_r^*E(a_{r-1}^* ,\ldots , b_{r-1})A(b_r)-A(a_r^*E(a_{r-1}^* ,\ldots , b_{r-1})b_r) \end{aligned}$$

which completes the proof with the help of (6.2). \(\square \)

The recursion formula immediately yields that

Lemma 6.8

Let \(k_0\) be as above. Then

$$\begin{aligned} \Gamma (x,y) =\sum _{k=1}^{k_0-1} \Gamma ^{(k)}[x,y]\,. \end{aligned}$$

where

$$\begin{aligned} \Gamma ^{(k)}[x,y] =a_{m}^* ,\ldots , a_{k+1}^*\Gamma [a_k,E(a_{k-1}^* ,\ldots , a_1^*b_1 ,\ldots , b_{k-1})b_k]b_{k+1} ,\ldots , b_n \,. \end{aligned}$$

In order to calculate \(\Gamma _2\), we have to analyze

$$\begin{aligned} \Gamma ^{(k)}[A(x),y]+\Gamma ^{(k)}[x,A(y)]-A\Gamma ^{(k)}[x,y] \,. \end{aligned}$$

Observe that for \(j<k\) all terms containing \(A(a_j^*)\) or \(A(b_j)\) appear inside the conditional expectation \(E\) in \(\Gamma ^{(k)}[A(x),y]+\Gamma ^{(k)}[x,A(y)]\) and there is no counterpart in \(A\Gamma ^{(k)}\). Hence we find

$$\begin{aligned} I^{(k)}(x,y)&= a_{m}^* ,\ldots , a_{k+1}^*\Gamma [a_k, E(A(a_{k-1}^* ,\ldots , a_1^*)b_1 ,\ldots , b_{k-1})b_k]b_{k+1} ,\ldots , b_n \\&\quad +\, a_{m}^* ,\ldots , a_{k+1}^*\Gamma [a_k, E(a_{k-1}^* ,\ldots , a_1^*A(b_1 ,\ldots , b_{k-1}))b_k]b_{k+1} ,\ldots , b_n. \end{aligned}$$

For \(k\le j<k_0\) we are left with the following terms

$$\begin{aligned} II^{(k)}(x,y)&= \sum _{j=k}^{k_0-1} a_m^* ,\ldots , A(a_j^*) ,\ldots , a_{k+1}^*\Gamma [a_k,E(a_{k-1}^* ,\ldots ,\\&\quad a_1^*b_1 ,\ldots , b_{k-1})b_k]b_{k+1} ,\ldots , b_n\\&\quad +\, \sum _{j=k}^{k_0-1} a_m^* ,\ldots , a_{k+1}^*\Gamma [a_k,E(a_{k-1}^* ,\ldots , a_1^*b_1, \\&\quad \ldots , b_{k-1})b_k]b_{k+1} ,\ldots , A(b_j) ,\ldots , b_n\\&\quad - a_m^* ,\ldots , A(a_{k_0-1}^* ,\ldots , a_{k+1}^*\Gamma [a_k,E(a_{k-1}^* ,\ldots ,\\&\quad a_1^*b_1 ,\ldots , b_{k-1})b_k]b_{k+1} ,\ldots , b_{k_0-1}) ,\ldots , b_n \end{aligned}$$

where we understand, when \(j=k\), that \(A(a_k)\) and \(A(b_k)\) are inside the \(\Gamma \)-form. Since

$$\begin{aligned} \Gamma [a_k,E(a_{k-1}^* ,\ldots , a_1^*b_1 ,\ldots , b_{k-1})b_k]\in {\mathcal {N}}_{i_k}, \end{aligned}$$

we are in the situation of Lemma 6.7. The recursion formula gives that

$$\begin{aligned} II^{(k)}(x,y)&= \sum _{j=k+1}^{k_0-1} F_{jk}(x,y)\\&\quad +\, 2a_{m}^* ,\ldots , a_{k+1}^*\Gamma _2[a_k,E(a_{k-1}^* ,\ldots , a_1^*b_1 ,\ldots , b_{k-1})b_k]b_{k+1} ,\ldots , b_n \end{aligned}$$

where

$$\begin{aligned} F_{jk}(x,y)&= a_{m}^* ,\ldots , a_{j+1}^* \Gamma [a_j,E(a_{j-1}^* ,\ldots , \Gamma [a_k,\\&\quad E(a_{k-1}^* ,\ldots , a_1^*b_1 ,\ldots , b_{k-1})b_k] ,\ldots , b_{j-1})b_j]b_{j+1} ,\ldots , b_n\,. \end{aligned}$$

Therefore we find \(\Gamma _2\).

Lemma 6.9

Using the above notation, we have

$$\begin{aligned} 2\Gamma _2(x,y)&= \sum _{k=1}^{k_0-1} \Gamma ^{(k)}[A(x),y]+\Gamma ^{(k)}[x,A(y)]-A\Gamma ^{(k)}[x,y] \\&= \sum _{k=1}^{k_0-1}I^{(k)}(x,y)+II^{(k)}(x,y) \end{aligned}$$

In order to show \(\Gamma _2\ge \alpha \Gamma \), we need a technical lemma which is an application of the Hilbert \(W^*\)-module theory; see [39, 46].

Lemma 6.10

Let \(\Phi : {\mathcal {A}}\times {\mathcal {A}}\rightarrow {\mathcal {N}}\) be a sesquilinear form where \({\mathcal {A}}\) is a separable *-algebra contained in the domain of \(\Phi \) and \({\mathcal {N}}\) is a von Neumann algebra. Then \(\Phi \) is a positive form if and only if there exists a map \(v:{\mathcal {A}}\rightarrow C({\mathcal {N}})\) such that \(\Phi (x,y)=v(x)^*v(y)\) for \(x,y\in {\mathcal {A}}\) where \(C({\mathcal {N}})=\ell _2 \otimes {\mathcal {N}}\) denotes the Hilbert \({\mathcal {N}}\)-module, or the column space of \({\mathcal {N}}\).

Proof

The sufficiency is obvious. Conversely, following the KSGNS construction [39, 46], we consider the algebraic tensor product \({\mathcal {A}}\otimes {\mathcal {N}}\) and define \(\langle \sum _i x_i\otimes a_i, \sum _j y_j\otimes b_j\rangle = \sum _{i,j}a_i^*\Phi (x_i,y_j)b_j\) for \(x_i,y_j\in {\mathcal {A}}\) and \(a_i,b_j\in {\mathcal {N}}\). Set \({\mathcal {K}}= \{x\in A\otimes {\mathcal {N}}: \langle x,x\rangle = 0\}\). Then \({\mathcal {A}}\otimes {\mathcal {N}}/{\mathcal {K}}\) is a pre-Hilbert \({\mathcal {N}}\)-module with \({\mathcal {N}}\)-valued inner product \(\langle x+{\mathcal {K}}, y+{\mathcal {K}}\rangle =\langle x, y\rangle \) for \(x,y\in {\mathcal {A}}\otimes {\mathcal {N}}\). Let \({\mathcal {A}}\otimes _\Phi {\mathcal {N}}\) be the completion of \({\mathcal {A}}\otimes {\mathcal {N}}/{\mathcal {K}}\). Then \({\mathcal {A}}\otimes _\Phi {\mathcal {N}}\) is a Hilbert \({\mathcal {N}}\)-module. Since \({\mathcal {A}}\) is separable, \({\mathcal {A}}\otimes _\Phi {\mathcal {N}}\) is countably generated. It follows from [39, Theorem 6.2] that there exists a right module map \(u: {\mathcal {A}}\otimes _\Phi {\mathcal {N}}\rightarrow \ell _2\otimes {\mathcal {N}}\) such that

$$\begin{aligned} \sum _{i,j}a_i^*\Phi (x_i,y_j)b_j =\left\langle u\left( \sum _i x_i\otimes a_i+{\mathcal {K}}\right) \!, u\left( \sum _j y_j\otimes b_j+{\mathcal {K}}\right) \right\rangle \,. \end{aligned}$$

In particular, \(\Phi (x,y)= u(x\otimes 1+{\mathcal {K}})^*u(y\otimes 1+{\mathcal {K}})\). Define \(v(x)=u(x\otimes 1+{\mathcal {K}})\). This is the desired map. \(\square \)

Remark 6.11

Since we only consider finitely many elements for the sake of positivity of a form in the following, the separability assumption on \({\mathcal {A}}\) in the previous lemma is automatically satisfied. If we want to remove separability, we can use the fact that every Hilbert right module over \({\mathcal {N}}\) embeds isometrically in a self-dual module. Indeed, this follows from [46]. Let us sketch the approach from [33]. Consider the \(L_{1/2}({\mathcal {N}})\) module

$$\begin{aligned} Y= X \otimes _{\mathcal {N}}L_1({\mathcal {N}}). \end{aligned}$$

Then the antilinear dual \(Y^*\) is self-dual and obviously contains \(X\) isometrically.

By an argument similar to (6.2), we have \(A^{1/2}(zx)=zA^{1/2}(x)\) for \(z\in {\mathcal {D}}\) and \(x\in {\mathcal {N}}\). Then for \(x,y\in {\mathcal {N}}\),

$$\begin{aligned} \tau (zE(A(x)y))=\tau (E(A(zx)y))=\tau (A^{1/2}(zx)A^{1/2}(y))=\tau (zE(A^{1/2}(x)A^{1/2}(y))). \end{aligned}$$

Hence, \(E(A(x)y)=E(A^{1/2}(x)A^{1/2}(y))\) and we find

$$\begin{aligned}&I^{(k)}(x,y)\\&\quad =2 a_{m}^* ,\ldots , \Gamma [a_k, E(A^{1/2}(a_{k-1}^* ,\ldots , a_1^*)A^{1/2}(b_1 ,\ldots , b_{k-1}))b_k]b_{k+1} ,\ldots , b_n. \end{aligned}$$

We claim that this is a positive form. Indeed, \(I^{(k)}\) is nontrivial only if \(a_k\) and \(b_k\) are in the same \({\mathcal {N}}_{i_k}\). Using Lemma 6.10 with \(\Phi =\Gamma \), we find \(\beta _k:{\mathcal {N}}_{i_k}\rightarrow C({\mathcal {N}}_{i_k})\) such that \(\Gamma (a,b)=\beta _k(a)^*\beta _k(b)\) for \(a,b\in {\mathcal {N}}_{i_k}\). Similarly with \(\Phi (x,y)=E(x^*y)\), we find \(v_k:{\mathcal {N}}\rightarrow C({\mathcal {D}})\) such that \(E(x^*y)=v_k(x)^*v_k(y)\) for \(x,y\in {\mathcal {N}}\). Define

$$\begin{aligned} u_k(b_1 ,\ldots , b_n)=e_{i_1, ,\ldots , ,i_k}\otimes (\beta _k\otimes Id)(v_k(A^{1/2}(b_1 ,\ldots , b_{k-1}))b_k)b_{k+1} ,\ldots , b_n\,. \end{aligned}$$

Note that by the module property (6.2) \(\Gamma (z^*a,b)=\Gamma (a,zb)\) for \(z\in {\mathcal {D}}, x,y\in {\mathcal {N}}_{i_k}\). Write \(v_k(x)=(v_k^l(x))_l\) where \(v_k^l(x)\in {\mathcal {D}}\). It follows that

$$\begin{aligned} I^{(k)}(x,y)&= 2 a_m^* ,\ldots , a_{k+1}^*\Gamma (a_k,~\sum _l v_k^l[A^{1/2}(a_1 ,\ldots , a_{k-1})]^*v_k^l\\&\quad \times \, [A^{1/2}(b_1 ,\ldots , b_{k-1})] b_k) b_{k+1} ,\ldots , b_n\\&= 2 \sum _l a_m^* ,\ldots , a_{k+1}^*\Gamma (v_k^l[A^{1/2}(a_1 ,\ldots , a_{k-1})]a_k, ~v_k^l\\&\quad \times \, [A^{1/2}(b_1 ,\ldots , b_{k-1})] b_k)b_{k+1} ,\ldots , b_n\\&= 2\sum _l a_m^* ,\ldots , a_{k+1}^*\beta _k(v_k^l[A^{1/2}(a_1 ,\ldots , a_{k-1})]a_k)^*\beta _k\\&\quad \times \,(v_k^l[A^{1/2}(b_1 ,\ldots , b_{k-1})]b_k)b_{k+1} ,\ldots , b_n\\&= 2u_k(a_1 ,\ldots , a_m)^*u_k(b_1 ,\ldots , b_n)\,. \end{aligned}$$

By Lemma 6.10, \(I^{(k)}\) is a positive form.

Now we claim that \(F_{jk}\) are positive forms for \(j=k+1,\ldots , k_0-1\). Indeed, define

$$\begin{aligned} u_{jk}(b_1 ,\ldots , b_n)&= e_{i_{k+1},\ldots ,i_j}\otimes (\beta _j\otimes Id)((v_j\otimes Id )[e_{i_1,\ldots ,i_k}\\&\quad \otimes (\beta _k\otimes Id)[v_k(b_1 ,\ldots , b_{k-1})b_k]b_{k+1} ,\ldots , b_{j-1}]b_j)b_{j+1} ,\ldots , b_n. \end{aligned}$$

Then similar to the argument for \(I^{(k)}\), we find \(F_{jk}(x,y)=u_{jk}(a_1 ,\ldots , a_m)^*u_{jk}(b_1 ,\ldots , b_n)\). By Lemma 6.10, \(F_{jk}\) is a positive form. Hence, we find

$$\begin{aligned} II^{(k)}(x,y)\ge 2a_{m}^* ,\ldots , a_{k+1}^*\Gamma _2[a_k,E(a_{k-1}^* ,\ldots , a_1^*b_1 ,\ldots , b_{k-1})b_k]b_{k+1} ,\ldots , b_n\,\!. \end{aligned}$$

Therefore we deduce the main result

Proposition 6.12

Let \(A_j\) be self-adjoint generators of standard nc-diffusion semigroups \((T_t^{A_j})\) and \(\Gamma _{A_j}^2\ge \alpha \Gamma _{A_j}\) respectively for \(j=1,\ldots ,n\) with the same constant \(\alpha >0\). Then the free product generator \(*A_j(a_1 ,\ldots , a_n)=\sum _{j} a_1 ,\ldots , a_{j-1}A(a_j) a_{j+1} ,\ldots , a_n\) generates a standard nc-diffusion semigroup \((T_t^{*A_j})\) with

$$\begin{aligned} \Gamma ^2_{*A_j}\ge \alpha \Gamma _{*A_j} \,\!. \end{aligned}$$

Example 6.13

The free product of all the examples considered so far satisfies the \(\Gamma _2\)-criterion. In particular, the free product of matrix algebra \(*_i M_n\) admits a standard nc-diffusion semigroup with the \(\Gamma _2\)-criterion.

Example 6.14

(Block length function) Consider the free product of groups \(G_i\), \(G=*_{i\in I} G_i\) with the block length function \(\psi \), i.e.

$$\begin{aligned} \psi (g_1^{k_1} ,\ldots , g_n^{k_n})=n \end{aligned}$$

for \(g_1\in G_{i_1},\ldots ,g_n\in G_{i_n}\), \(i_1\ne i_2\ne \cdots \ne i_n\) and \(k_i\in {\mathbb {Z}}\). Fix \(i\) and denote by \(\lambda \) the left regular representation of \(G_i\). Define the conditional expectation \(E: {\mathcal {L}}(G_i)\rightarrow {\mathbb {C}}1\) to be

$$\begin{aligned} E(\lambda (g))=\tau (\lambda (g))1={\left\{ \begin{array}{ll} 1,&{} \text{ if } g=e,\\ 0, &{} \text{ if } g\ne e. \end{array}\right. } \end{aligned}$$

Here \(e\) is the identity element of \(G_i\) and \(1\) is the identity element of \({\mathcal {L}}(G_i)\). Example 4.11 says that \(T_t \lambda (g)=e^{-t(I-E)}\lambda (g)\) is a standard nc-diffusion semigroup with \(\Gamma _2\ge \frac{1}{2}\Gamma \) where

$$\begin{aligned} T_t\lambda (g)={\left\{ \begin{array}{ll} \lambda (g),&{} \text{ if } g = e,\\ e^{-t} \lambda (g),&{} \text{ if } g\ne e. \end{array}\right. } \end{aligned}$$

Since \({\mathcal {L}}(G)=*_{i\in I}{\mathcal {L}}(G_i)\), using Proposition 6.12 and the relation \(\lambda (g_1 ,\ldots , g_n)=\lambda (g_1) ,\ldots , \lambda (g_n)\) for \(g_1\in G_{i_1},\ldots ,g_n\in G_{i_n}\) and \(i_1\ne i_2\ne \cdots \ne i_n\), we deduce that \((T_t^b)\) is a standard nc-diffusion semigroup acting on \({\mathcal {L}}(G)\) with \(\Gamma _2\ge \frac{1}{2} \Gamma \) where

$$\begin{aligned} T_t^b(\lambda (g_1^{k_1} ,\ldots , g_n^{k_n})) = e^{-tn}\lambda (g_1^{k_1} ,\ldots , g_n^{k_n}). \end{aligned}$$

Clearly, the infinitesimal generator of \(T_t^b\) is the block length function. In particular, for \(G_i={\mathbb {Z}}\) we find a standard nc-diffusion semigroup acting on \({\mathcal {L}}({\mathbb {F}}_n)\) which is different from the one considered in Sect. 5.1. In fact, our result applies even for free product of groups with amalgamation in general.

7 The classical diffusion processes

We consider classical diffusion semigroups in this section. As explained in Theorem 4.9, we have stronger results in the this setting thanks to the better constant in the commutative BDG inequality.

7.1 Ornstein–Uhlenbeck process in \({\mathbb {R}}^d\)

Let us start with Ornstein-Uhlenbeck process whose infinitesimal generator is \(-A=\Delta -x\cdot \nabla \) in \({\mathbb {R}}^d\). We refer the readers to e.g. [40] for the facts we state in this subsection. Let \(T_t=e^{-tA}\) be the semigroup generated by \(A\) and \(\gamma \) denote the canonical Gaussian measure on \({\mathbb {R}}^d\) with density \((2\pi )^{-d/2}e^{-|x|^2/2}\). It is well known that \(\gamma \) is an invariant measure of \(T_t\) and

$$\begin{aligned} T_tf(x)=\int _{{\mathbb {R}}^d}f(e^{-t}x+(1-e^{-2t})^{1/2}y)d\gamma (y). \end{aligned}$$

Let \({\mathcal {A}}=C_c^\infty ({\mathbb {R}}^d)\), the compactly supported smooth functions. Clearly \({\mathcal {A}}\) is weakly dense in \({\mathcal {N}}=L_\infty ({\mathbb {R}}^d,\gamma )\) and \(T_t\) is self-adjoint with respect to \(\gamma \). Clearly \({\mathcal {A}}\) is dense in \(\mathrm{Dom }(A^{1/2})\) in the graph norm. Note that \(\Gamma (f,f)=|\nabla f|^2\) and that for \(f\in \mathrm{Dom }(A^{1/2})\)

$$\begin{aligned} \Vert \Gamma (f,f)\Vert _1=\langle A^{1/2}f,A^{1/2} f\rangle _{L_2({\mathbb {R}}^d,\gamma )}\,. \end{aligned}$$

Therefore \((T_t)\) is a standard nc-diffusion semigroup satisfying the assumptions in Lemma 4.6. It is easy to check that

$$\begin{aligned} \Gamma _2(f,f)=|\nabla f|^2+\Vert \mathrm{Hess }f\Vert _{HS}^2\ge \Gamma (f,f),\quad f\in C_c^\infty ({\mathbb {R}}^d). \end{aligned}$$

Here \(\mathrm{Hess }f\) denotes the Hessian of \(f\) and \(\Vert \cdot \Vert _{HS}\) denotes the Hilbert–Schmidt norm. Note that \(Af=0\) only if \(f\) is a constant. Thus the fixed point algebra is trivial. Theorem 4.4 with \(\alpha =1\) and Theorem 4.9 immediately lead to the following result.

Corollary 7.1

Let \(2\le p<\infty \). Then there exist a constant \(C\) such that for all real valued functions \(f\in W^{1,p}({\mathbb {R}}^d,\gamma )\)

$$\begin{aligned} \Big \Vert f-\int f d\gamma \Big \Vert _{L_p({\mathbb {R}}^d,\gamma )}\le C \sqrt{p} \Vert \,|\nabla f|\,\Vert _{L_p({\mathbb {R}}^d,\gamma )} \end{aligned}$$
(7.1)

where \(W^{1,p}({\mathbb {R}}^d,\gamma )\) denotes the Sobolev space consisting of all \(L_p({\mathbb {R}}^d,\gamma )\) functions with first order weak derivatives also in \(L_p({\mathbb {R}}^d,\gamma )\).

This result can be generalized to infinite dimension. Let \((W,H,\mu )\) be an abstract Wiener space and \(L\) the Ornstein–Uhlenbeck operator on \(W\). Then it can be checked that the gradient form associated with \(L\) satisfies

$$\begin{aligned} \Gamma _2(F,F)=(\nabla F, \nabla F)+\Vert \nabla ^2 F\Vert ^2_{HS}\ge \Gamma (F,F) \end{aligned}$$

for \(F(w)\in \mathrm{Cylin}(W)\), the cylindrical functions on \(W\). Based on standard facts from Malliavin calculus, an argument similar to the \({\mathbb {R}}^d\) case shows that the Ornstein–Uhlenbeck semigroup \(T_t\) is a standard nc-diffusion semigroup satisfying the assumptions in Lemma 4.6. Moreover, the fixed point algebra \(\mathrm{Fix}\) is trivial. See [21, 44] for more details. Hence our Poincaré type inequality (7.1) holds in this setting.

7.2 Diffusion processes on Riemannian manifolds

Consider an elliptic differential operator \(-A\) on a connected smooth manifold \(M\) of dimension \(d\) with invariant probability measure \(\mu \) on Borel sets which is equivalent to Lebesgue measure. We can write it in a local coordinate chart as

$$\begin{aligned} -Af(x)=\sum _{i,j}g^{ij}\frac{\partial ^2f}{\partial x^i\partial x^j}(x) +\sum _ib^i \frac{\partial f}{\partial x^i}(x) \end{aligned}$$

where \(g^{ij}\) and \(b^i\) are smooth functions and \((g^{ij})\) is a nonnegative definite matrix. The inverse of \((g^{ij})\) then defines a Riemannian metric. It can be checked that

$$\begin{aligned} \Gamma (f,h)=\sum _{ij} g^{ij}\frac{\partial f}{\partial x^i}\frac{\partial h}{\partial x^j} \end{aligned}$$

for all \(f,h\in C_c^\infty (M)\). To give an example, we take \(-A=\Delta + Z\) where \(\Delta \) is the Laplace–Beltrami operator on a stochastically complete Riemannian manifold and \(Z\) is a \(C^1\)-vector field on a Riemannian manifold \(M\) such that

$$\begin{aligned} \mathrm{Ric}(X,X)-\langle \nabla _X Z, X\rangle \ge \alpha |X|^2, \quad X\in TM \end{aligned}$$
(7.2)

for some \(\alpha >0\). By the Bochner identity, this inequality is equivalent to (see e.g. [63])

$$\begin{aligned} \Gamma _2(f,f)\ge \alpha \Gamma (f,f), \quad f\in C^\infty (M). \end{aligned}$$

Take \({\mathcal {A}}=C_c^\infty (M)\) and \(T_t=e^{-tA}\). The following result follows from Theorem 4.9 and the martingale problem on differentiable manifolds [25].

Corollary 7.2

Assume (7.2) and the following conditions

(1):

\(\int T_t(f) g d\mu =\int f T_t (g) d\mu \) (i.e. \(T_t\) is symmetric);

(2):

\(|\nabla f|\in L_2(M, \mu )\) whenever \(\langle A^{1/2} f,A^{1/2} f\rangle <\infty \).

Then for all \(2\le p<\infty \) and real valued functions \(f\in W^{1,p}(M,\mu )\),

$$\begin{aligned} \Vert f-E_\mathrm{Fix}f\Vert _{L_p(M,\mu )}\le C \sqrt{p/\alpha } \Vert ~ |\nabla f|~ \Vert _{L_p(M,\mu )} \end{aligned}$$
(7.3)

where \(W^{1,p}(M,\mu )\) is the Sobolev space on the Riemannian manifold \(M\).

Remark 7.3

Since compactly supported smooth functions are dense in the graph norm of \(A^{1/2}\), the work of Cipriani and Sauvageot [12] shows that Condition (2) is automatically satisfied.

Functional inequalities related to diffusion processes on Riemannian manifolds have been studied extensively; see [64] for more details on this subject. To give an even more concrete example, let \(\nu \) be the normalized volume measure and \(\mu (dx)=e^{-V(x)}\nu (dx)\) a probability measure for \(V\in C^2(M)\). Suppose (7.2) holds. It is clear that the semigroup \(T_t\) with generator \(-A=\Delta -\nabla V\cdot \nabla \) fulfills the assumptions of Corollary 7.2 and the fixed point algebra is trivial. It follows that

$$\begin{aligned} \Bigg \Vert f-\int f d\mu \Bigg \Vert _{L_p(M,\mu )}\le C \sqrt{p/\alpha } \Vert ~ |\nabla f|~ \Vert _{L_p(M,\mu )}\,. \end{aligned}$$

This improves X.-D. Li’s result [41, Theorem 1.2, Theorem 5.2] for \(p\ge 2\) which was proved by using his sharp estimate of the \(L_p\)-norm of Riesz transform. Indeed, his Poincaré inequality has constant \(p/\sqrt{\alpha }\).

Remark 7.4

(7.3) is true only for scalar-valued functions. If one is interested in some noncommutative objects, e.g., matrix-valued functions on manifolds or free product of manifolds, one has to apply the noncommutative theory and then the Poincaré inequalities are in the form of Theorem 4.4. Of course, the deviation and the transportation inequalities still hold in all those situations.