KMS Dirichlet forms, coercivity and superbounded Markovian semigroups on von Neumann algebras

Abstract

We introduce a construction of Dirichlet forms on von Neumann algebras M associated to any eigenvalue of the Araki modular Hamiltonian of a faithful normal non-tracial state, providing also conditions by which the associated Markovian semigroups are GNS symmetric. The structure of these Dirichlet forms is described in terms of spatial derivations. Coercivity bounds are proved and the spectral growth is derived. We introduce a regularizing property of positivity preserving semigroups (superboundedness) stronger than hypercontractivity, in terms of the symmetric embedding of M into its standard space $L^2(M)$ and the associated noncommutative $L^p(M)$ spaces. We prove superboundedness for a special class of positivity preserving semigroups and that some of them are dominated by the Markovian semigroups associated to the Dirichlet forms introduced above, for type I factors M. These tools are applied to a general construction of the quantum Ornstein–Uhlembeck semigroups of the Canonical Commutation Relations CCR and some of their non-perturbative deformations.

1 Introduction and description of the results

The structure of completely Dirichlet forms with respect to lower semicontinuous, faithful traces on von Neumann algebras is well understood in terms of closable derivations taking values in Hilbert bimodules (see [14] and the recent [38, 39]). However, for applications to Quantum Statistical Mechanics (see [4,5,6,7, 10, 11, 22, 23, 25, 27,28,29,30, 40]) and Quantum Probability (see [12, 26]) or to deal with general Compact Quantum Groups, is unavoidable to consider quadratic forms which are Markovian with respect to non-tracial states or weights. Concerning the structure of Dirichlet forms of GNS-symmetric Markovian semigroups, one is invited to consult the recent [38, 39].

In QSM, for example, the relevant states one wishes to consider are the KMS equilibria of time evolution automorphisms which are non-tracial at finite temperature. In the CQGs situation, on the other hand, the Haar state is a trace only for the special subclass of CQGs of Kac type. In several most studied CQGs the Haar state is not a tracial state, as for examples for the special unitary CQGs $SU_q(N)$. In this framework a detailed understanding has been found for the completely Dirichlet forms generating translation invariant completely Markovian semigroups of Levy quantum stochastic processes. The construction relies on the Schürmann cocycle associated to the generating functional of the process (see [13]).

On the other hand, a general construction of completely Dirichlet forms on the standard form of a $\sigma$-finite von Neumann algebra with respect to a faithful, normal state in the sense of [8, 9, 16,17,18], has been introduced in [23, 24, 35] and by Y.M. Park and his school (see [4, 5, 28,29,30]) with applications to QSM of bosons and fermions system and their quasi-free states. In this approach the Dirichlet forms depend upon the explicitly knowledge of the modular automorphisms group of the state.

In this work we formulate a general and natural construction of a completely Dirichlet form, Markovian with respect to a fixed normal, faithful state $\omega _0$, associated to each non zero and not necessarily discrete eigenvalue of the Araki modular Hamiltonian $\ln \Delta _0$. Hence, by superposition, one has a malleable tool to construct completely Dirichlet forms and completely Markovian, modular symmetric, semigroups starting from the spectrum of the modular operator $\Delta _0$ or its associated Araki modular Hamiltonian $\ln \Delta _0$. Compared to Park’s approach, this has the advantage to avoid the explicit use of the modular automorphism group. The present method generalizes the construction of bounded Dirichlet form of [8] Proposition 5.3 and that of unbounded Dirichlet forms of [8] Proposition 5.4, removing the assumption of self-adjointness and affiliation to the centralizer for the coefficients.

The framework of the construction is that of Dirichlet forms and Markovian semigroups on standard forms $(M, L^2(M), L^2_+(M), J)$ of von Neumann algebras M as in [8] and related modular theory [1, 2, 6, 34, 36, 37]. In particular, we associate in Sect. 2, a one-parameter family of unbounded, J-real, non negative, densely defined, closed quadratic forms $({\mathcal {E}}_Y^\lambda ,{\mathcal {F}}_Y^\lambda )$ on $L^2(M)$ satisfying the first Beurling–Deny condition to each densely defined, closed operator (Y, D(Y)) affiliated to M, thus generating $C_0$-continuous, contractive semigroups on $L^2(M)$ which are positivity preserving (in the sense that they leave globally invariant the positive self-polar cone $L^2_+(M)$). Moreover, the quadratic form $({\mathcal {E}}_Y^\lambda ,{\mathcal {F}}_Y^\lambda )$ is Markovian with respect to the cyclic vector $\xi _0\in L^2(M)_+$ representing $\omega _0$, in the strong sense that ${\mathcal {E}}_Y^\lambda [\xi _0]=0$, if and only if $\xi _0$ lies in the domain both of Y and its adjoint $Y^*$ and $\xi :=Y\xi _0$ is an eigenvector of the modular operator $\Delta _0$ associated to the non zero eigenvalue $\lambda >0$. This construction applies, in particular, to any eigenvector $\xi$ of any non zero eigenvalue of $\Delta _0$.

Further, we investigate the fact that, by definitions, each $({\mathcal {E}}_Y^\lambda ,{\mathcal {F}}_Y^\lambda )$ is the quadratic form of an M-bimodule derivations $(d_Y^\lambda , D(d_Y^\lambda ))$ on the standard bimodule $L^2(M)$. In particular we show that in the Markovian case both $({\mathcal {E}}_Y^\lambda ,{\mathcal {F}}_Y^\lambda )$ and $(d_Y^\lambda , D(d_Y^\lambda ))$ are represented by the symmetric embedding on $L^2(M)$ of the unbounded, spatial derivations $\delta _Y:=i[Y,\cdot ]$ on M provided by the operator (Y, D(Y)) affiliated to M.

In the subsequent Sect. 3, we prove natural lower bounds for the Dirichlet form $({\mathcal {E}}_Y^\lambda ,{\mathcal {F}}_Y^\lambda )$ in terms of the quadratic forms of the affiliated operators $Y^*Y$, $YY^*$, $[Y,Y^*]$ and derive implications on the lower boundedness and discreteness of spectrum of $({\mathcal {E}}_Y^\lambda ,{\mathcal {F}}_Y^\lambda )$.

By the general theory, using the symmetric embeddings of the von Neumann algebra M into the standard Hilbert space $L^2(M)$ and the embedding of $L^2(M)$ into the predual space $M_*=L^1(M)$, provided by the modular theory of the state $\omega$, completely Markovian semigroups $T_t$ on $L^2(M)$ extend to completely (Markovian) contractive semigroups on M and on $L^1(M)$ (weak$^*$-continuous in the former case and strongly continuous in the latter one).

In Sect. 4, we introduce an extra regularity property of positivity preserving semigroups called superboundedness as the boundedness of $T_t$ from $L^2(M)$ to M for all $t>t_0$ and some $t_0\ge 0$. In case $t_0=0$ we call this property ultraboundedness. We prove that superboundedness holds true with respect to a finite temperature Gibbs state $\omega (\cdot )=\mathrm{Tr\,}(\cdot e^{-\beta _0 H_0})/\textrm{Tr }(e^{-\beta _0 H_0})$ on a type I$_{\infty }$ factor M, for the semigroup generated by the generalized sum $H_0\dot{+}JH_0J$ and that the property is stable with respect to domination of positivity preserving semigroups.

In Sect. 5 we apply the framework above to investigate a class of Dirichlet forms associated on a type I$_{\infty }$ factor which are Markovian with respect to a Gibbs state of the Number Operator of a representation of the CCR algebra. The construction fully generalizes that of Quantum Ornstein–Uhlenbeck semigroups introduced in [12]. In particular we prove the subexponential spectral growth rate of the generator and the domination of the Markovian semigroup with respect to the semigroup generated by $H_0\dot{+}JH_0J$ (this special class of semigroups is discussed in Appendix 7.1).

In Sect. 6 we apply the tools developed in the previous sections to construct Dirichlet forms associated with dynamics generated by deformations of the Number Operator.

In Appendix we represent the generators of a class of positivity preserving semigroups as generalized sums and we clarify superboundedness for abelian von Neumann algebras.

2 Dirichlet forms and derivations on von Neumann algebras standard forms

Let $(M,L^2(M),L^2_+(M),J)$ be a standard form of a $\sigma$-finite von Neumann algebra (for this subject and the related modular theory we refer to [6, 7, 36, 37]).

Let $\omega _0$ be the faithful normal state on M represented by the cyclic vector $\xi _0\in L^2_+(M)$ as

$$\begin{aligned} \omega _0(x)=\left( \xi _0|x\xi _0\right) _{L^2(M)}\qquad x\in M. \end{aligned}$$

The anti-linear, densely defined operator on $L^2(M)$ defined on the left Hilbert algebra by

$$\begin{aligned} M\xi _0\ni x\xi _0\mapsto x^*\xi _0\qquad x\in M, \end{aligned}$$

is closable. Its closure $S_0$ has a polar decomposition $S_0=J\Delta _0^{1/2}$ where the anti-unitary part J is called the modular conjugation and $\Delta _0:=S_0^*S_0$ is a densely defined, self-adjoint, positive operator on $L^2(M)$, called the modular operator of $\omega _0$, defining the modular automorphism group of M by $\sigma ^{\omega _0}_t(x):=\Delta _0^{it}x\Delta _0^{-it}$ for $x\in M$ and $t\in {\mathbb {R}}$. On the w$^*$-dense, involutive, sub-algebra of its analytic elements $M_0\subseteq M$, the modular group can be extended to any $t\in {\mathbb {C}}$. For any $x,y\in M_0$ and $z,w\in \mathbb {C}$, this extension satisfies

$$\begin{aligned} \sigma ^{\omega _0}_z(xy)=\sigma ^{\omega _0}_z(x)\sigma ^{\omega _0}_z(y),\quad \sigma ^{\omega _0}_{z+w}(x)= \sigma ^{\omega _0}_z(\sigma ^{\omega _0}_w(x)),\quad \left( \sigma ^{\omega _0}_z(x)\right) ^*=\sigma ^{\omega _0}_{{\bar{z}}}(x^*). \end{aligned}$$

We will make use of the symmetric embedding of M into its standard Hilbert space $L^2(M)$:

$$\begin{aligned} i_0:M\rightarrow L^2(M)\qquad i_0(x):=\Delta _0^{1/4}x\xi _0. \end{aligned}$$

Among its properties we recall that it is weak$^*$-continuous, injective with dense range and positivity preserving in the sense that $i_0(x)\in L^2(M)_+$ if and only if $x\in M_+$. Also it maps the closed and convex set of all $x\in M_+$ such that $0\le x\le 1$ onto the closed and convex set of all $\xi \in L^2_+(M)$ such that $0\le \xi \le \xi _0$. The projection of a J-real vector $\xi =J\xi \in L^2(M)$ onto the closed, convex set $\xi _0-L^2_+(M)$ wil be denoted by $\xi \wedge \xi _0$.

A Dirichlet form [8] Definition 4.8 with respect to $(M,\omega _0)$ is a lower bounded and lower semicontinuous quadratic form

$$\begin{aligned} {\mathcal {E}}:L^2(M)\rightarrow (-\infty ,+\infty ], \end{aligned}$$

with domain ${\mathcal {F}}:=\{\xi \in L^2(M): {\mathcal {E}}[\xi ]<+\infty \}$, satisfying the properties

• (i) ${\mathcal {F}}$ is dense in $L^2(M)$,

• (ii) ${\mathcal {E}}[J\xi ]={\mathcal {E}}[\xi ]$ for all $\xi \in L^2(M)$ (reality),

• (iii) ${\mathcal {E}}[\xi \wedge \xi _0]\le {\mathcal {E}}[\xi ]$ for all $\xi =J\xi \in L^2(M)$, (Markovianity).

$({\mathcal {E}},{\mathcal {F}})$ is said to be a completely Dirichlet form if its ampliation on the algebra

$(M\otimes M_n({\mathbb {C}}),\omega _0\otimes \textrm{tr}_n)$ defined by

$$\begin{aligned} {\mathcal {E}}^n:L^2(M\otimes M_n({\mathbb {C}}),\omega _0\otimes \textrm{tr}_n)\rightarrow [0,+\infty ]\qquad {\mathcal {E}}^n\left[ [\xi _{i,j}]_{i,j=1}^n\right] :=\sum _{i,j=1}^n{\mathcal {E}}[\xi _{i,j}] \end{aligned}$$

is a Dirichlet form for all $n\ge 1$ ($\textrm{tr}_n$ denotes the tracial state on the matrix algebra $M_n({\mathbb {C}})$).

A $C_0$-continuous, self-adjoint semigroup $\{T_t:t\ge 0\}$ on $L^2(M)$ is called

• (i) positivity preserving if $T_t\xi \in L^2_+(M)$ for all $\xi \in L^2_+(M)$ and $t\ge 0$;

• (ii) Markovian with respect to $\omega _0$ if it is positivity preserving and for $\xi =J\xi \in L^2(M)$

  $$\begin{aligned} 0\le \xi \le \xi _0\quad \implies \quad 0\le T_t \xi \le \xi _0\qquad t\ge 0; \end{aligned}$$

• (iii) completely positive (resp. Markovian) if the extensions $T^n_t:=T_t\otimes I_n$ to $L^2(M\otimes M_n({\mathbb {C}}),\omega _0\otimes \textrm{tr}_n)$ are positivity preserving (resp. Markovian) semigroups for all $n\ge 1$. In [8] Definition 2.8, property ii) above, Markovianity, was indicated as sub-Markovianity. As a result of the general theory, Dirichlet forms are automatically nonnegative and Markovian semigroups are automatically contractive see [8] Proposition 4.10 and Theorem 4.11. Dirichlet forms $({\mathcal {E}},{\mathcal {F}})$ are in one-to-one correspondence with Markovian semigroups $\{T_t:t\ge 0\}$: the self-adjoint, positive operator (H, D(H)) associated to $({\mathcal {E}},{\mathcal {F}})$ by ${\mathcal {E}}[\xi ]=\Vert \sqrt{H}\xi \Vert ^2_{L^2(M)}$ for all $\xi \in {\mathcal {F}}$, being the semigroup generator $T_t=e^{-tH}$, $t\ge 0$. $C_0$-continuous, self-adjoint, positivity preserving semigroups are in one-to-one correspondence with nonnegative, densely defined, real, lower semicontinuous quadratic forms satisfying the following first Beurling–Deny condition (weaker than Markovianity)

  $$\begin{aligned} \xi =J\xi \in {\mathcal {F}}\quad \Rightarrow \quad \xi _\pm \in {\mathcal {F}}\qquad \text {and}\qquad {\mathcal {E}}(\xi _+|\xi _-)\le 0, \end{aligned}$$

  equivalently stated (see [8] Proposition 4.5 and Theorem 4.7]) as

  $$\begin{aligned} \xi =J\xi \in {\mathcal {F}}\quad \Rightarrow \quad |\xi |\in {\mathcal {F}}\qquad \text {and}\qquad {\mathcal {E}}[|\xi |]\le {\mathcal {E}}[\xi ], \end{aligned}$$

  On the other hand, the first Beurling–Deny condition and the conservativeness condition

  $$\begin{aligned} \xi _0\in {\mathcal {F}},\qquad {\mathcal {E}}[\xi _0]=0 \end{aligned}$$

  together imply the Markovianity of closed forms $({\mathcal {E}},{\mathcal {F}})$ (see [8] Lemma 2.9 and Theorem 4.11).

2.1 Dirichlet forms associated to eigenvalues of the modular operators

The forthcoming construction of Dirichlet forms is based on the following well known fact (see [6] Proposition 2.5.9, [34, 37] page 19; see also [2] where von Neumann algebras with states having the logarithmic of the modular operators with spectrum consisting only of isolated eigenvalues are characterized).

We recall that a densely defined, closed operator (Y, D(Y)) on $L^2(M)$ is affiliated to M if for any $z'\in M'$ and any $\xi \in D(Y)$ one has $z'D(Y)\subseteq D(Y)$ and $Y(z'\xi )=z'(Y\xi )$ or, equivalently, if and only if its graph ${\mathcal {G}}(Y)\subset L^2(M)\oplus L^2(M)$ is left globally invariant $(z'\oplus z'){\mathcal {G}}(Y)\subseteq {\mathcal {G}}(Y)$ under the action of $z'\oplus z'\in M'\oplus M'$, for any $z'\in M'$ (see [36]).

For any operator (Y, D(Y)) affiliated to M, the operator $j(Y):=JYJ$ is affiliated to $M'$.

Lemma 2.1

For any $\xi \in D(S_0)=D(\Delta _0^{1/2})$ there exists a densely defined, closed operator (Y, D(Y)) affiliated to M such that

• (i) $\xi _0\in D(Y)\cap D(Y^*)$,

• (ii) $\xi =Y\xi _0$ and $S_0(\xi )=Y^*\xi _0$. iii) Among the operators (Y, D(Y)) with the properties i) and ii) above, there exists a minimal one $({\overline{Y_0}}, D({\overline{Y_0}}))$ obtained as the closure of the closable operator $(Y_0,D(Y_0))$ defined by

  $$\begin{aligned} D(Y_0):=M'\xi _0,\qquad Y_0(y'\xi _0):=y'\xi . \end{aligned}$$

Proof

The operator $(Y_0,D(Y_0))$ is affiliated to M because the action of any $w'\in M'$ leaves globally invariant the domain $M'\xi _0$ and $w'Y_0(y'\xi _0)=w'y'\xi =Y_0(w'y'\xi _0)$ for any $y'\in M'$. The operator $(Y_0,D(Y_0))$ is closable because it is in duality with the densely defined operator $Z_0:M'\xi _0\rightarrow L^2(M)$ given by $Z_0(z'\xi _0):=z'S_0\xi$ in the sense that

$$\begin{aligned} \begin{aligned} \left( z'\xi _0|Y_0\left( y'\xi _0\right) \right)&=\left( z'\xi _0|y'\xi \right) =\left( y'^*z'\xi _0|\xi \right) =\left( J\Delta _0^{-1/2}z'^*y'\xi _0|\xi \right) \\&=\left( J\xi |\Delta _0^{-1/2}z'^*y'\xi _0\right) =\left( z'J\Delta _0^{1/2}\xi |y'\xi _0\right) =\left( z'S_0\xi |y'\xi _0\right) \\&=\left( Z_0\left( z'\xi _0\right) |y'\xi _0\right) . \end{aligned} \end{aligned}$$

Clearly by definition $Y_0\xi _0=\xi$ and the calculation above implies $Y_0^*\xi _0=S_0\xi$. If (Y, D(Y)) is a closed operator affiliated to M with properties (i) and (ii) above, then, as $\xi _0\in D(Y)$, we have $y'\xi _0\in D(Y)$ for all $y'\in M'$ so that $M'\xi _0\subseteq D(Y)$ and $Y(y'\xi _0)=y'Y\xi _0=y'\xi =Y_0(y'\xi _0)$, which shows that (Y, D(Y)) is a closed extension of $({\overline{Y_0}}, D({\overline{Y_0}}))$. $\square$

This representation will be applied below to eigenvectors $\xi$ (if any) of the modular operator.

Lemma 2.2

Let (Y, D(Y)) be a densely defined, closed operator affiliated to M and $\mu ,\nu \ge 0$. Then defining $d_Y^{\mu ,\nu }:D(d_Y^{\mu ,\nu })\rightarrow L^2(M)$ as

$$\begin{aligned} d_Y^{\mu ,\nu }:=i\left( \mu Y-\nu j\left( Y^*\right) \right) \qquad D\left( d_Y^{\mu ,\nu }\right) :=D(Y)\cap JD\left( Y^*\right) , \end{aligned}$$

it results that $(d_Y^{\mu ,\nu },D(d_Y^{\mu ,\nu }))$ is a densely defined, closable operator on $L^2(M)$.

Proof

Since $J^2=I$, we have $D(j(Y^*))=JD(Y^*)$ so that $d_Y^{\mu ,\nu }$ is well defined on $D(d_Y^{\mu ,\nu })$. By hypotheses, $j(Y^*)$ is densely defined, closed and affiliated to the commutant von Neumann algebra $M'$. Hence Y and $j(Y^*)$ strongly commute and the contraction semigroup $e^{-t|Y|}\circ e^{-t|j(Y^*)|}=e^{-t|j(Y^*)|}\circ e^{-t|Y|}$ with parameter $t\ge 0$ strongly converges to the identity operator on $L^2(M)$ as $t\rightarrow 0^+$. Since $d_Y^{\mu ,\nu }\circ e^{-t|Y|}\circ e^{-t|j(Y^*)|}=i(\mu Y\circ e^{-t|Y|}\circ e^{-t|j(Y^*)|}-\nu e^{-t|Y|} \circ j(Y^*)\circ e^{-t|j(Y^*)|})$ is a bounded operator for any $t>0$, we have that $D(d_Y^{\mu ,\nu })$ is dense in $L^2(M)$.

To prove the statement concerning closability, observe that reasoning as above with $Y^*$ in place of Y and $Y^{**}=Y$ in place of $Y^*$, we have that $\mu Y^*-\nu j(Y)$ is densely defined on $D(Y^*)\cap JD(Y)$. Moreover, since

$$\begin{aligned} \begin{aligned} \left( d_Y^{\mu ,\nu }\eta |\zeta \right)&=\left( i\left( \mu Y-\nu j\left( Y^*\right) \right) \eta |\zeta \right) \\&=-i\mu \left( Y\eta |\zeta \right) +i\nu \left( j\left( Y^*\right) \eta |\zeta \right) \qquad \eta \in D\left( d_Y^{\mu ,\nu }\right) :=D(Y)\cap JD\left( Y^*\right) \\&=-i\mu \left( \eta |Y^*\zeta \right) +i\nu \left( \eta | j(Y)\zeta \right) \qquad \zeta \in D\left( Y^*\right) \cap JD(Y)\\&=\left( \eta |-i\left( \mu Y^*-\nu j(Y)\right) \zeta \right) ,\\ \end{aligned} \end{aligned}$$

the adjoint of $(d_Y^{\mu ,\nu }, D(d_Y^{\mu ,\nu }))$ is an extension of $(-i(\mu Y^*-\nu j(Y)), D(Y^*)\cap JD(Y))$. It is thus densely defined and consequently $(d_Y^{\mu ,\nu }, D(d_Y^{\mu ,\nu }))$ is closable. $\square$

Lemma 2.3

Let (Y, D(Y)) be a densely defined, closed operator affiliated to M. Then the J-real part of the domain D(Y) is invariant under the modulus map:

$$\begin{aligned} \xi \in D(Y)\,,\quad J\xi =\xi \qquad \Rightarrow \qquad |\xi |\in D(Y) \end{aligned}$$

and $\Vert Y|\xi |\Vert =\Vert Y\xi \Vert$. In particular, if $\xi =\xi _+-\xi _-$ is the polar decomposition of a J-real vector $\xi =J\xi \in D(Y)$, then $\xi _\pm =(|\xi |\pm \xi )/2\in D(Y)$ and

$$\begin{aligned} \Vert Y\xi _\pm \Vert \le \Vert Y\xi \Vert \,. \end{aligned}$$

Proof

Consider first the case where Y is bounded, and let $s'_\pm \in M'$ be the supports in $M'$ of the positive and negative parts $\xi _\pm$ of a J-real $\xi \in L^2(M)$. Then $(Y\xi _+|Y\xi _-)=(Ys'_+\xi _+|Ys'_-\xi _-)=(s'_+Y\xi _+|s'_-Y\xi _-)=0$ since $\xi _+\perp \xi _-$ imply $s'_+s'_-=0$, by [1] Theorem 4. Thus

$$\begin{aligned} \begin{aligned} \Vert Y\xi \Vert ^2&= \left( Y\xi |Y\xi \right) =\left( Y\xi _+-Y\xi _-|Y\xi _+-Y\xi _-\right) \\&=\left( Y\xi _++Y\xi _-|Y\xi _++Y\xi _-\right) =\Vert Y|\xi |\Vert ^2. \end{aligned} \end{aligned}$$

To deal with the general case, fix $\xi =J\xi \in D(Y)=D(|Y|)$ and consider the family of bounded operators $|Y|_\varepsilon :=|Y|(I+\varepsilon |Y|)^{-1}\in M$ for $\varepsilon >0$ as well as the spectral measure $E^{|Y|}$ of the self-adjoint operator |Y|. Applying the result concerning the bounded case, for all $\varepsilon >0$ we have

$$\begin{aligned} \int _0^{+\infty } E^{|Y|}_{|\xi |,|\xi |}(\textrm{d}\lambda )\frac{\lambda ^2}{\left( 1+\varepsilon \lambda \right) ^2}=\Vert |Y|_\varepsilon |\xi |\Vert ^2=\Vert |Y|_\varepsilon \xi \Vert ^2 =\int _0^{+\infty } E^{|Y|}_{\xi ,\xi }(\textrm{d}\lambda )\frac{\lambda ^2}{\left( 1+\varepsilon \lambda \right) ^2}. \end{aligned}$$

Letting $\varepsilon \downarrow 0$, by the Monotone Convergence Theorem we have $|\xi |\in D(|Y|)=D(Y)$ and $\Vert Y|\xi |\Vert =\Vert Y\xi \Vert$. $\square$

Lemma 2.4

Let (Y, D(Y)) be a densely defined, closed operator affiliated to M and $\mu ,\nu \ge 0$. Let $\xi \in D(d_Y^{\mu ,\nu }):=D(Y)\cap JD(Y^*)$ be a J-real vector with polar decomposition $\xi =\xi _+-\xi _-$. Then $\xi _\pm \in D(Y)\cap D(Y^*)$ and

$$\begin{aligned} \left( Y\xi _+|j\left( Y^*\right) \xi _-\right) \ge 0. \end{aligned}$$

Proof

If $Y\in M$ the assertion is true because in that case $Y^*j(Y^*)$ is positivity preserving. To deal with the general case, let $Y=U|Y|$ be the polar decomposition of Y. By the previous Lemma 2.3, since $\xi =J\xi \in D(Y)$, we have $\xi _\pm \in D(Y)=D(|Y|)$. Since also $Y^*$ is a densely defined, closed operator affiliated to M and, by assumption, $\xi =J\xi \in JJD(Y^*)=D(Y^*)$, again by Lemma 2.3 we have $\xi _-\in D(Y^*)$ too so that $(Y\xi _+|j(Y^*)\xi _-)=(|Y|\xi _+|j(|Y|)U^*j(U^*)\xi _-)$ (here we implicitly used the fact that $U^*j(U^*)\xi _-\in D(|Y|)$ since $\xi _-\in D(|Y|)$ by the previous Lemma 2.3, $U\in M$ and j(|Y|) is affiliated to the commutant algebra $M'$). Since $U\in M$ so that $U^*j(U^*)$ is positivity preserving, it is enough to prove that $(|Y|\eta |j(|Y|)\eta ')\ge 0$ for all $\eta \,, \eta '\in D(|Y|)\cap L^2_+(M)$. Since |Y| and j(|Y|) strongly commute, we can represent the value $(|Y|\eta |j(|Y|)\eta ')$ as an integral over the product of the spectral measures of the two operators

$$\begin{aligned} \left( |Y|\eta |j(|Y|)\eta '\right) =\int _{[0,+\infty )^2}\left( E^{|Y|}_\eta \times E^{j(|Y|)}_{\eta '}\right) \left( \textrm{d}\lambda ,\textrm{d}\lambda '\right) \lambda \cdot \lambda '. \end{aligned}$$

Setting $f_\varepsilon (\lambda ):=\lambda /(1+\varepsilon \lambda )$ we have $0\le f_\varepsilon (\lambda )\le \lambda$ and $\lim _{\varepsilon \rightarrow 0}f_\varepsilon (\lambda )=\lambda$. By the Dominated Convergence Theorem we have

$$\begin{aligned} \begin{aligned} \left( |Y|\eta |j(|Y|)\eta '\right)&=\int _{[0,+\infty )^2}\left( E^{|Y|}_\eta \times E^{j(|Y|)}_{\eta '}\right) \left( \textrm{d}\lambda ,\textrm{d}\lambda '\right) \lambda \cdot \lambda ' \\&=\lim _{\varepsilon \rightarrow 0} \int _{[0,+\infty )^2}\left( E^{|Y|}_\eta \times E^{j(|Y|)}_{\eta '}\right) \left( d\lambda ,d\lambda '\right) f_\varepsilon (\lambda )\cdot f_\varepsilon (\lambda ') \\&=\lim _{\varepsilon \rightarrow 0} \left( f_\varepsilon (|Y|)\eta |j\left( f_\varepsilon (|Y|)\right) \eta '\right) \\&=\lim _{\varepsilon \rightarrow 0} \left( \eta |f_\varepsilon (|Y|)j\left( f_\varepsilon (|Y|)\right) \eta '\right) \ge 0 \end{aligned} \end{aligned}$$

since $f_\varepsilon (|Y|)\in M$ and $f_\varepsilon (|Y|)j(f_\varepsilon (|Y|))$ is positivity preserving. $\square$

Theorem 2.5

Let (Y, D(Y)) be a densely defined, closed operator affiliated to M such that $\xi _0\in D(Y)\cap D(Y^*)$ and $\mu ,\nu >0$. Then the quadratic form ${\tilde{{\mathcal {E}}}}_Y^{\mu ,\nu }:{\tilde{{\mathcal {F}}}}_Y\rightarrow [0,+\infty )$ on $L^2(M)$

$$\begin{aligned} {\tilde{{\mathcal {E}}}}_Y^{\mu ,\nu }[\xi ]:=\left\| d_Y^{\mu ,\nu }\xi \right\| ^2_{L^2(M)} + \left\| d_{Y^*}^{\nu ,\mu }\xi \right\| ^2_{L^2(M)}\qquad {\tilde{{\mathcal {F}}}}_Y:=D\left( d_Y^{\mu ,\nu }\right) \cap D\left( d_{Y^*}^{\nu ,\mu }\right) \end{aligned}$$

is densely defined, closable, J-real (recall that $\Vert d_{Y^*}^{\nu ,\mu }\xi \Vert ^2_{L^2(M)}=\Vert d_{Y}^{\mu ,\nu }J\xi \Vert ^2_{L^2(M)}$) and satisfies the first Beurling–Deny condition

$$\begin{aligned} \xi =J\xi \in {\tilde{{\mathcal {F}}}}_Y\qquad \Rightarrow \qquad \xi _\pm \in {\tilde{{\mathcal {F}}}}_Y\qquad \text {and}\qquad {\tilde{{\mathcal {E}}}}_Y^{\mu ,\nu }\left( \xi _+|\xi _-\right) \le 0. \end{aligned}$$

Its closure $({\mathcal {E}}_Y^{\mu ,\nu },{\mathcal {F}}_Y^{\mu ,\nu })$ satisfies the first Beurling–Deny condition too and generates a contractive, positivity preserving semigroup $\{T_t:t\ge 0\}$. Moreover, $({\mathcal {E}}_Y^{\mu ,\nu },{\mathcal {F}}_Y^{\mu ,\nu })$ is a conservative, in the sense that

$$\begin{aligned} \xi _0\in {\mathcal {F}}_Y^{\mu ,\nu },\qquad {\mathcal {E}}_Y^{\mu ,\nu }[\xi _0]=0, \end{aligned}$$

completely Dirichlet form with respect to $(M,\omega _0)$ and the associated completely Markovian semigroup is conservative, in the sense that

$$\begin{aligned} T_t\xi _0=\xi _0\qquad t\ge 0, \end{aligned}$$

if and only if $Y\xi _0\in L^2(M)$ is an eigenvector of the modular operator corresponding to the eigenvalue $\mu /\nu$

$$\begin{aligned} Y\xi _0\in D\left( \Delta _0^{1/2}\right) ,\qquad \Delta _0^{1/2}Y\xi _0=(\mu /\nu )Y\xi _0. \end{aligned}$$

Proof

Since ${\tilde{{\mathcal {F}}}}_Y=D(Y)\cap JD(Y^*)\cap D(Y^*)\cap JD(Y)=D(Y)\cap D(Y^*)\cap J(D(Y)\cap D(Y^*))$, we have $J{\tilde{{\mathcal {F}}}}_Y={\tilde{{\mathcal {F}}}}_Y$ and $\xi _0=J\xi _0\in {\tilde{{\mathcal {F}}}}_Y$. Since $d_Y^{\mu ,\nu }J\xi =i(\mu Y-\nu JY^*J)J\xi =i(\mu YJ\xi -\nu JY^*\xi )=iJ(\mu JYJ\xi -\nu Y^*\xi )=Ji(\nu Y^*\xi -\mu JYJ\xi )=Jd_{Y^*}^{\nu ,\mu }\xi$ for all $\xi \in {\tilde{{\mathcal {F}}}}_Y$ and, exchanging the role of Y and $Y^*$, we have $d_{Y^*}^{\nu ,\mu }J\xi =Jd_Y^{\mu ,\nu }\xi$ too, we get

$$\begin{aligned} {\tilde{{\mathcal {E}}}}_Y^{\mu ,\nu }[J\xi ]= & {} \left\| d_Y^{\mu ,\nu }J\xi \right\| ^2 + \left\| d_{Y^*}^{\nu ,\mu }J\xi \right\| ^2=\left\| Jd_{Y^*}^{\nu ,\mu }\xi \right\| ^2 + \left\| Jd_Y^{\mu ,\nu }\xi \right\| ^2\\ {}= & {} {\tilde{{\mathcal {E}}}}_Y^{\mu ,\nu }[\xi ]\ \xi \in {\tilde{{\mathcal {F}}}}_Y, \end{aligned}$$

which proves that the quadratic form $({\tilde{{\mathcal {E}}}}_Y^{\mu ,\nu },{\tilde{{\mathcal {F}}}}_Y)$ is J-real.

Consider now a J-real vector $\xi \in {\tilde{{\mathcal {F}}}}_Y$, its polar decomposition $\xi =\xi _+-\xi _-$ with respect to the self-polar cone $L^2_+(M)$ and recall that, by definition, $|\xi |:=\xi _+ + \xi _-$. By the previous lemma, $|\xi |\in {\tilde{{\mathcal {F}}}}_Y$ so that $\xi _\pm =(|\xi |\pm \xi )/2\in {\tilde{{\mathcal {F}}}}_Y$. Then, if $s_\pm \in M$ (resp. $s'_\pm \in M'$) are the supports of $\xi _\pm$ in M (resp. $M'$), we have

$$\begin{aligned} \begin{aligned} \left( d_Y^{\mu ,\nu }\xi _+|d_Y^{\mu ,\nu }\xi _-\right)&=\left( (\mu Y\xi _+-\nu j(Y^*)\xi _+|(\mu Y\xi _- -\nu j(Y^*)\xi _-) \right. \\&\left. =\mu ^2(Ys'_+\xi _+| Ys'_-\xi _-) +\nu ^2(j(Y^*)s_+\xi _+|j(Y^*)s_-\xi _-) \right. \\&\left. \quad -\mu \nu \Bigl ((Y\xi _+|j(Y^*)\xi _-) + (j(Y^*)\xi _+|Y\xi _-)\right) \\&=-\mu \nu \Bigl ((Y\xi _+|j(Y^*)\xi _-) + (j(Y^*)\xi _+|Y\xi _-)\Bigr )\le 0 \end{aligned} \end{aligned}$$

by Lemma 2.4. Since, analogously, $(d_{Y^*}^{\nu ,\mu }\xi _+|d_{Y^*}^{\nu ,\mu }\xi _-)\le 0$ we have ${\tilde{{\mathcal {E}}}}_Y^{\mu ,\nu }(\xi _+|\xi _-)\le 0$ and consequently the first Beurling–Deny condition is satisfied by $({\tilde{{\mathcal {E}}}}_Y^{\mu ,\nu },{\tilde{{\mathcal {F}}}}_Y)$

$$\begin{aligned} \xi =J\xi \in {\tilde{{\mathcal {F}}}}_Y\qquad \Rightarrow \qquad |\xi |\in {\tilde{{\mathcal {F}}}}_Y\qquad \text {and}\qquad {\tilde{{\mathcal {E}}}}_Y^{\mu ,\nu }[|\xi |]\le {\tilde{{\mathcal {E}}}}_Y^{\mu ,\nu }[\xi ]. \end{aligned}$$

To establish the same condition for the closure $({\mathcal {E}}_Y^{\mu ,\nu },{\mathcal {F}}_Y^{\mu ,\nu })$, we adapt the proof of [8] Proposition 5.1 (according the suggestions which there precede it).

On one hand, since J is an isometry for the graph norm of $({\tilde{{\mathcal {E}}}}_Y^{\mu ,\nu },{\tilde{{\mathcal {F}}}}_Y)$ and ${\tilde{{\mathcal {F}}}}_Y$ is a core for $({\mathcal {E}}_Y^{\mu ,\nu },{\mathcal {F}}_Y^{\mu ,\nu })$,then J is an isometry for the latter form too and the closure form is J-real.

On the other hand, let $\xi =J\xi \in {\mathcal {F}}_Y^{\mu ,\nu }$ be a fixed J-real vector and let $\xi _n\in {\tilde{{\mathcal {F}}}}_Y$ be a sequence converging to it in the graph norm of $({\mathcal {E}}_Y^{\mu ,\nu },{\mathcal {F}}_Y^{\mu ,\nu })$. Since the Hilbert projection $\eta \mapsto \eta _+$ of the J-real part of $L^2(M)$ onto the closed, convex cone $L^2_+(M)$ is norm continuous and $|\eta |=2\eta _+-\eta$, it follows that the modulus map $\eta \mapsto |\eta |$ is norm continuous too. Then, since the form ${\mathcal {E}}_Y^{\mu ,\nu }$ is norm lower semicontinuous on $L^2(M)$, it follows that

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_Y^{\mu ,\nu }[|\xi |]&\le \liminf _n{\mathcal {E}}_Y^{\mu ,\nu }[|\xi _n|]=\liminf _n{\tilde{{\mathcal {E}}}}_Y^{\mu ,\nu }[|\xi _n|] \le \liminf _n{\tilde{{\mathcal {E}}}}_Y^{\mu ,\nu }[\xi _n] \\&=\liminf _n{\mathcal {E}}_Y^{\mu ,\nu }[\xi _n]={\mathcal {E}}_Y^{\mu ,\nu }[\xi ]. \end{aligned} \end{aligned}$$

The first Beurling–Deny condition is thus verified and, by [8] Proposition 4.10, it follows that the semigroup $\{T_t:t\ge 0\}$ has the desired properties.

Concerning the conservativeness property, notice that $\xi _0\in {\tilde{{\mathcal {F}}}}_Y\subseteq {\mathcal {F}}_Y^{\mu ,\nu }$. If ${\mathcal {E}}_Y^{\mu ,\nu }[\xi _0]=0$ then ${\tilde{{\mathcal {E}}}}_Y[\xi _0]=0$ so that $d_Y^{\mu ,\nu }\xi _0=0$ which implies $\mu Y\xi _0=\nu j(Y^*)\xi _0=\nu J Y^*\xi _0$ and, for any $x\in M$, $\mu (x\xi _0|Y\xi _0)=\nu (x\xi _0|J Y^*\xi _0)=\nu (Y^*\xi _0|J x\xi _0)=\nu (Y^*\xi _0|J xJ\xi _0)=\nu (J x^*J\xi _0|Y\xi _0)=\nu (J x^*\xi _0|Y\xi _0)=\nu (\Delta _0^{1/2}x\xi _0|Y\xi _0)$ since Y is affiliated to M and $Jx^*J\in M'$. Setting $\lambda ^2:=\mu /\nu$, this in turn implies $((\Delta _0^{1/2}-\lambda ^2 I)x\xi _0|Y\xi _0)=0$ for all $x\in M$ and, since $M\xi _0$ is a core for $\Delta _0^{1/2}$, it follows that $Y\xi _0\in D(\Delta _0^{1/2})$ and $\Delta _0^{1/2} Y\xi _0=(\mu /\nu ) Y\xi _0$, i.e. $Y\xi _0$ is an eigenvalue of the modular operator with eigenvalue $\mu /\nu$.

On the other hand, if $Y\xi _0\in D(\Delta _0^{1/2})$ and $\Delta _0^{1/2}Y\xi _0=(\mu /\nu )Y\xi _0$, using the identities above, it follows that $d_Y^{\mu ,\nu }\xi _0=0$ and $d_{Y^*}^{\nu ,\mu }\xi _0=d_Y^{\mu ,\nu }J\xi _0=d_Y^{\mu ,\nu }\xi _0=0$ so that ${\mathcal {E}}_Y^{\mu ,\nu }\xi _0={\tilde{{\mathcal {E}}}}_Y[\xi _0]=0$, i.e. $({\mathcal {E}}_Y^{\mu ,\nu },{\mathcal {F}}_Y^{\mu ,\nu })$ is conservative. By [8] Proposition 4.10, given conservativness, the first Beurling–Deny property and Markovianity are equivalent for quadratic forms as well as the positivity preserving property and the Markovianity are equivalent for the associated semigroups.

Concerning the complete Markovianity of the Dirichlet form, we notice that for any $n\ge 1$, the ampliation $({\mathcal {E}}^\lambda _Y)^n:L^2(M\otimes M_n({\mathbb {C}}),\omega _0\otimes \textrm{tr}_n)\rightarrow [0,+\infty ]$, defined as $({\mathcal {E}}^\lambda _Y)^n[[\xi _{i,j}]_{i,j=1}^n]:=\sum _{i,j=1}^n{\mathcal {E}}[\xi _{i,j}]$, has the same structure as ${\mathcal {E}}^\lambda _Y$. More precisely, a closed operator $Y^n:=Y\otimes I_n$ is densely defined on $D(Y^n):=D(Y)\otimes _{\textrm{alg}}L^2(M_n({\mathbb {C}}),tr_n)\subset L^2(M\otimes M_n({\mathbb {C}}),\omega _0\otimes tr_n)$ and one may check that $({\mathcal {E}}^\lambda _Y)^n={\mathcal {E}}^\lambda _{Y^n}$. If $\xi _0\in D(Y)\cap D(Y^*)$ and $Y\xi _0\in L^2(M)$ is an eigenvector of the modular operator $\Delta _0^{1/2}$ of the state $\omega _0$ on M, corresponding to the eigenvalue $\mu /\nu$, then, denoting by $\zeta _n\in L^2(M_n({\mathbb {C}}),tr_n)$ the unit vector representing the trace state, it easily verified that $\xi _0\otimes \zeta _n\in D(Y^n)\cap D((Y^n)^*)$ and that $Y^n(\xi _0\otimes \zeta _n)=Y\xi _0\otimes \zeta _n$ is an eigenvalue of the modular operator of the state $\omega _0\otimes tr_n$ on $M\otimes M_n({\mathbb {C}})$, corresponding to the same eigenvalue $\mu /\nu$. Applying the results obtained above to the form $({\mathcal {E}}^\lambda _Y)^n$ in place of ${\mathcal {E}}^\lambda _Y$, we get its Markovianity for any $n\ge 1$ and complete Markovianity of the associated semigroup. $\square$

Notation. If $\lambda ^2\in \textrm{Sp}(\Delta _0^{1/2}){\setminus }\{0\}$ is a strictly positive eigenvalue of the modular operator and $\mu /\nu =\lambda ^2$, then $d_Y^{\mu ,\nu }=\sqrt{\mu \nu } d_Y^{\lambda ,\lambda ^{-1}}$, $d_{Y^*}^{\nu ,\mu }=\sqrt{\mu \nu } d_{Y^*}^{\lambda ^{-1},\lambda }$ and ${\mathcal {E}}_Y^{\mu ,\nu }=\mu \nu \cdot {\mathcal {E}}_Y^{\lambda ,\lambda ^{-1}}$. Since now on, we will adopt the simplified notation

$$\begin{aligned} {\mathcal {E}}_Y^\lambda :={\mathcal {E}}_Y^{\lambda ,\lambda ^{-1}}. \end{aligned}$$

Remark. To any eigenvector $\xi \in D(S_0)$ of $\Delta _0^{1/2}$ or, equivalently, of the Araki Hamiltonian $\ln \Delta _0$, we associate a completely Dirichlet form ${\mathcal {E}}^\lambda _Y$ choosing a densely defined, closed operator (Y, D(Y)) as in Lemma 2.1. For this choice there exists a canonical candidate, namely $({\overline{Y_0}}, D({\overline{Y_0}}))$. In general $({\mathcal {E}}^\lambda _Y, {\mathcal {F}}^\lambda _Y)$ may depend upon the operator (Y, D(Y)) and not only upon the eigenvector $\xi =Y\xi _0$ it represents. The next result shows how this is connected to the GNS symmetry of the Markovian semigroup.

Theorem 2.6

(GNS symmetry) Let (Y, D(Y)) be a densely defined, closed operator affiliated to M, $\mu ,\nu >0$ such that $\xi _0\in D(Y)\cap D(Y^*)$ and $Y\xi _0\in L^2(M)$ is an eigenvector of $(\Delta _0^{1/2},D(\Delta _0^{1/2}))$ for the eigenvalue $\lambda ^2:=\mu /\nu$. Then, for any $t\in {\mathbb {R}}$,

• (i) the densely defined, closed operator $(Y_t,D(Y_t)):=(\Delta _0^{it}Y\Delta _0^{-it},\Delta _0^{it}D(Y))$, affiliated to M, verifies $\xi _0\in D(Y_t)\cap D(Y_t^*)$, $Y_t\xi _0=\lambda ^{4ti}Y\xi _0\in L^2(M)$ and $Y_t=\lambda ^{4ti}Y$ on the subspace $M'\xi _0$;

• (ii) $({\mathcal {E}}^\lambda _{Y_t},{\mathcal {F}}^\lambda _{Y_t})$ is a Dirichlet form with respect to $(M,\xi _0)$ coinciding with

  $$\begin{aligned} {\mathcal {F}}^\lambda _{Y_t}=\Delta _0^{it}({\mathcal {F}}^\lambda _Y)\qquad {\mathcal {E}}^\lambda _{Y_t}[\eta ]={\mathcal {E}}^\lambda _Y[\Delta _0^{-it}\eta ]. \end{aligned}$$

  If, moreover, $M'\xi _0\subseteq D(Y)$ is a core for (Y, D(Y)), then, for any $t\in {\mathbb {R}}$, we have

• (iii) $(Y_t,D(Y_t))=(\lambda ^{4it}\cdot Y,D(Y))$, for any $t\in {\mathbb {R}}$;

• (iv) $({\mathcal {E}}^\lambda _{Y_t},{\mathcal {F}}^\lambda _{Y_t})=({\mathcal {E}}^\lambda _Y,{\mathcal {F}}^\lambda _Y)$, the associated Markovian semigroup is symmetric

  $$\begin{aligned} (T_t(x\xi _0)|y\xi _0)=(x\xi _0|T_t(y\xi _0))\qquad x,y\in M,\,\, t\ge 0 \end{aligned}$$

  and, in particular, it commutes with $\{\Delta _0^{it}:t\in {\mathbb {R}}\}$;

• v) The semigroup generated by $({\mathcal {E}}^\lambda _{\overline{Y}_0},{\mathcal {F}}^\lambda _{{\overline{Y}}_0})$ is GNS symmetric (notations of Lemma 2.1).

Proof

(i) Since, for any $t\in {\mathbb {R}}$, one has $\Delta _0^{it}\xi _0=\xi _0$, it follows that $\xi _0\in D(Y_t)\cap D(Y_t^*)$, $Y_t\xi _0= \Delta _0^{it}Y\xi _0=\lambda ^{4ti}Y\xi _0\in L^2(M)$ and $Y_t(z'\xi _0)=z'Y_t\xi _0=\lambda ^{4ti}\cdot z'Y\xi _0=\lambda ^{4ti}\cdot Y (z'\xi _0)$ for any $z'\in M'$; (ii) thus $Y_t\xi _0$ is an eigenvector of $(\Delta _0^{1/2},D(\Delta _0^{1/2}))$ for the eigenvalue $\lambda ^2$ and, by Theorem 2.5, $({\mathcal {E}}^\lambda _{Y_t},{\mathcal {F}}^\lambda _{Y_t})$ is a well defined Dirichlet form. The displayed identity follows from the identities $d^{\mu ,\nu }_{Y_t}=\Delta _0^{it}\circ d^{\mu ,\nu }_Y\circ \Delta _0^{-it}$, $d^{\mu ,\nu }_{Y_t^*}=\Delta _0^{it}\circ d^{\mu ,\nu }_{Y^*}\circ \Delta _0^{-it}$, valid, for any $t\in {\mathbb {R}}$, on ${\tilde{{\mathcal {F}}}}_{Y_t}={\tilde{{\mathcal {F}}}}_Y$ and the fact that this space is a form core for $\eta \mapsto {\mathcal {E}}^\lambda _{Y_t}[\eta ]$ and $\eta \mapsto {\mathcal {E}}^\lambda _Y[\Delta _0^{-it}\eta ]$.

(iii) Since the core $M'\xi _0$ for (Y, D(Y)) is invariant under the group $\{\Delta _0^{it}:t\in {\mathbb {R}}\}$, it is a core also for $(Y_t,D(Y_t))$, for any fixed $t\in {\mathbb {R}}$. Since, by i), $Y_t=\lambda ^{4ti}\cdot Y$ on this common core, we have $D(Y_t)=D(Y)$ and $Y_t=\lambda ^{4ti}\cdot Y$, for any $t\in {\mathbb {R}}$; iv) since, by iii), $({\mathcal {E}}^\lambda _{Y_t},{\mathcal {F}}^\lambda _{Y_t})=({\mathcal {E}}^\lambda _Y,{\mathcal {F}}^\lambda _Y)$ for any $t\in {\mathbb {R}}$, ii) implies that $({\mathcal {E}}^\lambda _Y,{\mathcal {F}}^\lambda _Y)$ is invariant under the unitary group $\{\Delta _0^{it}:t\in {\mathbb {R}}\}$ so that the Markovian semigroup it generates commutes with $\{\Delta _0^{it}:t\in {\mathbb {R}}\}$ and it is GNS symmetric by [8] Theorem 6.6; v) follows from iv) as, by definition, $M'\xi _0$ is a core for $(\overline{Y_0},D(\overline{Y_0}))$. $\square$

2.2 Representation of Dirichlet forms as square of commutators

In this section we show how to represent the Dirichlet forms on $L^2(M)$ constructed above, in terms of generalized commutators, i.e. unbounded spatial derivations on M.

We recall that $(S_0,D(S_0))$ is an unbounded conjugation, i.e. anti-linear and idempotent on its domain. Thus $S_0^2$ is the identity operator on $D(S_0)$ or, more explicitly, that $\xi \in D(S_0)$ implies $S_0\xi \in D(S_0)$ and $S_0(S_0\xi )=\xi$. In other terms, the image of $S_0$ coincides with its domain and $S_0=S_0^{-1}$ holds true as an identity between densely defined, closed operators. In terms of the polar decomposition $S_0=J\Delta _0^{1/2}$ we have $J\Delta _0^{1/2}=\Delta _0^{-1/2}J$ as an identity between densely defined, closed operators. This means, in particular, that the modular conjugation exchanges domains as follows $JD(\Delta _0^{1/2})=D(\Delta _0^{-1/2})$, $D(\Delta _0^{1/2})=JD(\Delta _0^{-1/2})$. More in general, one has the intertwining relation ${{\bar{f}}}(\Delta _0^{-1})=Jf(\Delta _0) J$ between closed operators valid for any Borel measurable function $f:[0,+\infty )\rightarrow \mathbb {C}$ (see Introduction to Chapter 10 in [36]). The relation, which is equivalent to $JD({\bar{f}}(\Delta _0^{-1}))=D(f(\Delta _0) )$ and ${\bar{f}}(\Delta _0^{-1})\xi =Jf(\Delta _0) J\xi$ for all $\xi \in D({\bar{f}}(\Delta _0^{-1}))$, will be mostly used for power functions f.

Among its consequences, we will make use of the following:

• (a) for any $\alpha \in {\mathbb {R}}$, the closed operator $J\Delta _0^{\alpha }$ is an unbounded conjugation on its domain $D(\Delta _0^{\alpha })$;

• (b) $S_0=J\Delta _0^{1/2}=\Delta _0^{-1/4}J\Delta _0^{1/4}$ is an identity between densely defined, closed operators: in fact, $D(\Delta _0^{-1/4}J\Delta _0^{1/4}):=\{\xi \in D(\Delta _0^{1/4}):J\Delta _0^{1/4}\xi \in D(\Delta _0^{-1/4})\}$ but since $D(\Delta _0^{-1/4})=JD(\Delta _0^{1/4})$ one has $D(\Delta _0^{-1/4}J\Delta _0^{1/4})=\{\xi \in D(\Delta _0^{1/4}):\Delta _0^{1/4}\xi \in D(\Delta _0^{1/4})\}=D(\Delta _0^{1/2})$ and, for all $\xi \in D(\Delta _0^{1/2})$,

  $$\begin{aligned} \left( \Delta _0^{-1/4}J\Delta _0^{1/4}\right) \xi =\left( \Delta _0^{-1/4}J\right) \Delta _0^{1/4}\xi =\left( J\Delta _0^{1/4}\right) \Delta _0^{1/4}\xi =J\Delta _0^{1/2}\xi ; \end{aligned}$$

• (c) $(J\Delta _0^{1/4},D(\Delta _0^{1/4}))$ is a closed extension of the densely defined operator $(\Delta _0^{1/4}S_0,D(S_0))$: in fact, the latter operator is well defined since $\zeta \in D(S_0)$ implies $S_0\zeta \in D(S_0)=D(\Delta _0^{1/2})\subset D(\Delta _0^{1/4})$ and also $\Delta _0^{1/4}S_0\zeta =\Delta _0^{1/4}S_0^{-1}\zeta =\Delta _0^{1/4}\Delta _0^{-1/2}J\zeta =\Delta _0^{-1/4}J\zeta =J\Delta _0^{1/4}\zeta$;

• (d) $(\Delta _0^{-1/4}J,D(\Delta _0^{1/4}))=(J\Delta _0^{1/4},D(\Delta _0^{1/4}))$ is a closed extension of the densely defined operator $(S_0\Delta _0^{-1/4},D(\Delta _0^{-1/4})\cap D(\Delta _0^{1/4}))$: in fact $D(S_0\Delta _0^{-1/4}):=\{\zeta \in D(\Delta _0^{-1/4}): \Delta _0^{-1/4}\zeta \in D(S_0)\}=\{\zeta \in D(\Delta _0^{-1/4}): \Delta _0^{-1/4}\zeta \in D(\Delta _0^{1/2})\}=D(\Delta _0^{-1/4})\cap D(\Delta _0^{1/4})$ and $J\Delta _0^{1/4}\zeta =J\Delta _0^{1/2}\Delta _0^{-1/4}\zeta =S_0\Delta _0^{-1/4}\zeta$ for all $\zeta \in D(\Delta _0^{-1/4})\cap D(\Delta _0^{1/4})$;

• (e) Let $M_0\subseteq M$ be the involutive w$^*$-dense sub-algebra of analytic vectors of the group $\sigma ^{\omega _0}$. For any $y\in M_0$, the operator $\Delta _0^{1/4}y\Delta _0^{-1/4}$ on $L^2(M)$ is densely defined on $i_0(M_0)$ and closable. Its closure is a bounded operator belonging to M, which coincides with the analytic extension of the map ${\mathbb {R}}\ni t\mapsto \sigma ^{\omega _0}_t(y)\in M_0\subset M$ evaluated at $t=-i/4$

  $$\begin{aligned} \overline{\left( \Delta _0^{1/4}y\Delta _0^{-1/4}\right) }=\sigma ^{\omega _0}_{-i/4}(y)\in M_0\subset M \end{aligned}$$

  and $\sigma ^{\omega _0}_{-i/4}(y)i_0(x)=i_0(yx)$ for all $x\in M_0$;

• (f) by Proposition in Section 9.24 in [36], for any $y\in M_0$ and any $\alpha \in \mathbb {C}$ one has the important identity

  $$\begin{aligned} D\left( \Delta _0^{\alpha }y\Delta _0^{-\alpha }\right) =D\left( \Delta _0^{-\alpha }\right) \end{aligned}$$

  and the boundedness of the operator $\Delta _0^{\alpha }y\Delta _0^{-\alpha }$ on $D(\Delta _0^{-\alpha })$. Since $JD(\Delta _0^{1/2})=D(\Delta _0^{-1/2})$, the case $\alpha =1/2$ implies that $D(S_0yS_0)=D(S_0)$ and the boundedness of the operator $S_0yS_0$ on $D(S_0)$;

• g) the involutive sub-algebra $M_0':=JM_0J\subset M'$ coincides with the set of analytic vectors of the modular group of the commutant $M'$ associated to the state determined by $\xi _0\in L^2(M)$. The left Hilbert sub-algebra $M_0\xi _0\subset M\xi _0\subset L^2(M)$ is dense in $L^2(M)$ and it coincides with the symmetric embedding of the algebra of analytic elements

  $$\begin{aligned} M_0\xi _0=i_0(M_0), \end{aligned}$$

  as it results from the identity $i_0(y)=\sigma ^{\omega _0}_{-i/4}(y)\xi _0$ valid for all $y\in M_0$. Also, $M_0\xi _0$ is J-invariant

  $$\begin{aligned} Ji_0(y)=i_0(y^*)\qquad y\in M_0. \end{aligned}$$

Lemma 2.7

If $\eta \in D(S_0)$, the densely defined operator $({\mathcal {L}}_\eta ,D({\mathcal {L}}_\eta ))$ given by

$$\begin{aligned} D({\mathcal {L}}_\eta ):=i_0(M_0)\ni i_0(y)\qquad {\mathcal {L}}_\eta i_0(y):=J\sigma ^{\omega _0}_{-i/4}(y^*)J\eta \end{aligned}$$

is closable since its adjoint is an extension of the densely defined operator $B:D(B)\rightarrow L^2(M)$

$$\begin{aligned} D(B):=M'\xi _0\ni z'\xi _0\qquad B\left( z'\xi _0\right) :=z'S_0\eta . \end{aligned}$$

The densely defined operator $({\mathcal {R}}_\eta ,D({\mathcal {R}}_\eta ))$ given by

$$\begin{aligned} D({\mathcal {R}}_\eta ):=i_0(M_0)\ni i_0(y)\qquad {\mathcal {R}}_\eta i_0(y):=\sigma ^{\omega _0}_{-i/4}(y)J\eta \end{aligned}$$

satisfies the relation ${\mathcal {R}}_\eta =J{\mathcal {L}}_\eta J$ from which it follows that it is closable too.

Proof

Since $\xi _0=J\xi _0\in D(\Delta _0^{-1/4})=D(\Delta _0^{1/4}y\Delta _0^{-1/4})$, $\Delta _0^{1/4}J=J\Delta _0^{-1/4}$, $y\xi _0\in D(\Delta _0^{1/2})$ and $\Delta _0^{1/2}y\xi _0\in D(\Delta _0^{-1/4})$, we have

$$\begin{aligned} \begin{aligned} J\sigma ^{\omega _0}_{-i/4}(y^*)J\xi _0&=J\left( \Delta _0^{1/4}y^*\Delta _0^{-1/4}\right) \xi _0=J\Delta _0^{1/4}y^*\xi _0 \\&=J\Delta _0^{1/4}J\Delta _0^{1/2}y\xi _0=\Delta _0^{1/4}y\xi _0=i_0(y). \end{aligned} \end{aligned}$$

Since, moreover, $w'^*\xi _0=\Delta _0^{1/2}Jw'\xi _0$ for all $w\in M'$ and $J\sigma ^{\omega _0}_{-i/4}(y^*)J\in M'$ for all $y\in M_0$, for $z'\in M'$ we have

$$\begin{aligned} \begin{aligned} \left( z'\xi _0|{\mathcal {L}}_\eta i_0(y)\right)&=\left( z'\xi _0|J\sigma ^{\omega _0}_{-i/4}(y^*)J\eta \right) =\left( \left( J\sigma ^{\omega _0}_{-i/4}(y^*)J\right) ^*z'\xi _0|\eta \right) \\&=\left( \Delta _0^{1/2}Jz'^*J\sigma ^{\omega _0}_{-i/4}(y^*)J\xi _0|\eta \right) =\left( Jz'^*i_0(y)|\Delta _0^{1/2}\eta \right) \\&=\left( J\Delta _0^{1/2}\eta |z'^*i_0(y)\right) =\left( z'S_0\eta |i_0(y)\right) =\left( B\left( z'\xi _0\right) |i_0(y)\right) . \end{aligned} \end{aligned}$$

The relation between the operators ${\mathcal {L}}_\eta$ and ${\mathcal {R}}_\eta$ follows from the identities $i_0(y^*)=\Delta _0^{1/4}y^*\xi _0=\Delta _0^{1/4}S_0(y\xi _0) =\Delta _0^{1/4}\Delta _0^{-1/2}J(y\xi _0)=\Delta _0^{-1/4}J(y\xi _0)=J\Delta _0^{1/4}y\xi _0=Ji_0(y)$ for all $y\in M_0$ and the fact that J is idempotent and it leaves $D({\mathcal {L}}_\eta )=D({\mathcal {R}}_\eta )=i_0(M_0)$ globally invariant: $J{\mathcal {L}}_\eta Ji_0(y)=J{\mathcal {L}}_\eta i_0(y^*)=JJ\sigma ^\omega _{-i/4}(y)J\eta ={\mathcal {R}}_\eta i_0(y)$. $\square$

Lemma 2.8

Let $\xi \in D(S_0)$ and fix, by Lemma 2.1, a densely defined, closed operator (X, D(X)) affiliated to M such that

$$\begin{aligned} \xi _0\in D(X)\cap D(X^*),\qquad \xi =X\xi _0,\quad S_0(X\xi _0)=X^*\xi _0. \end{aligned}$$

Then the following properties hold true:

• (i) the intersection of domains $D(X)\cap D(X^*)$ contains $M_0\xi _0$;

• (ii) the images of $M_0\xi _0$ under (X, D(X)) and $(X^*,D(X^*))$ are contained in $D(S_0)$

  $$\begin{aligned} X(y\xi _0)\in D(S_0),\quad X^*(y^*\xi _0)\in D(S_0),\qquad \text {for all}\quad y\in M_0 \end{aligned}$$

  and

  $$\begin{aligned} S_0(Xy\xi _0)=y^*X^*\xi _0,\quad S_0(X^*y^*\xi _0)=yX\xi _0\qquad \text {for all}\quad y\in M_0; \end{aligned}$$

  Consider the densely defined operators on $L^2(M)$ given by

  $$\begin{aligned} L_\xi i_0(y):=J(\Delta _0^{1/4}y^*\Delta _0^{-1/4})J\Delta _0^{1/4}\xi \qquad i_0(y)\in i_0(M_0)=:D(L_\xi ), \\ R_\xi i_0(y):=(\Delta _0^{1/4}y\Delta _0^{-1/4})J\Delta _0^{1/4}S_0(\xi ) \qquad i_0(y)\in i_0(M_0)=:D(R_\xi ). \end{aligned}$$

  These are closable, by Lemma 2.7, since $L_\xi ={\mathcal {L}}_\eta$ and $R_\xi ={\mathcal {R}}_\eta$ for $\eta :=\Delta _0^{1/4}\xi \in D(S_0)$ and

• (iii) for any $y\in M_0$ we have

  $$\begin{aligned} Xy\xi _0\in D\left( \Delta _0^{1/4}\right) ,\qquad L_\xi i_0(y)=\Delta _0^{1/4}Xy\xi _0; \end{aligned}$$

• (iv) for any $y\in M_0$ we have

  $$\begin{aligned} yX\xi _0\in D\left( \Delta _0^{1/4}\right) ,\qquad R_\xi i_0(y)=\Delta _0^{1/4}yX\xi _0; \end{aligned}$$

• (v) $\overline{L_\xi }$ is affiliated with M, $\overline{R_\xi }$ is affiliated with $M'$ and $\overline{R_\xi }=J\overline{L_\xi } J$;

• (vi) the operator $\Delta _0^{1/4}X\Delta _0^{-1/4}$ is well defined on $i_0(M_0)$ and there it coincides with $L_\xi$;

• (vii) the operator $J\Delta _0^{1/4}X^*\Delta _0^{-1/4}J$ is well defined on $i_0(M_0)$ and there it coincides with $R_\xi$;

• (viii) If $\xi =X\xi _0$ is an eigenvector of $\Delta _0^{1/2}$ corresponding to the eigenvalue $\lambda ^2>0$, with $\lambda >0$,

  $$\begin{aligned} \Delta _0^{1/2}X\xi _0=\lambda ^2\cdot X\xi _0, \end{aligned}$$

  then $L_\xi =\lambda X$ and $R_\xi =\lambda ^{-1} JX^*J$ on $i_0(M_0)$.

Proof

(i) As X and $X^*$ are affiliated to M and $\xi _0\in D(X)\cap D(X^*)$, it follows that $M'\xi _0\subset D(X)\cap D(X^*)$ and, a fortiori, that $M_0\xi _0=M'_0\xi _0\subset M'\xi _0\subset D(X)\cap D(X^*)$.

(ii) Since $M_0\xi _0$ is a core for $(S_0,D(S_0))$, there exists a sequence $x_n\in M_0$ such that $\Vert x_n\xi _0-X\xi _0\Vert \rightarrow 0$ and $\Vert x_n^*\xi _0-X^*\xi _0\Vert \rightarrow 0$. As mentioned at item f) of the introduction of the present section, since $y\in M_0$, the operator $S_0y^*S_0$ is bounded on $D(S_0)$ and then on $M_0\xi _0\subset D(S_0)$. Thus $x_ny\xi _0\in D(S_0)$ is a Cauchy sequence in $L^2(M)$ as

$$\begin{aligned} \begin{aligned} \Vert x_ny\xi _0-x_my\xi _0\Vert&=\left\| S_0\left( y^*x_n^*\xi _0-y^*x_m^*\xi _0\right) \right\| =\left\| S_0y^*S_0\left( x_n\xi _0-x_m\xi _0\right) \right\| \\&\le \left\| S_0y^*S_0\right\| \cdot \left\| x_n\xi _0-x_m\xi _0\right\| . \end{aligned} \end{aligned}$$

Analogously, $S_0(x_ny\xi _0)=y^*x_n^*\xi _0\in L^2(M)$ is a Cauchy sequence too as

$$\begin{aligned} \left\| S_0\left( x_ny\xi _0\right) -S_0\left( x_my\xi _0\right) \right\| =\left\| y^*x_n^*\xi _0-y^*x_m^*\xi _0\right\| \le \Vert y^*\Vert \cdot \left\| x_n^*\xi _0-x_m^*\xi _0\right\| . \end{aligned}$$

Hence $x_ny\xi _0\in D(S_0)$ is a Cauchy sequence in the graph norm of the closed operator $(S_0,D(S_0))$ and the image of $\eta :=\lim _n x_n y\xi _0\in D(S_0)$ is given by $S_0\eta =\lim _ny^*x_n^*\xi _0=y^*X^*\xi _0$. Since X is affiliated to M, $X\xi _0-x_n\xi _0\in D(S_0)$ and $S_0y^*S_0$ is bounded on $D(S_0)$, for $z'\in M'$ we have

$$\begin{aligned} \begin{aligned} \left( z'\xi _0|Xy\xi _0-x_ny\xi _0\right)&=\left( X^*\xi _0-x_n^*\xi _0|yz'^*\xi _0\right) =\left( y^*S_0\left( X\xi _0-x_n\xi _0\right) |S_0^*(z'\xi _0)\right) \\&=\left( \left( S_0y^*S_0\right) \left( X\xi _0-x_n\xi _0\right) |z'\xi _0)\right) \\&\le \Vert S_0y^*S_0\Vert \cdot \Vert X\xi _0-x_n\xi _0\Vert \cdot \Vert z'\xi _0\Vert . \end{aligned} \end{aligned}$$

By the density of $M'\xi _0$ in $L^2(M)$, it follows that $\Vert Xy\xi _0-x_ny\xi _0\Vert \le \Vert S_0y^*S_0\Vert \cdot \Vert X\xi _0-x_n\xi _0\Vert \rightarrow 0$ as $n\rightarrow \infty$ and we have $Xy\xi _0=\eta \in D(S_0)$ and $S_0(Xy\xi _0)=S_0(\eta )=y^*X^*\xi _0$ for any $y\in M_0$.

As $S_0^2$ is the identity operator on $D(S_0)$, from $S_0(X\xi _0)=X^*\xi _0$ it follows that $X^*\xi _0\in D(S_0)$ and $S_0(X^*\xi _0)=X\xi _0$. Thus $(X^*,D(X^*))$ satisfies the same hypotheses as (X, D(X)) and the statements involving $(X^*,D(X^*))$ can be deduced from those involving (X, D(X)) proved above, by substitution and the fact that the sub-algebra $M_0$ is involutive.

The operator $L_\xi$ is well defined since $\xi \in D(S_0)=D(\Delta _0^{1/2})$ implies $\Delta _0^{1/4}\xi \in D(\Delta _0^{1/4})$ and $J\Delta _0^{1/4}\xi \in JD(\Delta _0^{1/4})=D(\Delta _0^{-1/4})=D(\Delta _0^{1/4}y\Delta _0^{-1/4})$. Since $\xi \in D(S_0)$ implies $S_0\xi \in D(S_0)$, analogous relations imply that $J\Delta _0^{1/4}S_0\xi \in D(\Delta _0^{1/4}y\Delta _0^{-1/4})$ so that $R_\xi$ is well defined too.

As first step to prove (iii), we show that $D(\Delta _0^{1/4})$ is a left $M_0$-module, i.e. $y\zeta \in D(\Delta _0^{1/4})$ for any $y\in M_0$ and $\zeta \in D(\Delta _0^{1/4})$ (a fact probably known in literature). Notice first that since $\sigma ^{\omega _0}_{-i/4}(y)\in M_0$ we have

$$\begin{aligned} \sigma ^{\omega _0}_{-i/4}(y)\xi _0 \in D\left( \Delta _0^{1/4}\right) =D\left( \Delta _0^{-1/4}y^*\Delta _0^{1/4}\right) , \end{aligned}$$

which means, in particular, that $y^*\Delta _0^{1/4}\sigma ^{\omega _0}_{-i/4}(y)\xi _0\in D(\Delta _0^{-1/4})$ and implies

$$\begin{aligned} \begin{aligned} |\sigma ^{\omega _0}_{-i/4}(y)|^2\xi _0&=(\sigma ^{\omega _0}_{-i/4}(y))^*\sigma ^{\omega _0}_{-i/4}(y)\xi _0=\sigma ^{\omega _0}_{i/4}(y^*)\sigma ^{\omega _0}_{-i/4}(y)\xi _0\\&=\overline{\Delta _0^{-1/4}y^*\Delta _0^{1/4}}\sigma ^{\omega _0}_{-i/4}(y)\xi _0=\Delta _0^{-1/4}y^*\Delta _0^{1/4}\sigma ^{\omega _0}_{-i/4}(y)\xi _0\\&=\Delta _0^{-1/4}y^*\Delta _0^{1/4}\sigma ^{\omega _0}_{-i/4}(y)\Delta _0^{1/4}\xi _0\\&=\Delta _0^{-1/4}y^*\sigma ^{\omega _0}_{-i/4}(\sigma ^{\omega _0}_{-i/4}(y))\xi _0\\&=\Delta _0^{-1/4}y^*\sigma ^{\omega _0}_{-i/2}(y)\xi _0\\&=\Delta _0^{-1/4}y^*\Delta _0^{1/2}y\xi _0\\&=\Delta _0^{-1/4}y^*Jy^*J\xi _0.\\ \end{aligned} \end{aligned}$$

Since $i_0$ is positivity preserving, setting $c:=\Vert \sigma ^{\omega _0}_{-i/4}(y)\Vert ^2$, we thus obtain the bound

$$\begin{aligned} y^*Jy^*J\xi _0=\Delta _0^{1/4}\left( |\sigma ^{\omega _0}_{-i/4}(y)|^2\xi _0\right) =i_0\left( |\sigma ^{\omega _0}_{-i/4}(y)|^2\right) \le c\cdot \xi _0. \end{aligned}$$

Consider now a sequence $x_n\xi _0\in M\xi _0$ converging to $\zeta \in D(\Delta _0^{1/4})$ in the graph norm of $(\Delta _0^{1/4},D(\Delta _0^{1/4}))$. Then $\lim _n\Vert yx_n\xi _0-y\zeta \Vert \le \Vert y\Vert \cdot \lim _n\Vert x_n\xi _0-\zeta \Vert =0$. Since $\Delta _0^{^1/4}x_n\xi _0\in L^2(M)$ is a Cauchy sequence, we have $((x_n-x_m)J(x_n-x_m)J\xi _0|\xi _0)=(x_n\xi _0-x_m\xi _0|J(x_n^*-x_m^*)J\xi _0)=(x_n\xi _0-x_m\xi _0|\Delta _0^{1/2}(x_n-x_m)\xi _0)=\Vert \Delta _0^{1/4}x_n\xi _0-\Delta _0^{1/4}x_m\xi _0\Vert ^2\rightarrow 0$ and, by analogous identities, the self-polarity of $L^2_+(M)$ and the bound above, we get $\Vert \Delta _0^{1/4}yx_n\xi _0-\Delta _0^{1/4}yx_m\xi _0\Vert ^2=((x_n-x_m)J(x_n-x_m)J\xi _0|y^*Jy^*J\xi _0)\le c\cdot ((x_n-x_m)J(x_n-x_m)J\xi _0|\xi _0)\rightarrow 0$. Thus $yx_n\xi _0\in D(\Delta _0^{1/4})$ converges in the graph norm of $(\Delta _0^{1/4},D(\Delta _0^{1/4}))$ to $y\zeta \in D(\Delta _0^{1/4})$. The arbitrariness of $y\in M_0$ and $\zeta \in D(\Delta _0^{1/4})$ implies that $D(\Delta _0^{1/4})$ is an $M_0$-module.

Coming back to the proof of (iii), notice that, by the identity $J\Delta _0^{1/4}=\Delta _0^{-1/4}J$, one has $J\Delta _0^{1/4}=(J\Delta _0^{1/4})^{-1}$ so that the closed operator $J\Delta _0^{1/4}$ is idempotent on its domain:

$$\begin{aligned} \left( J\Delta _0^{1/4}\right) \xi \in D\left( \Delta _0^{1/4}\right) ,\qquad \left( J\Delta _0^{1/4}\right) ^2\zeta =\zeta ,\qquad \forall \,\zeta \in D\left( \Delta _0^{1/4}\right) . \end{aligned}$$

Thus, for $y\in M_0$ and $\zeta \in D(\Delta _0^{1/4})$, we have $y^*\zeta =y^*(J\Delta _0^{1/4})^2\zeta$ and, since $y^*\in M_0$ implies $y^*\zeta \in D(\Delta _0^{1/4})$, we have $y^*\zeta =(J\Delta _0^{1/4})^2y^*(J\Delta _0^{1/4})^2\zeta$ too. Applying this identity to $\zeta :=J\Delta _0^{1/4}\eta$ for any $\eta \in D(\Delta _0^{1/4})$, we have $(J\Delta _0^{1/4})^{-1}y^*(J\Delta _0^{1/4})\eta =(J\Delta _0^{1/4})y^*\left( J\Delta _0^{1/4}\right) ^{-1}\eta$, i.e.

$$\begin{aligned} \Delta _0^{-1/4}Jy^*J\Delta _0^{1/4}\eta =J\Delta _0^{1/4}y^*\Delta _0^{-1/4}J\eta \qquad \eta \in D(\Delta _0^{1/4}). \end{aligned}$$

Since, by hypotheses, $X\xi _0\in D(\Delta _0^{1/2})$, we may apply the identity to $\eta :=\Delta _0^{1/4}X\xi _0\in D(\Delta _0^{1/4})$, to get

$$\begin{aligned} \begin{aligned} \Delta _0^{-1/4}Jy^*J\Delta _0^{1/4}\left( \Delta _0^{1/4}X\xi _0\right)&=J\Delta _0^{1/4}y^*\Delta _0^{-1/4}J\Delta _0^{1/4}X\xi _0 \\&=J\sigma ^\omega _{-i/4}(y^*)J\Delta _0^{1/4}\xi =L_\xi i_0(y). \end{aligned} \end{aligned}$$

Since, by (ii), $Xy\xi _0\in D(S_0)=D(\Delta _0^{1/2})\subseteq D(\Delta _0^{1/4})$ and $S_0^{-1}=\Delta _0^{-1/2}J$, we have

$$\begin{aligned} Xy\xi _0=S_0^{-1}(S_0(Xy\xi _0))=S_0^{-1}(y^*X^*\xi _0)=\Delta _0^{-1/2}Jy^*J\Delta _0^{1/2}X\xi _0 \end{aligned}$$

and we conclude the proof of (iii) by

$$\begin{aligned} \Delta _0^{1/4}Xy\xi _0=\Delta _0^{-1/4}Jy^*J\Delta _0^{1/4}\left( \Delta _0^{1/4}X\xi _0\right) =L_\xi i_0(y). \end{aligned}$$

To prove (iv), notice first that, since $S_0^2$ is the identity operator on $D(S_0)$ and $X\xi _0\in D(S_0)$, we have $X^*\xi _0=S_0(X\xi _0)\in D(S_0)=D(\Delta _0^{1/2})$, $X\xi _0=S_0(X^*\xi _0)$, $\Delta _0^{1/4}X^*\xi _0\in D(\Delta _0^{1/4})$ and, for all $y\in M$, $J\Delta _0^{1/4}X^*\xi _0\in D(\Delta _0^{-1/4})=D(\Delta _0^{1/4}y\Delta _0^{-1/4})$. Since, as shown above, $yX\xi _0\in D(\Delta _0^{1/4})$ for all $y\in M_0$ as $X\xi _0\in D(\Delta _0^{1/2})\subset D(\Delta _0^{1/4})$ and $J\Delta _0^{1/2}=\Delta _0^{-1/4}J\Delta _0^{1/4}$ as closed operators, we have

$$\begin{aligned} \begin{aligned} \Delta _0^{1/4}yX\xi _0&=\Delta _0^{1/4}yS_0(X^*\xi _0)=\Delta _0^{1/4}yJ\Delta _0^{1/2}X^*\xi _0=\Delta _0^{1/4}y\Delta _0^{-1/4}J\Delta _0^{1/4}S_0(\xi )\\&=\sigma ^{\omega _0}_{-i/4}(y)J\Delta _0^{1/4}S_0(\xi )=R_\xi i_0(y). \end{aligned} \end{aligned}$$

To prove (v), i.e. that $\overline{L_\xi }$ is affiliated with M, let us start to notice that for $z'\in M_0'$ and $y\in M_0$, setting $z:=J\Delta _0^{-1/4}z'^*\Delta _0^{1/4}J=\Delta _0^{1/4}Jz'^*J\Delta _0^{-1/4}=\sigma ^{\omega _0}_{-1/4}(Jz'^*J)\in M$, since $\Delta _0^{-1/4}J=J\Delta _0^{1/4}$ and $\Delta _0^{-1/2}z'\xi _0\in D(\Delta _0^{1/2})\subset D(\Delta _0^{1/4})=D(J\Delta _0^{1/4})$, we have $z\xi _0=J\Delta _0^{-1/4}z'^*\Delta _0^{1/4}J\xi _0=J\Delta _0^{-1/4}z'^*\xi _0=J\Delta _0^{-1/4}J\Delta _0^{-1/2}z'\xi _0==JJ\Delta _0^{1/4}\Delta _0^{-1/2}z'\xi _0=\Delta _0^{-1/4}z'\xi _0$, $z'\xi _0=\Delta _0^{1/4}z\xi _0=\sigma ^{\omega _0}_{-i/4}(z)\xi _0$ and

$$\begin{aligned} \begin{aligned} z'i_0(y)&=z'\Delta _0^{1/4}y\Delta _0^{-1/4}\xi _0=z'\sigma ^{\omega _0}_{-i/4}(y)\xi _0=\sigma ^{\omega _0}_{-i/4}(y)z'\xi _0 \\&=\sigma ^{\omega _0}_{-i/4}(y)\sigma ^{\omega _0}_{-i/4}(z)\xi _0=\sigma ^{\omega _0}_{-i/4}(yz)\xi _0=i_0(yz) \end{aligned} \end{aligned}$$

so that $z'i_0(y)\in M_0\xi _0=D(L_\xi )$. Since $\sigma ^{\omega _0}_{i/4}(z')\in M'$, $y\in M$, X is affiliated to M, $y\xi _0\in D(X)$, $yz\xi _0\in D(X)$ by i), using iii) we have

$$\begin{aligned} \begin{aligned} L_\xi z'i_0(y)=L_\xi i_0(yz)&=\Delta _0^{1/4}Xyz\xi _0\\&=\Delta _0^{1/4}Xy\Delta _0^{-1/4}z'\xi _0\\&=\Delta _0^{1/4}Xy\Delta _0^{-1/4}z'\Delta _0^{1/4}\xi _0\\&=\Delta _0^{1/4}Xy\sigma ^{\omega _0}_{i/4}\left( z'\right) \xi _0\\&=\Delta _0^{1/4}\sigma ^{\omega _0}_{i/4}\left( z'\right) Xy\xi _0\\&=\Delta _0^{1/4}\sigma ^{\omega _0}_{i/4}\left( z'\right) \Delta _0^{-1/4}\Delta _0^{1/4}Xy\xi _0\\&=\sigma ^{\omega _0}_{-i/4}\left( \sigma ^{\omega _0}_{i/4}\left( z'\right) \right) L_\xi i_0(y)\\&=z'L_\xi i_0(y). \end{aligned} \end{aligned}$$

Since by Lemma 2.7${\mathcal {R}}_\eta =J{\mathcal {L}}_\eta J$, for $\eta \in D(S_0)$ we have $R_\xi =JL_\xi J$ for $\xi \in D(\Delta _0^{1/2})$. This is equivalent to $(J\oplus J){\mathcal {G}}(R_\xi )={\mathcal {G}}(L_\xi )$ and implies $(J\oplus J){\mathcal {G}}(\overline{R_\xi })={\mathcal {G}}(\overline{L_\xi })$, i.e. $\overline{R_\xi }=J\overline{L_\xi } J$ as an identity between densely defined closed operators.

To prove (vi), notice that, by (i) we have $i_0(y)\in D(\Delta _0^{-1/4})$ and $\Delta _0^{-1/4}i_0(y)=y\xi _0\in D(X)$ and by ii) we have that $X\Delta _0^{-1/4}i_0(y)=Xy\xi _0\in D(S_0)\subseteq D(\Delta _0^{1/4})$. Hence $i_0(y)\in D(\Delta _0^{1/4}X\Delta _0^{-1/4})$ and $(\Delta _0^{1/4}X\Delta _0^{-1/4})i_0(y)= \Delta _0^{1/4}Xy\xi _0=L_\xi i_0(y)$.

To prove (vii), notice that, since $y\xi _0\in D(S_0)$ and $S_0=\Delta _0^{-1/4}J\Delta _0^{1/4}$ on $D(S_0)$, we have $y^*\xi _0=S_0(y\xi _0)=\Delta _0^{-1/4}J\Delta _0^{1/4}y\xi _0=\Delta _0^{-1/4}Ji_0(y)$. By i) and ii), $y^*\xi _0\in D(X^*)$, $X^*y^*\xi _0\in D(S_0)$ and $yX\xi _0=S_0(X^*y^*\xi _0)\in D(S_0)\subset D(\Delta _0^{1/4})$ so that by (iv), $R_\xi i_0(y)=\Delta _0^{1/4}yX\xi _0=(\Delta _0^{1/4}S_0)(X^*y^*\xi _0)$. Since $\Delta _0^{1/4}S_0=J\Delta _0^{1/4}$ on $D(S_0)$, we have

$$\begin{aligned} R_\xi i_0(y)=\left( \Delta _0^{1/4}S_0\right) \left( X^*y^*\xi _0\right) =\left( J\Delta _0^{1/4}\right) \left( X^*y^*\xi _0\right) =\left( J\Delta _0^{1/4}X^*\Delta _0^{-1/4}J\right) i_0(y) \end{aligned}$$

showing that $J\Delta _0^{1/4}X^*\Delta _0^{-1/4}J$ is densely defined on $i_0(M_0)$ and there it coincides with $R_\xi$.

To prove the first identity in (viii), notice that, by the Spectral Theorem, $\xi =X\xi _0$ is an eigenvector of $\Delta _0^{1/4}$ with eigenvalue $\lambda >0$: $\Delta _0^{1/4}X\xi _0=\lambda \cdot X\xi _0$. By the density of $M_0'\xi _0$ in $L^2(M)$ and for all $z'\in M_0'$ we then have

$$\begin{aligned} \begin{aligned} \left( z'\xi _0|L_\xi i_0(y)\right)&=\left( z'\xi _0|\Delta _0^{1/4}Xy\xi _0\right) \\&=\left( \Delta _0^{1/4}z'\Delta _0^{-1/4}\xi _0|Xy\xi _0\right) \\&=\left( \Delta _0^{1/4}z'\Delta _0^{-1/4}X^*\xi _0|y\xi _0\right) \\&=\left( \Delta _0^{1/4}z'\Delta _0^{-1/4}J\Delta _0^{1/2}X\xi _0|y\xi _0\right) \\&=\lambda ^2\cdot \left( \Delta _0^{1/4}z'\Delta _0^{-1/4}JX\xi _0|y\xi _0\right) \\&=\lambda ^2\cdot \left( \Delta _0^{1/4}z'J\Delta _0^{1/4}X\xi _0|y\xi _0\right) \\&=\lambda ^2\cdot \left( z'J\Delta _0^{1/4}X\xi _0|\Delta _0^{1/4}y\xi _0\right) \\&=\lambda ^2\cdot \left( z'J\Delta _0^{1/4}X\xi _0|i_0(y)\right) \\&=\lambda ^{3}\cdot \left( z'JX\xi _0|i_0(y)\right) \\&=\lambda \cdot \left( z'X^*\xi _0|i_0(y)\right) \\&=\left( z'\xi _0|\lambda Xi_0(y)\right) . \end{aligned} \end{aligned}$$

To prove the second identity in (viii), we first need to show that the adjoint of the densely defined operator $(\Delta _0^{1/4}X\Delta _0^{-1/4},i_0(M_0))$ (which is closable by ( operator $(\Delta _0^{-1/4}X^*\Delta _0^{1/4},M_0\xi _0)$. By ii), for $x\in M_0$ and since $i_0(x)=\sigma ^{\omega _0}_{-i/4}(x)\xi _0\in M_0\xi _0\subset D(X^*)$, we have $JX^*i_0(x)=JX^*JJi_0(x)=JX^*Ji_0(x^*)= JX^*J\sigma ^{\omega _0}_{-i/4}(x^*)\xi _0$. Since $JX^*J$ is affiliated to $M'$, $\sigma ^{\omega _0}_{-i/4}(x^*)\in M$ and by ii) $\sigma ^{\omega _0}_{-1/4}(x^*)\xi _0\in D(JX^*J)$, we have

$$\begin{aligned} \begin{aligned} JX^*i_0(x)&=\sigma ^{\omega _0}_{-i/4}(x^*)JX^*J\xi _0=\sigma ^{\omega _0}_{-i/4}(x^*)JX^*\xi _0 =\sigma ^{\omega _0}_{-i/4}(x^*)JS_0(X\xi _0) \\&=\sigma ^{\omega _0}_{-i/4}(x^*)\Delta _0^{1/2}(X\xi _0). \end{aligned} \end{aligned}$$

The hypothesis that $X\xi _0$ is an eigenvalue of $\Delta _0^{1/2}$ then implies $JX^*i_0(x)=\lambda ^2\cdot \sigma ^{\omega _0}_{-i/4}(x^*)X\xi _0$ which in turn, by ii), implies $JX^*i_0(x)= \lambda ^2\cdot S_0(X^*\sigma ^{\omega _0}_{i/4}(x)\xi _0)\in D(S_0)=D(\Delta _0^{1/2})\subset D(\Delta _0^{1/4})$ so that $X^*\Delta _0^{1/4}x\xi _0=X^*i_0(x)\in JD(\Delta _0^{1/4})=D(\Delta _0^{-1/4})$ and $x\xi _0\in D(\Delta _0^{-1/4}X^*\Delta _0^{1/4})$. For all $y\in M_0$ we may then compute

$$\begin{aligned} \begin{aligned} \left( x\xi _0|\left( \Delta _0^{1/4}X\Delta _0^{-1/4}\right) i_0(y)\right)&=\left( \Delta _0^{1/4}x\xi _0|\left( X\Delta _0^{-1/4}\right) i_0(y)\right) \\&=\left( X^*\Delta _0^{1/4}x\xi _0|\Delta _0^{-1/4}i_0(y)\right) \\&=\left( \left( \Delta _0^{-1/4}X^*\Delta _0^{1/4}\right) x\xi _0|i_0(y)\right) \end{aligned} \end{aligned}$$

so that $x\xi _0\in D((\Delta _0^{1/4}X\Delta _0^{-1/4})^*)$ and $(\Delta _0^{-1/4}X^*\Delta _0^{1/4})x\xi _0= (\Delta _0^{1/4}X\Delta _0^{-1/4})^*x\xi _0$. Since by vi) and the first identity proved above we have

$$\begin{aligned} \left( \Delta _0^{1/4}X\Delta _0^{-1/4}\right) x\xi _0=L_\xi x\xi _0=\lambda \cdot Xx\xi _0 \end{aligned}$$

for all $x\xi _0\in M_0\xi _0=i_0(M_0)=D(L_\xi )\subset D(\Delta _0^{1/4}X\Delta _0^{-1/4})\cap D(X)$, by i) we then have

$$\begin{aligned} \left( \Delta _0^{-1/4}X^*\Delta _0^{1/4}\right) x\xi _0=\lambda X^* x\xi _0\qquad x\xi _0\in M_0\xi _0\subset D\left( \Delta _0^{-1/4}X^*\Delta _0^{1/4}\right) \cap D(X^*). \end{aligned}$$

To finalize the proof of the second identity in viii), rewrite the eigenvalue equation satisfied by $\xi =X\xi _0$ as $JX^*J\xi _0=\Delta _0^{1/2}X\xi _0=\lambda ^2 X\xi _0$ so that, for all $y\in M_0$, we have

$$\begin{aligned} \begin{aligned} R_\xi i_0(y)&=\Delta _0^{1/4}yX\xi _0\\&=\lambda ^{-2}\cdot \Delta _0^{1/4}y(JX^*J)\xi _0\\&=\lambda ^{-2}\cdot \Delta _0^{1/4}(JX^*J)y\xi _0\\&=\lambda ^{-2}\cdot J\left( \Delta _0^{-1/4}X^*\Delta _0^{1/4}\right) J\Delta _0^{1/4}y\xi _0\\&=\lambda ^{-2}\cdot J\left( \Delta _0^{-1/4}X^*\Delta _0^{1/4}\right) Ji_0(y)\\&=\lambda ^{-2}\cdot J\left( \Delta _0^{-1/4}X^*\Delta _0^{1/4}\right) i_0(y^*)\\&=\lambda ^{-2}\cdot J\left( \Delta _0^{-1/4}X^*\Delta _0^{1/4}\right) \sigma ^{\omega _0}_{-i/4}(y^*)\xi _0\\&=\lambda ^{-2}\cdot JX^*\sigma ^{\omega _0}_{-i/4}(y^*)\xi _0\\&=\lambda ^{-2}\cdot JX^*Ji_0(y). \end{aligned} \end{aligned}$$

$\square$

Lemma 2.9

Let $\xi \in D(S_0)$ be eigenvector of $\Delta _0^{1/2}$ corresponding to the eigenvalue $\lambda ^2>0$ and fix a densely defined, closed operator (X, D(X)) affiliated to M such that

$$\begin{aligned} \xi _0\in D(X)\cap D(X^*),\qquad \xi =X\xi _0,\quad S_0(X\xi _0)=X^*\xi _0. \end{aligned}$$

Then $S_0\xi \in D(S_0)=D(\Delta _0^{1/2})$ is eigenvector of $\Delta _0^{1/2}$ corresponding to the eigenvalue $\lambda ^{-2}$.

Proof

On one hand we have $S_0\xi =J\Delta _0^{1/2}\xi =\lambda ^2\cdot J\xi$. On the other hand, since $JD(\Delta _0^{1/2})=D(\Delta _0^{-1/2})$ and $S_0=S_0^{-1}$ on $D(\Delta _0^{1/2})=D(S_0)=D(S_0^{-1})$ we have $S_0\xi =S_0^{-1}\xi =\Delta _0^{-1/2}J\xi$ so that $\Delta _0^{1/2}S_0\xi =J\xi =\lambda ^{-2}\cdot S_0\xi$. $\square$

Combining the results obtained, we have

Corollary 2.10

Let $\xi \in D(S_0)$ be an eigenvector of $\Delta _0^{1/2}$ corresponding to the eigenvalue $\lambda ^2>0$ and fix a densely defined, closed operator (X, D(X)) affiliated to M such that

$$\begin{aligned} \xi _0\in D(X)\cap D(X^*),\qquad \xi =X\xi _0,\quad S_0(X\xi _0)=X^*\xi _0. \end{aligned}$$

Then for all $y\in M_0$ we have

$$\begin{aligned} (L_\xi -R_\xi )i_0(y)=\Delta _0^{1/4}(Xy-yX)\xi _0 \end{aligned}$$

(2.1)

and

$$\begin{aligned} (L_{S_0\xi }-R_{S_0\xi })i_0(y)=\Delta _0^{1/4}(X^*y-yX^*)\xi _0. \end{aligned}$$

(2.2)

The commutator $[X,y]:=Xy-yX$ is in general only densely defined if X is affiliated to M but, within the hypotheses assumed at the beginning of this section, the vector $\xi _0$ belongs to the domain of [X, y] and its image $[X,y]\xi _0$ belongs to $D(\Delta _0^{1/4})$. This may justify the notation

$$\begin{aligned} i_0([X,y]):=\Delta _0^{1/4}(Xy-yX)\xi _0\qquad y\in M_0. \end{aligned}$$

In the following we will use the notation $j(X^*):=JX^*J$.

Next result shows that the symmetric embedding $i_0$ intertwines the unbounded spatial derivations $\delta _X$, $\delta _{X^*}$ on M with the unbounded bimodule derivations $d^\lambda _X$, $d^{\lambda ^{-1}}_{X^*}$ on $L^2(M)$.

Proposition 2.11

(Bimodule derivations and spatial derivations) Let $\xi \in D(S_0)$ be eigenvector of $\Delta _0^{1/2}$ corresponding to the eigenvalue $\lambda ^2>0$ and fix a densely defined, closed operator (X, D(X)) affiliated to M such that

$$\begin{aligned} \xi _0\in D(X)\cap D(X^*),\qquad \xi =X\xi _0,\quad S_0(X\xi _0)=X^*\xi _0. \end{aligned}$$

Then, setting $d_X^{\lambda }:=i(\lambda X-\lambda ^{-1} j(X^*))$ and $d_{X^*}^{\lambda ^{-1}}:=i(\lambda ^{-1} X^*-\lambda j(X))$, we have

$$\begin{aligned} d_X^{\lambda } i_0(y)= i_0(i[X,y])\qquad d_{X^*}^{\lambda ^{-1} } i_0(y)= i_0(i[X^*,y])\qquad y\in M_0. \end{aligned}$$

Otherwise stated, setting $\delta _X(y):=i[X,y]$ for any $y\in M_0$, on the $^*$-algebra $M_0$ we have

$$\begin{aligned} d_X^{\lambda }\circ i_0= i_0\circ \delta _X\qquad d_{X^*}^{\lambda ^{-1}}\circ i_0= i_0\circ \delta _{X^*}. \end{aligned}$$

Proof

For $y\in M_0$ we have $\sigma ^{\omega _0}_{-i/4}(y)\in M_0$, $J\sigma ^{\omega _0}_{-i/4}(y)J\in M_0'$ and

$$\begin{aligned} J\sigma ^{\omega _0}_{-i/4}(y)J\xi _0=J\sigma ^{\omega _0}_{-i/4}(y)\xi _0=J\Delta _0^{1/4}y\xi _0=Ji_0(y). \end{aligned}$$

Since $X^*$ is affiliated with M and $\xi _0\in D(X^*)$, we have $(J\sigma ^{\omega _0}_{-i/4}(y)J)\xi _0\in D(X^*)$ and

$$\begin{aligned} \begin{aligned} j(X^*)i_0(y)&=JX^*Ji_0(y) \\&=JX^*\left( J\sigma ^{\omega _0}_{-i/4}(y)J\right) \xi _0\\&=J\left( J\sigma ^{\omega _0}_{-i/4}(y)J\right) X^*\xi _0 \\&=\sigma ^{\omega _0}_{-i/4}(y)JX^*\xi _0 \\&=\sigma ^{\omega _0}_{-i/4}(y)\Delta _0^{1/2}X\xi _0 \\&=\Delta _0^{1/4}y\Delta _0^{1/4}X\xi _0 \\&=\lambda \cdot \Delta _0^{1/4}yX\xi _0. \end{aligned} \end{aligned}$$

Since $\lambda Xi_0(y)=L_\xi i_0(y)=\Delta _0^{1/4}Xy\xi _0$ we have too $d_X^\lambda i_0(y):=i(\lambda X-\lambda ^{-1} j(X^*))i_0(y)=i\Delta _0^{1/4}(Xy\xi _0-yX\xi _0)=i_0(i[X,y])=i_0(\delta _X(y))$. The proof of the second identity is similar. $\square$

Theorem 2.12

Let $\xi \in D(S_0)$ be an eigenvector of $\Delta _0^{1/2}$ corresponding to the eigenvalue $\lambda ^2>0$ and (X, D(X)) a densely defined, closed operator affiliated to M such that

$$\begin{aligned} \xi _0\in D(X)\cap D(X^*),\qquad \xi =X\xi _0,\quad S_0(\xi )=X^*\xi _0. \end{aligned}$$

Then the completely Dirichlet form $({\mathcal {E}}_X^\lambda , {\mathcal {F}}_X^\lambda )$ constructed above may be represented as

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_X^\lambda [i_0(y)]&=\Vert i_0([X,y])\Vert ^2_{L^2(M)} + \Vert i_0([X^*,y])\Vert ^2_{L^2(M)}\qquad y\in M_0\\&=\Vert i_0(\delta _X(y))\Vert ^2_{L^2(M)} + \Vert i_0(\delta _{X^*}(y))\Vert ^2_{L^2(M)} \end{aligned} \end{aligned}$$

on the $L^2(M)$-dense, J-invariant subspace $M_0\xi _0=i_0(M)\subset {\tilde{{\mathcal {F}}}}_X\subset {\mathcal {F}}^\lambda _X$.

Remark 2.13

These results prove a fortiori that and under the stated assumptions, the form

$$\begin{aligned} i_0(y)\mapsto \lambda ^2\Vert i_0([X,y])\Vert ^2_{L^2(M)} + \Vert i_0([X^*,y])\Vert ^2_{L^2(M)} \end{aligned}$$

extends to a completely Dirichlet form on $L^2(M)$ with respect to the cyclic vector $\xi _0\in L^2_+(M)$. If $\xi _0$ would be the vector representing a finite, normal, faithful trace state $\omega _0$, this result would follow from the general theory relating completely Dirichlet forms and closable bimodule derivations on von Neumann algebras with trace (see [14]).

3 Coercivity of Dirichlet forms

In this section we still keep the assumption that $\xi \in D(S_0)$ is an eigenvector of $\Delta _0^{1/2}$ corresponding to the eigenvalue $\lambda ^2>0$ (we still assume $\lambda >0$) and (X, D(X)) a densely defined, closed operator affiliated to M such that

$$\begin{aligned} \xi _0\in D(X)\cap D(X^*),\qquad \xi =X\xi _0,\quad S_0(\xi )=X^*\xi _0. \end{aligned}$$

We prove below natural lower bounds on the Dirichlet form $({\mathcal {E}}_X^\lambda , {\mathcal {F}}_X^\lambda )$ constructed in Sect. 2, which lead to coercivity. Recall that $({\mathcal {E}}_X^\lambda , {\mathcal {F}}_X^\lambda )$ is defined as the closure of the densely defined, J-real, closable quadratic form ${\tilde{{\mathcal {E}}}}_X^\lambda :{\tilde{{\mathcal {F}}}}_X\rightarrow [0,+\infty )$ on $L^2(M)$ given by

$$\begin{aligned} {\tilde{{\mathcal {E}}}}_X^\lambda [\eta ]:=\left\| d_X^\lambda \eta \right\| ^2_{L^2(M)} + \left\| d_{X^*}^{\lambda ^{-1}}\eta \right\| ^2_{L^2(M)}\qquad \eta \in {\tilde{{\mathcal {F}}}}_X=D(d_X^\lambda )\cap D(d_{X^*}^{\lambda ^{-1}}), \end{aligned}$$

where $d_X^{\lambda }:=i(\lambda X-\lambda ^{-1} j(X^*))$ and $d_{X^*}^{\lambda ^{-1}}:=i(\lambda ^{-1} X^*-\lambda j(X))$ are defined on the domain

$$\begin{aligned} {\tilde{{\mathcal {F}}}}_X=D(X)\cap D(X^*)\cap J\left( D(X)\cap D(X^*)\right) \end{aligned}$$

containing the $L^2(M)$-dense, J-invariant subspace $i_0(M_0)=M_0\xi _0\subset {\tilde{{\mathcal {F}}}}_X$. Obviously ${\tilde{{\mathcal {F}}}}_X$ is a form core for $({\mathcal {E}}^\lambda _X,{\mathcal {F}}^\lambda _X)$ and on it ${\mathcal {E}}_X^{\lambda }$ and ${\tilde{{\mathcal {E}}}}_X^\lambda$ coincide.

We start showing an alternative representation of the Dirichlet form.

Theorem 3.1

The following representation holds true for the quadratic form $({\mathcal {E}}_X^\lambda , {\tilde{{\mathcal {F}}}}_X)$:

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_X^\lambda [\eta ]&=\lambda ^2\left( \Vert X\eta \Vert ^2+\Vert XJ\eta \Vert ^2\right) +\lambda ^{-2}\left( \Vert X^*\eta \Vert ^2+\Vert X^*J\eta \Vert ^2\right) \\&\quad -2\bigl [(X\eta |JX^*J\eta )+(X^*\eta |JXJ\eta )\bigr ]\qquad \qquad \qquad \qquad \eta \in {\tilde{{\mathcal {F}}}}_X. \end{aligned} \end{aligned}$$

(3.1)

Proof

In the following, we repeatedly use the fact that if $N\subseteq B(h)$ is a von Neumann algebra acting on a Hilbert space h and (A, D(A)), (B, D(B)) are densely defined, closed operator on h affiliated to N and $N'$, respectively, then

$$\begin{aligned} \left( A\eta |B\zeta \right) =\left( B^*\eta |A^*\zeta \right) \qquad \eta \in D(A)\cap D(B^*),\quad \zeta \in D(B)\cap D(A^*). \end{aligned}$$

This identity follows directly if $B\in M'$ is bounded since then $B^*\in M'$ and $\eta \in D(A)$ implies $B^*\eta \in D(A)$ and $AB^*\eta =B^*A\eta$ so that $(A\eta |B\zeta )=(B^*A\eta |\zeta )=(AB^*\eta |\zeta )=(B^*\eta |A^*\zeta )$. In general we may approximate B weakly by $B_\varepsilon :=B(I+\varepsilon |B|)^{-1}\in M'$ as $\varepsilon \downarrow 0$.

We start the proof of the result setting

$$\begin{aligned} d_X:=i\left( X-j(X^*)\right) ,\qquad V_X^\lambda :=i\left( 1-\lambda ^{-1}\right) \left( \lambda X+j(X^*)\right) \end{aligned}$$

and using the splittings

$$\begin{aligned} d_X^\lambda =d_X+V_X^\lambda \qquad d_{X^*}^{\lambda ^{-1}}=d_{X^*}+V_{X^*}^{\lambda ^{-1}}, \end{aligned}$$

for any $\eta \in {\tilde{{\mathcal {F}}}}_X$ to have the representation

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_X^\lambda [\eta ]&:=\left\| d_X\eta \right\| ^2 + \left\| d_{X^*}\eta \right\| ^2 + \left\| V_X^\lambda \eta \right\| ^2 + \left\| V_{X^*}^{\lambda ^{-1}}\eta \right\| ^2 +\\&\quad \left( d_X\eta | V_X^\lambda \eta \right) + \left( V_X^\lambda \eta |d_X\eta \right) + \left( d_{X^*}\eta | V_{X^*}^{\lambda ^{-1}}\eta \right) + \left( V_{X^*}^{\lambda ^{-1}}\eta |d_{X^*}\eta \right) . \end{aligned} \end{aligned}$$

(3.2)

Since

$$\begin{aligned} \begin{aligned} \left\| d_X\eta \right\| ^2&=\left\| (X-j(X^*))\eta \right\| ^2=\Vert X\eta \Vert ^2+\Vert j(X^*)\eta \Vert ^2 \\&\quad -\left[ (X\eta |j(X^*)\eta )+(X^*\eta | j(X)\eta )\right] ,\\ \Vert d_{X^*}\eta \Vert ^2&=\Vert (X^*-j(X))\eta \Vert ^2=\Vert X^*\eta \Vert ^2+\Vert j(X)\eta \Vert ^2 \\&\quad -\left[ (X\eta |j(X^*)\eta )+(X^*\eta | j(X)\eta )\right] \end{aligned} \end{aligned}$$

the sum of the first two addends in (3.2) equals

$$\begin{aligned} \begin{aligned} \Vert d_X\eta \Vert ^2+\Vert d_{X^*}\eta \Vert ^2&=\Vert X\eta \Vert ^2+\Vert j(X^*)\eta \Vert ^2+\Vert X^*\eta \Vert ^2+\Vert j(X)\eta \Vert ^2\\&\quad -2[(X\eta |j(X^*)\eta )+(X^*\eta | j(X)\eta )]. \end{aligned} \end{aligned}$$

(3.3)

Since also

$$\begin{aligned} \begin{aligned} \Vert V_X^\lambda \eta \Vert ^2&=(1-\lambda ^{-1})^2((\lambda X+j(X^*))\eta |(\lambda X+j(X^*))\eta )\\&=(1-\lambda ^{-1})^2\bigl [\lambda ^2 \Vert X\eta \Vert ^2+\Vert j(X^*)\eta \Vert ^2 +\\&\quad \lambda ((X\eta |j(X^*)\eta )+(X^*\eta |j(X)\eta )\bigr ],\\ \Vert V_{X^*}^{\lambda ^{-1}}\eta \Vert ^2&=(1-\lambda )^2((\lambda ^{-1} X^*+j(X))\eta |(\lambda ^{-1} X^*+j(X))\eta )\\&=(1-\lambda )^2\bigl [\lambda ^{-2} \Vert X^*\eta \Vert ^2+\Vert j(X)\eta \Vert ^2+ \\&\quad \lambda ^{-1} ((X^*\eta |j(X)\eta )+(X\eta |j(X^*)\eta )\bigr ] \end{aligned} \end{aligned}$$

the sum of the third and fourth addends in (3.2) equals

$$\begin{aligned} \begin{aligned}&\Vert V_X^\lambda \eta \Vert ^2+\Vert V_{X^*}^{\lambda ^{-1}}\eta \Vert ^2= \\&(\lambda -1)^2(\Vert X\eta \Vert ^2+\Vert j(X)\eta \Vert ^2)+(\lambda ^{-1}-1)^2(\Vert X^*\eta \Vert ^2+\Vert j(X^*)\eta \Vert ^2)+\\&[(1-\lambda ^{-1})^2\lambda +(1-\lambda )^2\lambda ^{-1}]((X\eta |j(X^*)\eta )+(X^*\eta |j(X)\eta ))=\\&(\lambda -1)^2(\Vert X\eta \Vert ^2+\Vert j(X)\eta \Vert ^2)+(\lambda ^{-1}-1)^2(\Vert X^*\eta \Vert ^2+\Vert j(X^*)\eta \Vert ^2)+\\&2(\lambda -1)^2\lambda ^{-1}\bigl ((X\eta |j(X^*)\eta )+(X^*\eta |j(X)\eta )\bigr ). \end{aligned} \end{aligned}$$

(3.4)

Since we have too

$$\begin{aligned} \begin{aligned}&(d_X\eta | V_X^\lambda \eta ) + (V_X^\lambda \eta |d_X\eta )= \\&(1-\lambda ^{-1})\bigl [((X-j(X^*))\eta |(\lambda X+j(X^*))\eta )+((\lambda X+j(X^*))\eta |(X-j(X^*))\eta )\bigr ]=\\&(1-\lambda ^{-1})\bigl [\lambda \Vert X\eta \Vert ^2+(X\eta |j(X^*)\eta )-\lambda (X^*\eta |j(X)\eta )-\Vert j(X^*)\eta \Vert ^2+\\&\lambda \Vert X\eta \Vert ^2-\lambda (X\eta | j(X^*)\eta )+(X^*\eta |j(X)\eta )-\Vert j(X^*)\eta \Vert ^2\bigr ], \end{aligned} \end{aligned}$$

the sum of the fifth and sixth addends in (3.2) equals

$$\begin{aligned} \begin{aligned}&(d_X\eta | V_X^\lambda \eta ) + (V_X^\lambda \eta |d_X\eta )=\\&(1-\lambda ^{-1})\bigl [2\lambda \Vert X\eta \Vert ^2-2\Vert j(X^*)\eta \Vert ^2+\\&(1-\lambda )((X\eta |j(X^*)\eta ) + (X^*\eta |j(X)\eta ))\bigr ] \end{aligned} \end{aligned}$$

(3.5)

and, analogously, the sum of the seventh and eighth addends in (3.2) equals

$$\begin{aligned} \begin{aligned}&\left( d_{X^*}\eta | V_{X^*}^{\lambda ^{-1}}\eta \right) + \left( V_{X^*}^{\lambda ^{-1}}\eta |d_{X^*}\eta \right) \\&\quad =(1-\lambda )\bigl [2\lambda ^{-1} \Vert X^*\eta \Vert ^2-2\Vert j(X)\eta \Vert ^2\\&\qquad +(1-\lambda ^{-1})((X^*\eta |j(X)\eta ) + (X\eta |j(X^*)\eta ))\bigr ]. \end{aligned} \end{aligned}$$

(3.6)

By substitution of (3.6), (3.5) and (3.4) in (3.2) we obtain

$$\begin{aligned}&\quad {\mathcal {E}}_X^\lambda [\eta ]-\bigl (\Vert d_X\eta \Vert ^2 + \Vert d_{X^*}\eta \Vert ^2\bigr )\\&\quad = (\lambda -1)^2(\Vert X\eta \Vert ^2+\Vert j(X)\eta \Vert ^2)+(\lambda ^{-1}-1)^2(\Vert X^*\eta \Vert ^2+\Vert j(X^*)\eta \Vert ^2)\\&\qquad + 2(\lambda -1)^2\lambda ^{-1}((X\eta |j(X^*)\eta )+(X^*\eta |j(X)\eta )\\&\qquad + (1-\lambda ^{-1})\bigl [2\lambda \Vert X\eta \Vert ^2-2\Vert j(X^*)\eta \Vert ^2\\&\qquad +(1-\lambda )((X\eta |j(X^*)\eta ) + (X^*\eta |j(X)\eta ))\bigr ]\\&\qquad +(1-\lambda )\bigl [2\lambda ^{-1} \Vert X^*\eta \Vert ^2-2\Vert j(X)\eta \Vert ^2\\&\qquad +(1-\lambda ^{-1})((X^*\eta |j(X)\eta ) + (X\eta |j(X^*)\eta ))\bigr ]\\&\quad =\bigl [(\lambda -1)^2+2(\lambda -1)\bigr ](\Vert X\eta \Vert ^2+\Vert j(X)\eta \Vert ^2)\\&\qquad +\bigl [(\lambda ^{-1}-1)^2+2(\lambda ^{-1}-1)\bigr ](\Vert X^*\eta \Vert ^2+\Vert j(X^*)\eta \Vert ^2)\\&\qquad +\bigl [2(1-\lambda )(1-\lambda ^{-1})+2(\lambda -1)^2\lambda ^{-1}\bigr ]((X\eta |j(X^*)\eta )+(X^*\eta |j(X)\eta ))\\&\quad =(\lambda ^2-1)(\Vert X\eta \Vert ^2+\Vert j(X)\eta \Vert ^2)+(\lambda ^{-2}-1)(\Vert X^*\eta \Vert ^2+\Vert j(X^*)\eta \Vert ^2) \\&\qquad +\bigl [2(1-\lambda )(\lambda -1)\lambda ^{-1}+2(\lambda -1)^2\lambda ^{-1}\bigr ]((X\eta |j(X^*)\eta )+(X^*\eta |j(X)\eta ))\\&\quad =(\lambda ^2-1)(\Vert X\eta \Vert ^2+\Vert j(X)\eta \Vert ^2)+(\lambda ^{-2}-1)(\Vert X^*\eta \Vert ^2+\Vert j(X^*)\eta \Vert ^2)\\&\qquad +\bigl [-2(\lambda -1)^2\lambda ^{-1}+2(\lambda -1)^2\lambda ^{-1}\bigr ]((X\eta |j(X^*)\eta )+(X^*\eta |j(X)\eta ))\\&\quad =(\lambda ^2-1)(\Vert X\eta \Vert ^2+\Vert j(X)\eta \Vert ^2)+(\lambda ^{-2}-1)(\Vert X^*\eta \Vert ^2+\Vert j(X^*)\eta \Vert ^2) \end{aligned}$$

and then, by (3.3), we finally obtain (3.1) for any $\eta \in {\tilde{{\mathcal {F}}}}_X$

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_X^\lambda [\eta ]&=\bigl (\Vert d_X\eta \Vert ^2 + \Vert d_{X^*}\eta \Vert ^2\bigr )+(\lambda ^2-1)(\Vert X\eta \Vert ^2+\Vert j(X)\eta \Vert ^2)+\\&\quad (\lambda ^{-2}-1)(\Vert X^*\eta \Vert ^2+\Vert j(X^*)\eta \Vert ^2)\\&\quad =\lambda ^2(\Vert X\eta \Vert ^2+\Vert XJ\eta \Vert ^2)+\lambda ^{-2}(\Vert X^*\eta \Vert ^2+\Vert X^*J\eta \Vert ^2)-\\&\quad 2\bigl [(X\eta |JX^*J\eta )+(X^*\eta |JXJ\eta )\bigr ]. \end{aligned} \end{aligned}$$

$\square$

Corollary 3.2

(Lower bound) The following lower bounds hold true for any $\varepsilon ,\delta >0$ and any $\eta \in {\tilde{{\mathcal {F}}}}_X$

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_X^\lambda [\eta ]&\ge \left( \lambda ^2-\varepsilon ^2\right) \Vert X\eta \Vert ^2 + \left( \lambda ^2-\delta ^{-2}\right) \Vert XJ\eta \Vert ^2+(\lambda ^{-2}-\delta ^2)\Vert X^*\eta \Vert ^2 +\\&\quad (\lambda ^{-2}-\varepsilon ^{-2})\Vert X^*J\eta |\Vert ^2. \end{aligned} \end{aligned}$$

(3.7)

In particular, for $\varepsilon =\delta =1$ and any $\eta \in \tilde{\mathcal {F}}_X$ we have

$$\begin{aligned} {\mathcal {E}}_X^\lambda [\eta ]\ge (\lambda ^2-1)\left( \Vert X\eta \Vert ^2 + \Vert XJ\eta \Vert ^2\right) +(\lambda ^{-2}-1)\left( \Vert X^*\eta \Vert ^2 + \Vert X^*J\eta |\Vert ^2\right) . \end{aligned}$$

(3.8)

Proof

The result follows from (3.1) and the identities, valid for $\varepsilon ,\delta >0$,

$$\begin{aligned} \varepsilon ^2\Vert X\eta \Vert ^2 +\varepsilon ^{-2}\left\| j(X^*)\eta \right\| ^2-\left[ (X\eta |j(X^*)\eta )+(X^*\eta |j(X)\eta )\right] =\Vert d_X^\varepsilon \eta \Vert ^2\ge 0 \\ \delta ^2\left\| X^*\eta \right\| ^2 +\delta ^{-2}\left\| j(X)\eta \right\| ^2-\left[ (X^*\eta |j(X)\eta )+(X\eta |j(X^*)\eta )\right] =\Vert d_{X^*}^\delta \eta \Vert ^2\ge 0. \end{aligned}$$

$\square$

We address now the problem to find conditions on (X, D(X)) sufficient to guarantee that the lower bounds above are coercive for our Dirichlet form. By this we mean bounds in which the Dirichlet form dominates a quadratic form with a certain degree of discreteness of the spectrum such as existence and finite degeneracy of a ground state, spectral gaps or emptiness of essential spectrum. The conditions will be formulated in terms of relative smallness of the quadratic form of the self-commutator $[X,X^*]$ with respect to the quadratic form of $X^*X$ and they will be exploited in Sect. 5 when M is a type I$_\infty$ factor.

Let us denote by $(t_X,D(t_X))$ and $(t_{X^*},D(t_X^*))$, the densely defined, positive, closed quadratic forms defined as

$$\begin{aligned} \begin{aligned} t_X[\eta ]&:=\Vert X\eta \Vert ^2\qquad \eta \in D(t_X):=D(X),\\ t_{X^*}[\eta ]&:=\Vert X^*\eta \Vert ^2\qquad \eta \in D(t_{X^*}):=D(X^*), \end{aligned} \end{aligned}$$

whose associated positive, self-adjoint operators are $(X^*X,D(X^*X))$ and $(XX^*,D(XX^*))$.

Consider also the quadratic form $(\tilde{q}_X^\lambda ,D({{\tilde{q}}}_X^\lambda ))$ given by

$$\begin{aligned} \tilde{q}_X^\lambda [\eta ]:=(\lambda ^2-1)\Vert X\eta \Vert ^2+(\lambda ^{-2}-1)\Vert X^*\eta \Vert ^2\qquad \eta \in D({{\tilde{q}}}_X^\lambda ):=D(X)\cap D(X^*). \end{aligned}$$

By the densely defined quadratic form $(q_0, D(q_0))$ defined as

$$\begin{aligned} q_0[\eta ]:=t_{X^*}[\eta ]-t_X[\eta ]=\Vert X^*\eta \Vert ^2-\Vert X\eta \Vert ^2\qquad \eta \in D(q_0):=D(X)\cap D(X^*), \end{aligned}$$

on $D({{\tilde{q}}}_X^\lambda )=D(X)\cap D(X^*)$ we can write

$$\begin{aligned} {{\tilde{q}}}_X^\lambda =\left( \lambda -\lambda ^{-1}\right) ^2\cdot t_X+\left( \lambda ^{-2}-1\right) \cdot q_0=\left( \lambda -\lambda ^{-1}\right) ^2\cdot t_{X^*}+\left( 1-\lambda ^2\right) \cdot q_0 \end{aligned}$$

and regard ${{\tilde{q}}}_X^\lambda$ as a perturbation of a multiple of $t_X$ or $t_{X^*}$ by a multiple of $q_0$. Notice that $q_0$ is the form of the self-commutator $[X,X^*]=XX^*-X^* X$, at least on $D(X^*X)\cap D(XX^*)$.

Using the quadratic form $(\tilde{{\mathcal {Q}}}^\lambda _X,{\tilde{{\mathcal {F}}}}_X)$ given by

$$\begin{aligned} \tilde{{\mathcal {Q}}}^\lambda _X[\eta ]:={{\tilde{q}}}_X^\lambda [\eta ]+\tilde{q}_X^\lambda [J\eta ]\qquad \eta \in {\tilde{{\mathcal {F}}}}_X=D(X)\cap D(X^*)\cap J(D(X)\cap D(X^*)), \end{aligned}$$

the lower bound (3.8) can be written as

$$\begin{aligned} \tilde{{\mathcal {Q}}}^\lambda _X[\eta ]\le {\tilde{{\mathcal {E}}}}^\lambda _X[\eta ]\qquad \eta \in {\tilde{{\mathcal {F}}}}_X. \end{aligned}$$

(3.9)

Although $\tilde{{\mathcal {Q}}}^\lambda _X$ is densely defined, since $i_0(M_0)=M_0\xi _0\subset {\tilde{{\mathcal {F}}}}_X$ by Lemma 2.8 ii), it is not necessarily lower bounded, closable or a proper functional.

For sake of clarity, we recall some definition we will use concerning lower bounded quadratic forms $({\mathcal {A}},D({\mathcal {A}}))$, $({\mathcal {B}},D({\mathcal {B}}))$ and their associated self-adjoint operators (A, D(A)), (B, D(B)) on a Hilbert space h (see [19]):

• (i) $({\mathcal {A}},D({\mathcal {A}}))$ is $\varepsilon$-bounded w.r.t. $({\mathcal {B}},D({\mathcal {B}}))$ for $\varepsilon >0$, if $D({\mathcal {B}})\subseteq D({\mathcal {A}})$ and ${\mathcal {A}}[\xi ]\le \varepsilon \cdot {\mathcal {B}}[\xi ]+b_\varepsilon \cdot \Vert \xi \Vert ^2$ for some $b_\varepsilon \ge 0$ and all $\xi \in D({\mathcal {B}})$; the infimum of all such $\varepsilon$ is the form bound of $({\mathcal {A}},D({\mathcal {A}}))$ w.r.t. $({\mathcal {B}},D({\mathcal {B}}))$;

• (ii) $({\mathcal {A}},D({\mathcal {A}}))$ is small (resp. infinitesimally small) w.r.t. $({\mathcal {B}},D({\mathcal {B}}))$ if its form bound is strictly less than one (resp. vanishes);

• (iii) (A, D(A)) is $\varepsilon$-bounded w.r.t. (B, D(B)) for $\varepsilon >0$, if $D(B)\subseteq D(A)$ and $\Vert A\xi \Vert ^2\le \varepsilon \cdot \Vert B\xi \Vert ^2+b_\varepsilon \cdot \Vert \xi \Vert ^2$ for some $b_\varepsilon \ge 0$ and all $\xi \in D(B)$; the infimum of all such $\varepsilon$ is the operator bound of (A, D(A)) w.r.t. (B, D(B));

• (iv) (A, D(A)) is small (resp. infinitesimally small) w.r.t. (B, D(B)) if its operator bound is strictly less than one (resp. vanishes);

• (v) $({\mathcal {A}},D({\mathcal {A}}))$ is said an infinitesimal perturbation of $({\mathcal {B}},D({\mathcal {B}}))$ if $D({\mathcal {B}})\subseteq D({\mathcal {A}})$ and $({\mathcal {A}}-{\mathcal {B}},D({\mathcal {B}}))$ is infinitesimally small w.r.t. $({\mathcal {B}},D({\mathcal {B}}))$;

• (vi) (A, D(A)) is said infinitesimally perturbation of (B, D(B)) if $D(B)\subseteq D(A)$ and $(A-B,D(B))$ is infinitesimally small with respect to (B, D(B)); It is well known that (iii) implies (i), (iv) implies (ii) and (vi) implies (v);

• (vii) (A, D(A)) has purely discrete spectrum if this is made by discrete eigenvalues only (isolated eigenvalues of finite degeneracy); by the Min-Max Theorem this holds true if and only if $({\mathcal {A}},D({\mathcal {A}}))$ is proper in the sense that $\{\xi \in D({\mathcal {A}}): \Vert \xi \Vert \le 1,{\mathcal {A}}[\xi ]\le 1\}$ is relatively compact in h.

Theorem 3.3

(Coercivity) Assume $({{\tilde{q}}}_X^\lambda ,D(X)\cap D(X^*))$ to be lower bounded and closable, denote by $(q^\lambda _X,D(q^\lambda _X))$ its closure and by $(Q^\lambda _X,D(Q^\lambda _X))$ the associated lower bounded, self-adjoint operator. Then

• (i) $(\tilde{{\mathcal {Q}}}^\lambda _X,{\tilde{{\mathcal {F}}}}_X)$ is lower bounded, closable and its closure $({{\mathcal {Q}}}^\lambda _X,D({{\mathcal {Q}}}^\lambda _X))$ bounds the Dirichlet form

  $$\begin{aligned} {{\mathcal {Q}}}^\lambda _X[\eta ]\le {\mathcal {E}}^\lambda _X[\eta ]\qquad \eta \in {\mathcal {F}}^\lambda _X\subseteq D({{\mathcal {Q}}}^\lambda _X); \end{aligned}$$

  (3.10)

  if moreover, the self-adjoint operator associated to $({{\mathcal {Q}}}^\lambda _X,D({{\mathcal {Q}}}^\lambda _X))$ has discrete spectrum, then the spectrum of the self-adjoint operator $(H^\lambda _X,D(H^\lambda _X))$ associated to $({\mathcal {E}}^\lambda _X,{\mathcal {F}}^\lambda _X)$ is discrete too.

• (ii) $(Q^\lambda _X,D(Q^\lambda _X))$ is affiliated to M, $(j(Q^\lambda _X),JD(Q^\lambda _X))$ is affiliated to $M'$ and $D(Q^\lambda _X)\cap JD(Q^\lambda _X)$ is dense in $L^2(M)$. Assume now on $D(X)=D(X^*)$, $D(X^*X)=D(XX^*)$ and the quadratic form $(q_0, D(q_0))$ to be infinitesimally small with respect to $(t_X,D(t_X))$. Then

• (iii) the form $({{\tilde{q}}}_X^\lambda ,D(X))$ is lower bounded, closed and $(Q^\lambda _X,D(Q^\lambda _X))$ equals the Friedrichs extension of the lower bounded, densely defined, symmetric operator

  $$\begin{aligned} \begin{aligned} D(N^\lambda _X)&:=D(X^*X)=D(XX^*)\\ N^\lambda _X&:=(\lambda -\lambda ^{-1})^2\cdot X^*X+(\lambda ^{-2}-1)\cdot [X,X^*]; \end{aligned} \end{aligned}$$

  (3.11)

• (iv) in particular, the conclusions in (iii) subsist if the self-commutator $([X,X^*],D(X^*X))$ is infinitesimally small w.r.t. $(X^*X,D(X^*X))$ and in this case

  $$\begin{aligned} \left( Q^\lambda _X,D(Q^\lambda _X)\right) =\left( N^\lambda _X,D(X^*X)\right) . \end{aligned}$$

  (3.12)

  If moreover the spectrum of $(X^*X,D(X^*X))$ is discrete, then the spectrum of the generator $(H^\lambda _X,D(H^\lambda _X))$ of the Dirichlet form is discrete too.

Proof

• (i) Since $({{\tilde{q}}}_X^\lambda ,D(X)\cap D(X^*))$ is lower bounded and closable and J is isometric, the quadratic form $J(D(X)\cap D(X^*))\ni \eta \mapsto {{\tilde{q}}}_X^\lambda [J\eta ]$ is densely defined, lower bounded and closable too. This implies that $(\tilde{{\mathcal {Q}}}^\lambda _X,{\tilde{{\mathcal {F}}}}_X)$ is lower bounded and closable as a sum of forms sharing these same properties. The lower bound (3.10) follows from (3.9) and the lower boundedness of $(\tilde{{\mathcal {Q}}}^\lambda _X,{\tilde{{\mathcal {F}}}}_X)$. The last assertion concerning discreteness of spectra follows from the Min-Max Theorem.

• (ii) Since (X, D(X)) and $(X^*,D(X^*))$ are closed operators affiliated to M, it follows that for any unitary $u'\in M'$ we have $u'(D(X)\cap D(X^*))\subset D(X)\cap D(X^*)$ and $\tilde{q}_X^\lambda [u'\eta ]={{\tilde{q}}}_X^\lambda [\eta ]$ for any $\eta \in D(X)\cap D(X^*)$. By approximation, these invariance still hold true for the closure $(q^\lambda _X,D(q^\lambda _X))$ and implies that for all unitaries $u'\in M'$ and all $\eta \in D(Q^\lambda _X)$ one has $u'\eta \in D(Q^\lambda _X)$ and $Q^\lambda _Xu'\eta =u' Q^\lambda _X\eta$. Hence $(Q^\lambda _X,D(Q^\lambda _X))$ is affiliated to M, $(j(Q^\lambda _X),D(j(Q^\lambda _X)))$ is affiliated to $M'$, the operators strongly commute and have a common dense core.

• (iii) Since $D(X)=D(X^*)$ and $(q_0,D(X)\cap D(X^*))=(q_0,D(X))$ is infinitesimally small with respect to $(t_X,D(X))$, the sum $\tilde{q}_X^\lambda =(\lambda -\lambda ^{-1})^2\cdot t_X+(\lambda ^{-2}-1)\cdot q_0$ is lower bounded and closed since $(t_X,D(X))$ is lower bounded and closed. Since $q_0=t_{X^*}-t_X$ is infinitesimally small with respect to $t_X$ on the common domain $D(X)=D(X^*)$, we have that $t_{X^*}$ is relatively bounded with respect to $t_X$ and that $t_{X}$ is relatively bounded with respect to $t_{X^*}$. As $D(X^*X)=D(XX^*)$ by assumption, the symmetric operator $(N^\lambda _X,D(N^\lambda _X))$ is densely defined and lower bounded since its quadratic form is the restriction of the lower bounded form $({{\tilde{q}}}_X^\lambda ,D(X))$ to $D(X^*X)$, i.e. $(\eta |N^\lambda _X\eta )={{\tilde{q}}}_X^\lambda [\eta ]$ for all $\eta \in D(X^*X)$. Since $D(X^*X)$ is form core for $(t_X,D(X))$ and $({{\tilde{q}}}_X^\lambda ,D(X))$ is an infinitesimal perturbation of a multiple of it, $D(X^*X)$ is a form core for $({{\tilde{q}}}_X^\lambda ,D(X))$ too. Since, by definition, the Friedrichs extension of $(N^\lambda _X,D(X^*X))$ is the self-adjoint operator associated to the closure of its quadratic form $(\tilde{q}_X^\lambda ,D(X^*X))$, it results that $(Q^\lambda _X,D(Q^\lambda _X))$ coincides with it.

• iv) In this case the operator $(N^\lambda _X,D(N^\lambda _X))$ is an infinitesimal symmetric perturbation of a multiple of the self-adjoint operator $(X^*X,D(X^*X))$ and it is self-adjoint by the Kato-Rellich Theorem. Since it is also lower bounded, it has to coincides with its Friedrichs extension $(Q^\lambda _X,D(Q^\lambda _X))$. To prove the last assertion, recall that the spectrum of a lower bounded self-adjoint operator is discrete if and only if its associated quadratic form is proper (see [15]). Now, by a general corollary of the Min-Max Theorem, if the spectrum of $(\lambda -\lambda ^{-1})^2X^*X$ is discrete, then the spectrum of $N^\lambda _X$ is discrete too, as the latter operator is the sum of the former and the lower bounded self-adjoint operator $(\lambda ^{-2}-1)[X,X^*]$, all with domain $D(X^*X)$. Hence, the lower bounded, closed quadratic form $({{\tilde{q}}}_X^\lambda ,D(X))$ of $(N^\lambda _X,D(X^*X))$ is a proper functional and consequently the lower bounded, closed form $({{\mathcal {Q}}}^\lambda _X,D({{\mathcal {Q}}}^\lambda _X))$ is proper too, as a sum of proper functionals. The lower bound (3.10) then implies that the Dirichlet form is a proper functional.

$\square$

4 Superboundedness of a class of semigroups on type I von Neumann algebras

In this section we introduce a further continuity property, called superboundedness, for positivity preserving semigroups on standard forms of $\sigma$-finite von Neumann algebras, showing that the property is owned by a class of semigroups on type I$_\infty$ factors. Also we show how this property persists under domination of positivity preserving semigroups.

As usual, $i_0:M\rightarrow L^2(M)$ denotes the symmetric embedding of a $\sigma$-finite von Neumann algebra M endowed with a faithful normal state $\omega _0\in M_{*+}$ represented by $\xi _0\in L^2_+(M)$.

Definition 4.1

(Excessive vectors and superboundedness)

• (i) The vector $\xi _0\in L^2_+(M)$ is $(\gamma _0,t_0)$-excessive or excessive, for some $\gamma _0, t_0\ge 0$, with respect to a positivity preserving semigroup $\{T_t:t\ge 0\}$ on $L^2(M)$ if the maps $e^{-\gamma _0 t}T_t$ are Markovian w.r.t. $\xi _0$ for any $t>t_0$. Markovian semigroups are just those for which $\xi _0$ is (0, 0)-excessive;

• ii) a positivity preserving semigroup $\{T_t:t\ge 0\}$ is superbounded if for some $\gamma _0, t_0\ge 0$

• (a) $\xi _0\in L^2_+(M)$ is $(\gamma _0,t_0)$-excessive,

• (b) $T_t(L^2(M))\subseteq i_0(M)$ for all $t>t_0$.

If we endow the subspace $i_0(M)\subseteq L^2(M)$ by the norm of the von Neumann algebra, i.e. $\Vert i_0(x)\Vert _M:=\Vert x\Vert _M$ for $x\in M$, then superboundedness implies the boundedness of $T_t$ as a map from $(L^2(M),\Vert \cdot \Vert _2)$ to $(i_0(M),\Vert \cdot \Vert _M)$ for all $t>t_0$. In fact, by the norm continuity of the symmetric embedding $i_0:M\rightarrow L^2(M)$, the norm $\Vert \cdot \Vert _M$ is stronger than the Hilbert norm $\Vert \cdot \Vert _2$ so that the continuous maps $T_t:L^2(M)\rightarrow L^2(M)$ are closed when considered from the Hilbert space $L^2(M)$ to the Banach space $(i_0(M),\Vert \cdot \Vert _M)$ and, by the Closed Graph Theorem, they result to be bounded (notice that this involves only condition (b) in Definition 4.1).

We shall refer to part (b) of superboundedness writing $\Vert T_t\Vert _{L^2(M)\rightarrow M}<+\infty$ for all $t>t_0$ and to part b) of supercontractivity writing $\Vert T_t\Vert _{L^2(M)\rightarrow M}\le 1$ for all $t>t_0$.

By the Markovianity of $e^{-\gamma _0 t} T_t$ required in (i), bounded, positivity preserving maps $S_t:M\rightarrow M$ satisfying the relations $i_0(S_t(x))=T_t(i_0(x))$ for $x\in M$ are well defined and one has, for suitable scalars $b_t\ge 0$,

$$\begin{aligned} \Vert S_t\Vert \le e^{\gamma _0 t},\qquad \Vert S_t x\Vert _M\le b_t\cdot \Vert i_0(x)\Vert _{L^2(M)}\qquad x\in M,\quad t>t_0. \end{aligned}$$

Consider the noncommutative spaces $L^p(M,\omega _0)$ for $p\in [2,+\infty ]$ defined by the symmetric embedding $i_0:M\rightarrow L^2(M)$ (see [21]). By complex interpolation it follows that a superbounded semigroup is hypercontractive too in the sense that there exists $T_0\ge 0$ such that $T_t$ is bounded from $L^2(M)$ to $L^4(M,\omega _0)$ for $t>T_0$.

The following observation will be useful later on.

Lemma 4.2

(Superboundedness by domination) Let $\{e^{-tG_0}:t\ge 0\}$ be a superbounded semigroup on $L^2(M)$ such that, for some $\gamma _0, t_0\ge 0$

$$\begin{aligned} \xi _0\in L^2_+(M)\,\, \text {is }(\gamma _0,t_0)\text {-excessive}. \end{aligned}$$

Let $\{e^{-tG_1}:t\ge 0\}$ be a $C_0$-continuous, self-adjoint, positivity preserving semigroup such that, for some $\gamma _1, t_1\ge 0$

$$\begin{aligned} \xi _0\in L^2_+(M)\,\, \text {is }(\gamma _1,t_1)\text {-excessive.} \end{aligned}$$

If the semigroup $\{e^{-tG_1}:t\ge 0\}$ is dominated by the semigroup $\{e^{-tG_0}:t\ge 0\}$ in the sense

$$\begin{aligned} e^{-tG_1}\eta \le e^{-tG_0}\eta \qquad \eta \in L^2_+(M),\quad t\ge 0, \end{aligned}$$

(4.1)

then $\{e^{-tG_1}:t\ge 0\}$ is superbounded with

$$\begin{aligned} \left\| e^{-tG_1}\eta \right\| _M\le \left\| e^{-tG_0}\right\| _{L^2(M)\rightarrow M}\cdot \Vert \eta \Vert _2,\qquad \eta \in L^2_+(M),\quad t>t_0\vee t_1. \end{aligned}$$

(4.2)

Proof

The superboundedness of $\{e^{-tG_0}:t\ge 0\}$ and the domination (4.1) imply that $e^{-tG_1}(L^2_+(M))\subset i_0(M_+)$ for any $t>t_0\vee t_1$. Since $L^2_+(M)$ linearly generates $L^2(M)$, it follows that $e^{-tG_1}(L^2(M))\subseteq i_0(M)$ for all $t>t_0\vee t_1$ so that $\{e^{-tG_1}:t\ge 0\}$ is superbounded. The bound (4.2) follows from the domination (4.1) and the superboundedness of $\{e^{-tG_0}:t\ge 0\}$. $\square$

4.1 A class of superbounded Markovian semigroups on a type I$_\infty$ factor

Let h be a Hilbert space and consider the type I factor $M:=B(h)$. Its (Hilbert-Schmidt) standard representation acts, by left composition, on the space $L^2(M)=L^2(h)$ of Hilbert-Schmidt operators on h, where the standard cone $L^2_+(M)=L^2_+(h)$ is that of operators in $L^2(h)$. The modular involution is given by the operator adjoint: $J\xi :=\xi ^*$ for $\xi \in L^2(h)$ and the right representation of B(h) on $L^2(h)$ is given by right composition.

Let $H_0$ be a lower bounded, self-adjoint operator affiliated to B(h) (i.e. any self-adjoint, lower bounded operator on h) and consider the strongly continuous semigroup on $L^2(h)$ given by

$$\begin{aligned} T_t\eta =e^{-tH_0}J\left( e^{-tH_0}J(\eta )\right) =e^{-tH_0}\circ \eta \circ e^{-tH_0}\qquad \eta \in L^2(h). \end{aligned}$$

Its self-adjoint generator $G_0$ on $L^2(h)$, defined by $G_0(\xi ):=\lim _{t\rightarrow 0} t^{-1}(\xi -T_t\xi )$ on the subspace $D(G_0)\subset L^2(h)$ for whose vectors the limit exists, coincides with the generalized sum $H_0\dot{+}JH_0J$ (see [19]) of the closed operators $H_0$ and $JH_0J$, affiliated to the commuting von Neumann algebras given by the left and right representations of B(h) on $L^2(h)$ (see Lemma 7.1 in Appendix). The operator $H_0$, resp. $JH_0J$, is considered here as acting on a suitable dense subspace of the Hilbert–Schmidt space $L^2(h)$ by left, resp. right, composition. For example, $G_0(\xi )=\overline{H_0\circ \xi } + \overline{\xi \circ H_0}\in L^2(h)$ for those $\xi \in L^2(h)$ such that the operators $H_0\circ \xi$ and $\xi \circ H_0$ are densely defined, closable and bounded on their domains and their closures are Hilbert-Schmidt operators. To ease notation, the operators $H_0\circ \xi , \xi \circ H_0$ will be represented by the juxtaposition $H_0\xi ,\xi H_0$ of the symbols of the operators $H_0$ and $\xi$ so that, the formula above appears $G_0(\xi )=\overline{H_0\xi }+\overline{\xi H_0}$. For further details on Hilbert-Schmidt standard form we refer to [12] Section 2.

Lemma 4.3

If $H_0$ has discrete spectrum $\textrm{Sp}(H_0):=\{\lambda _j:j\in {\mathbb {N}}\}$¹ with the increasing eigenvalues written with repetitions according to the their multiplicity, then

• (i) $G_0$ has discrete spectrum too given by $\textrm{Sp}(G_0):=\{\lambda _j+\lambda _k\in {\mathbb {R}}:(j,k)\in {\mathbb {N}}\times {\mathbb {N}}\}$;

• (ii) if $n_{H_0}(\lambda ):=\natural \{j\in {\mathbb {N}}:\lambda _j\le \lambda \}$ is the eigenvalue counting function of $H_0$, then the eigenvalue counting function of $G_0$ is bounded by $n_{G_0}(\lambda )\le (n_{H_0}(\lambda -\lambda _0))^2$, $\lambda \in {\mathbb {R}}$.

Proof

Let $H_0=\sum _{k=0}^\infty \lambda _k P_k$ be the spectral decomposition of $H_0$ as an operator acting on h. Then the spectral decomposition of $G_0$ is given by

$$\begin{aligned} G_0=\sum _{j,k=0}^\infty (\lambda _j+\lambda _k) P_j JP_kJ, \end{aligned}$$

since $\{P_j JP_kJ:j,k\ge 0\}$ is a complete family of mutually orthogonal projections acting on the standard Hilbert space $L^2(h)$ such that

$$\begin{aligned} \begin{aligned} (H_0+JH_0J)P_jJP_kJ&=H_0P_jJP_kJ+P_jJH_0P_kJ=\lambda _jP_jJP_kJ+\lambda _k P_j P_k J \\&=(\lambda _j+\lambda _k)P_j JP_kJ. \end{aligned} \end{aligned}$$

Thus $G_0$ has the discrete spectrum indicated in the statement and since $\lambda _j+\lambda _k\le \lambda$ implies both $\lambda _j+\lambda _0\le \lambda$ and $\lambda _0+\lambda _k\le \lambda$, the bound $n_{G_0}(\lambda )\le n_N(\lambda -\lambda _0)^2$ holds true for $\lambda \in {\mathbb {R}}$. $\square$

Suppose now the lower bounded, self-adjoint operator $H_0$ on h to have a discrete spectrum $\textrm{Sp}(H_0):=\{\lambda _j:j\in {\mathbb {N}}\}$ such that, for some $\beta >0$,

$$\begin{aligned} \textrm{Tr}(e^{-\beta H_0})=\sum _{k=0}^\infty e^{-\beta \lambda _k}<+\infty , \end{aligned}$$

so that the Gibbs state on B(h) with density matrix $\rho _\beta :=e^{-\beta H_0}/\textrm{Tr}(e^{-\beta H_0})$ is well defined

$$\begin{aligned} \omega _\beta (x):=\textrm{Tr}(x\rho _\beta )\qquad x\in B(h) \end{aligned}$$

and its representative positive vector is given by $\xi _0:=\rho _\beta ^{1/2}\in L^2_+(h)$. Recall that in this case the symmetric embedding $i_0:B(h)\rightarrow L^2(h)$ is given by $i_0(x)=\rho _\beta ^{1/4}x\rho _\beta ^{1/4}$ for $x\in B(h)$.

Theorem 4.4

• (i) The $C_0$-continuous, self-adjoint semigroup $\{e^{-tG_0}:t>0\}$ is positive preserving and $\xi _0:=\rho _\beta ^{1/2}\in L^2_+(h)$ is $(-2(\lambda _0\wedge 0),0)$-excessive;

• (ii) the semigroup $\{e^{-tG_0}:t>0\}$ is superbounded with

  $$\begin{aligned} \Vert e^{-tG_0}\Vert _{L^2(M)\rightarrow M}\le e^{-(2t-\beta /2)\lambda _0}\qquad t>\beta /4. \end{aligned}$$

  In particular, if $\lambda _0\ge 0$, the semigroup is Markovian and supercontractive.

Proof

Replacing $H_0$ with $H_0+\beta ^{-1}\ln \textrm{Tr}(e^{-\beta H_0})$, we may just consider the case $\textrm{Tr}(e^{-\beta H_0})=1$.

• (i) If $\xi \in L^2_+(h)$, since $e^{-tH_0}$ is self-adjoint, we have $e^{-tG_0}\xi =(e^{-tH_0})^*\xi e^{-tH_0}\in L^2_+(h)$ for any $t\ge 0$, showing that the semigroup is positivity preserving. Since $\lambda _0\le H_0$ and $\beta >0$, for any $t\ge 0$ we have

  $$\begin{aligned} \begin{aligned} e^{-tG_0}\xi _0&=e^{-tH_0}\rho _\beta ^{1/2}e^{-tH_0}=e^{-\beta H_0/4}e^{-2tH_0}e^{-\beta H_0/4}\\&\le e^{-2t\lambda _0}e^{-\beta H_0/4}e^{-\beta H_0/4}= e^{-2t\lambda _0}\xi _0 \end{aligned} \end{aligned}$$

  so that $\xi _0$ is $(-2(\lambda _0\wedge 0),0)$-excessive.

• (ii) For $\xi \in L^2(h)$ and $t>\beta /4$ we have $x:=e^{-(t-\beta /4)H_0}\xi e^{-(t-\beta /4)H_0}\in B(h)$ and

  $$\begin{aligned} \begin{aligned} i_0(x)&=\rho _\beta ^{1/4}x\rho _\beta ^{1/4}=e^{-\beta H_0/4}e^{-(t-\beta /4)H_0}\xi e^{-(t-\beta /4)H_0} e^{-\beta H_0/4} \\&=e^{-tH_0}\xi e^{-tH_0}=e^{-tG_0}\xi . \end{aligned} \end{aligned}$$

  Since for $t>\beta /4$ we have $\Vert e^{-(t-\beta /4)H_0}\Vert _{B(h)}\le e^{-(t-\beta /4)\lambda _0}$, we get

  $$\begin{aligned} \begin{aligned} \Vert x\Vert _{B(h)}&\le \Vert x\Vert _{L^2(h)}=\Vert e^{-(t-\beta /4)H_0}\xi e^{-(t-\beta /4)H_0}\Vert _{L^2(h)} \\&\le \Vert e^{-(t-\beta /4)H_0}\Vert _{B(h)}\Vert \xi \Vert _{L^2(h)}\Vert e^{-(t-\beta /4)H_0}\Vert _{B(h)} \\&\le e^{-(2t-\beta /2)\lambda _0}\cdot \Vert \xi \Vert _{L^2(h)}. \end{aligned} \end{aligned}$$

$\square$

5 General quantum Ornstein–Uhlenbeck semigroups

In this section we apply the above framework to construct a family of Dirichlet forms and Markovian semigroups, a special case of which is the quantum Ornstein–Uhlenbeck semigroup studied in [12]. While in [12] we computed explicitly the spectrum of the generator and proved the Feller property with respect to the algebra of compact operators, here we prove, for each semigroups we construct, subexponential spectral growth rate and domination with respect to positivity preserving semigroups belonging to a natural related class (see Appendix 7.1).

On the Hilbert space $h:=l^2({\mathbb {N}})$, consider the C$^*$-algebra of compact operators ${\mathcal {K}}(h)$. The Number Operator (N, D(N)), defined by the natural basis $e:=\{e_k\in l^2({\mathbb {N}}):k\in \mathbb {N}\}$ as

$$\begin{aligned} D(N):=\left\{ \sum _{k\in \mathbb {N}}c_k\cdot e_k:\sum _{k\in \mathbb {N}} k^2\cdot |c_k|^2<+\infty \right\} \qquad Ne_k:=ke_k\qquad k\in {\mathbb {N}}, \end{aligned}$$

generates the $C_0$-continuous group of automorphisms $\alpha :=\{\alpha _t\in \textrm{Aut}({\mathcal {K}}(h)):t\in {\mathbb {R}}\}$

$$\begin{aligned} \alpha _t(B):=e^{itN}Be^{-itN}\qquad B\in {\mathcal {K}}(h),\quad t\in {\mathbb {R}}. \end{aligned}$$

For any $\beta >0$ there exists a unique $(\alpha ,\beta )$-KMS state $\omega _\beta$, satisfying the KMS condition

$$\begin{aligned} \omega _\beta (A\alpha _{i\beta }(B))=\omega _\beta (BA) \end{aligned}$$

for $\alpha$-analytic elements A, B, given by, in terms of the density matrix,

$$\begin{aligned} \rho _\beta :=\left( 1-e^{-\beta }\right) e^{-\beta N}=\left( 1-e^{-\beta }\right) \sum _{k\in {\mathbb {N}}}e^{-\beta k} p_k,\quad \omega _\beta (A):=\textrm{Tr}(A\rho _\beta ),\quad A\in {\mathcal {K}}(h) \end{aligned}$$

($p_k$ being the projection onto ${\mathbb {C}}e_k$). The von Neumann algebra M generated by the GNS representation of $\omega _\beta$ can be identified with B(h) and the normal extension of $\omega _\beta$ on it is still given by the formula above for any $A\in B(h)$. The extension of the automorphisms group $\alpha$ to a $C_0^*$-continuous group on B(h) is given by the same formula above on ${\mathcal {K}}(h)$.

In the Hilbert–Schmidt standard form of $M:=B(h)$ described in Sect. 4.1, the cyclic and separating vector representing $\omega _\beta$ is given by

$$\begin{aligned} \xi _0:=\rho _\beta ^{1/2}=\sqrt{1-e^{-\beta }}e^{-\beta N/2}\in L^2_+(h). \end{aligned}$$

The action of the Hilbert algebra unbounded conjugation operator $S_0$ on $L^2(h)$, characterized as $S_0(x\xi _0):=x^*\xi _0$ for $x\in B(h)$, can be identified on a suitable domain $D(S_0)\subset L^2(h)$ with

$$\begin{aligned} S_0(\eta )=\overline{\rho _\beta ^{-1/2}\eta ^*\rho _\beta ^{1/2}} \end{aligned}$$

and its polar decomposition $S_0=J\Delta _0^{1/2}$ is provided by the modular operator

$$\begin{aligned} \Delta _0^{1/2}(\eta )=\overline{\rho _\beta ^{1/2}\eta \rho _\beta ^{-1/2}}=\overline{e^{-\beta N/2}\eta e^{\beta N/2}}\qquad \eta \in D(S_0). \end{aligned}$$

The modular group of $\omega _\beta$, satisfying the modular condition $\omega _\beta (A\sigma ^{\omega _\beta }_{-i}(B))=\omega _\beta (BA)$, for analytics elements A, B, is then given by $\sigma ^{\omega _\beta }_t=\alpha _{-\beta t}$ for $t\in {\mathbb {R}}$. Regarding the Number Operator N as an operator affiliated to B(h) in its normal representation on $L^2(h)$ (i.e. acting, on a suitable domain of the Hilbert–Schmidt operators, by left composition), we have that the modular (Araki) Hamiltonian is given by the strong sum of the densely defined, self-adjoint operators N and $-JNJ$ (belonging to commuting von Neumann algebras)

$$\begin{aligned} -\ln \Delta _0=\beta \overline{N-JNJ} \end{aligned}$$

and its (discrete) spectrum is given by $\textrm{Sp}(-\ln \Delta _0)=\beta {\mathbb {Z}}$. Consequently $\textrm{Sp}(\Delta _0^{1/2})=e^{\beta {\mathbb {Z}}/2}$ with uniform multiplicity one.

Let us consider the annihilation and creation operators $(A,D(A)), (A^*,D(A^*))$ on h, defined on the domain $D(A):=D(\sqrt{N})=:D(A^*)$ as

$$\begin{aligned} Ae_0:=0,\quad Ae_k:=\sqrt{k}e_{k-1}\quad \hbox { if}\ k\ge 1,\qquad A^*e_k:=\sqrt{k+1}e_{k+1}\qquad k\in \mathbb {N}. \end{aligned}$$

They satisfy the Canonical Commutation Relations $AA^*=A^* A+I$, as closed operators defined on D(N), and allow to represent the Number Operator as $N=A^*A$. All these operators and their functional calculi are understood as affiliated to B(h) acting by left composition on operators belonging to the Hilbert-Schmidt class $L^2(h)$.

Let us consider the family of operators affiliated to B(h)

$$\begin{aligned} D(X_m)=D\left( N^{m/2}\right) \qquad X_m:=(A^*)^m\qquad m\in {\mathbb {N}}\setminus \{0\}. \end{aligned}$$

(5.1)

Lemma 5.1

• (i) For any $m\ge 1$ and $\lambda _m^2:=e^{-m\beta /2}$ we have

  $$\begin{aligned} \begin{aligned} D(X_m)&=D\left( X_m^*\right) =D\left( N^{m/2}\right) ,\quad D\left( X_m^*X_m\right) = D\left( X_mX_m^*\right) =D(N^m)\\ X_{m}^*X_{m}&=A^{m}(A^*)^{m}=(N+m)(N+m-1)\cdots (N+2)(N+1)\\ X_mX_m^*&= (A^*)^{m}A^{m} = N(N-1)(N-2)\cdots (N-(m-1)) \end{aligned} \end{aligned}$$

  (5.2)

  and the self-commutator $([X_m,X^*_m], D(N^{m}))$ is a self-adjoint operator, infinitesimally small with respect to $(X_m^*X_m,D(N^m))$ and $(N^m,D(N^m))$;

• (ii) $X_m\xi _0=(A^*)^m\xi _0\in L^2(h)$ is an eigenvector of $\Delta _0^{1/2}$ corresponding to the eigenvalue $\lambda _{m}^{2}$, $X_m^*\xi _0=(A)^m\xi _0\in L^2(h)$ is an eigenvector of $\Delta _0^{1/2}$ corresponding to the eigenvalue $\lambda _{m}^{-2}$.

Proof

(i) Formulae (5.2) follow by induction starting from the case $m=1$. They show that the self-commutator is a polynomial in N of degree $(m-1)$ and this implies the remaining conclusion. (ii) Since $A^*e_k:=\sqrt{k+1}e_{k+1}$ for $k\in {\mathbb {N}}$, we have $(A^*\xi _0)e_k=A^*(\rho _\beta ^{1/2}(e_k))=(1-e^{-\beta })^{1/2}e^{-\beta k/2}A^*e_k=(1-e^{-\beta })^{1/2}e^{-\beta k/2}\sqrt{k+1}e_{k+1}$ and then for, any $m\in {\mathbb {N}}$, we have too $((A^*)^m\xi _0)e_k=(1-e^{-\beta })^{1/2}e^{-\beta k/2}(A^*)^m e_k=(1-e^{-\beta })^{1/2}e^{-\beta k/2}\sqrt{k+1}\cdots \sqrt{k+m}\,e_{k+m}$ so that

$$\begin{aligned} \begin{aligned} \left( \Delta _0^{1/2}\left( (A^*)^m\xi _0\right) \right) e_k&=(\rho _\beta ^{1/2} \left( (A^*)^m\rho _\beta ^{1/2}\right) \rho _\beta ^{-1/2})e_k=\rho _\beta ^{1/2}\left( (A^*)^me_k\right) \\&=\sqrt{k+1}\cdots \sqrt{k+m}\,\rho _\beta ^{1/2}e_{k+m}\\&=\left( 1-e^{-\beta }\right) ^{1/2}e^{-\beta (k+m)/2}\sqrt{k+1}\cdots \sqrt{k+m}\,e_{k+m}\\&=e^{-m\beta /2}\left( (A^*)^m\xi _0\right) e_k. \end{aligned} \end{aligned}$$

Hence $(A^*)^m\xi _0\in L^2(h)$ is eigenvector of $\Delta _0^{1/2}$ corresponding to the eigenvalue $\lambda _m^2:=e^{-m\beta /2}$. The other series of eigenvalues follow from Lemma 2.9. $\square$

We are now in position to apply Theorem 2.5 with $Y=X_m$, $\lambda =e^{-m\beta /4}$, $\xi _0=\rho _\beta ^{1/2}\in L^2(l^2(\mathbb {N}))$

and consider the Dirichlet form $({\mathcal {E}}^{\lambda _m}_{X_m},{\mathcal {F}}^{\lambda _m}_{X_m})$ on $L^2(l^2(\mathbb {N}))$ and its generator $(H^{\lambda _m}_{X_m},D(H^{\lambda _m}_{X_m}))$.

The following result generalizes, in particular, some of those obtained in [12] for the quantum Ornstein–Uhlebeck semigroup, corresponding to the present parameter $m=1$.

Theorem 5.2

(Spectral growth rate) For $m\ge 1$ and $\lambda _m^2:=e^{-m\beta /2}$, the operator $(H^{\lambda _m}_{X_m},D(H^{\lambda _m}_{X_m}))$ has discrete spectrum and subexponential spectral growth rate

$$\begin{aligned} \textrm{Tr}(e^{-tH_{X_m}^{\lambda _{m}}})<+\infty \qquad t>0. \end{aligned}$$

Proof

By Lemma 5.1 (i) above, the self-adjoint operator

$$\begin{aligned} N_{X_m}^{\lambda _{m}}:=\left( \lambda _m-\lambda _m^{-1}\right) ^2 X_m^* X_m + \left( \lambda _m^{-2}-1\right) \left[ X_m, X_m^*\right] \end{aligned}$$

has, on its domain, the following explicit form

$$\begin{aligned} N_{X_m}^{\lambda _{m}}&=\left( \lambda _m^2-1\right) A^m (A^*)^m + \left( \lambda _m^{-2}-1\right) (A^*)^m A^m\nonumber \\&=\left( \lambda _m^2-1\right) (N+1)\cdots (N+m) + \left( \lambda _m^{-2}-1)N(N-1)\cdots (N-(m-1)\right) \nonumber \\&= \left( \lambda _m^2+\lambda _m^{-2}-2\right) N^m +p_{m-1}(N)= \left( \lambda _m-\lambda _m^{-1}\right) ^2N^m +p_{m-1}(N), \end{aligned}$$

(5.3)

where $p_{m-1}:{\mathbb {R}}\rightarrow {\mathbb {R}}$ is a suitable polynomial of degree $(m-1)$ with real coefficients. Since for any $\varepsilon \in (0,1)$ one has $b_m^\varepsilon :=\inf _{s\ge 0}(\varepsilon (\lambda _m-\lambda _m^{-1})^2\,s^m+p_{m-1}(s))>-\infty$ and $(\lambda _m-\lambda _m^{-1})^2=(2\sinh (m\beta /4))^2>0$, $(N_{X_m}^{\lambda _{m}},D(N^m))$ is lower bounded, self-adjoint with subexponential spectral growth rate

$$\begin{aligned} \begin{aligned} \textrm{Tr}\left( e^{-tN_{X_m}^{\lambda _{m}}}\right)&\le e^{-tb_m^\varepsilon }\textrm{Tr}\left( e^{-t(1-\varepsilon )\left( \lambda _m-\lambda _m^{-1}\right) ^2N^m}\right) \\&=e^{-tb_m^\varepsilon }\sum _{k=1}^\infty e^{-t(1-\varepsilon )\left( \lambda _m-\lambda _m^{-1}\right) ^2k^m},\qquad t>0, \end{aligned} \end{aligned}$$

by [31] Proposition 1.2.15. Applying Lemma 4.3, these same properties (having discrete spectrum and sub-exponential spectral growth rate) hold true for the sum $N_{X_m}^{\lambda _{m}}\dot{+}j(N_{X_m}^{\lambda _{m}})$. Also, since $D(X_m)=D(X_m^*)=D(N^{m/2})$ and $D(X_m^*X_m)=D(X_mX_m^*)=D(N^m)$, by Theorem 3.3 (iii) and (iv) with the notations there introduced, we deduce spectrum discreteness and growth rate for $H_{X_m}^{\lambda _m}$ too:

$$\begin{aligned} \begin{aligned} \textrm{Tr}(e^{-tH_{X_m}^{\lambda _{m}}})&\le \textrm{Tr}\left( e^{-t(N_{X_m}^{\lambda _{m}}\dot{+}j(N_{X_m}^{\lambda _{m}})}\right) = \textrm{Tr}\left( e^{-tN_{X_m}^{\lambda _{m}}}Je^{-tN_{X_m}^{\lambda _{m}}}J\right) \\&=\Bigl (\textrm{Tr}(e^{-tN_{X_m}^{\lambda _{m}}})\bigr )^2,\qquad t>0. \end{aligned} \end{aligned}$$

$\square$

Theorem 5.3

(Domination) For $m\ge 1$ and $\lambda _m^2:=e^{-m\beta /2}$, the following properties hold:

the Markovian semigroup $\{e^{-tH_{X_m}^{\lambda _{m}}}:t\ge 0\}$, associated to the Dirichlet form $({\mathcal {E}}^{\lambda _m}_{X_m},{\mathcal {F}}^{\lambda _m}_{X_m})$ dominates the Markovian semigroup $\{e^{-tG_1}:t\ge 0\}$, generated by the closed, self-adjoint operator $(G_1,D(G_1))$ on $L^2(h)$ given by

$$\begin{aligned} \begin{aligned} D(G_1)&:=D(N^m)\cap JD(N^m)\\ G_1&:=\left( \lambda _m^2\cdot X_m^*X_m + \lambda _m^{-2}\cdot X_mX_m^*\right) \dot{+} j\left( \lambda _m^2\cdot X_m^*X_m + \lambda _m^{-2}\cdot X_mX_m^*\right) , \end{aligned} \end{aligned}$$

which can be expressed as

$$\begin{aligned} e^{-tG_1}(\eta )=e^{-tB_m}j\left( e^{-tB_m}\right) (\eta )=e^{-tB_m}\eta e^{-tB_m}\qquad \eta \in L^2(M), \end{aligned}$$

by the self-adjoint, positive operator $(B_m,D(B_m)):=(\lambda _m^2\cdot X_m^*X_m + \lambda _m^{-2}\cdot X_mX_m^*,D(N^m))$, affiliated to $M:=B(h)$ in its left action on $L^2(M)=L^2(h)$.

Proof

Set $(q_0,D(q_0)):=({\mathcal {E}}^{\lambda _m}_{X_m},{\mathcal {F}}^{\lambda _m}_{X_m})$ and consider the forms $(q_1,D(q_1))$, (w, D(w)) given by $D(q_1):={\tilde{{\mathcal {F}}}}_{X_m}=:D(w)$ and

$$\begin{aligned} \begin{aligned} q_1[\eta ]&:=\lambda _m^2\left( \Vert X_m\eta \Vert ^2+\Vert X_mJ\eta \Vert ^2\right) \Bigg ) + \lambda _m^{-2}\left( \Vert X_m^*\eta \Vert ^2+\Vert X_m^*J\eta \Vert ^2\right) ,\\ w[\eta ]&:=2\left[ \left( X_m\eta |JX_m^*J\eta \right) +\left( X_m^*\eta |JX_mJ\eta \right) \right] , \end{aligned} \end{aligned}$$

so that on $D(q_1):={\tilde{{\mathcal {F}}}}_{X_m}$, the representation (3.1) of the form $({\mathcal {E}}^{\lambda _m}_{X_m},{\mathcal {F}}^{\lambda _m}_{X_m})$ can be written as

$$\begin{aligned} q_1[\eta ]=q_0[\eta ]+w[\eta ]\qquad \eta \in D(q_1). \end{aligned}$$

As, by definition, $(q_0,D(q_0))$ is a Dirichlet form, its associated self-adjoint operator

$$\begin{aligned} \left( G_0,D(G_0)\right) :=\left( H^{\lambda _m}_{X_m},D\left( H^{\lambda _m}_{X_m}\right) \right) \end{aligned}$$

generates a Markovian, hence a $C_0$-continuous, self-adjoint, positivity preserving, semigroup $\{e^{-tG_0}:t\ge 0\}$. Since, by definition (see statement and proof of Theorem 2.5) and (5.1), ${\tilde{{\mathcal {F}}}}_{X_m}=D(X_m)\cap D(X_m^*)\cap J(D(X_m)\cap D(X_m^*))=D(N^{m/2})\cap JD(N^{m/2})$, the quadratic form $(q_1,D(q_1))$ is closed and the associated self-adjoint operator is just $(G_1,D(G_1))$. Since $\{e^{-tG_1}:t\ge 0\}$ is positivity preserving (see Appendix 7.1), to apply Lemma 4.2, we exploit the characterization of domination between positivity preserving semigroups on standard forms of von Neumann algebras, established in [3] Theorem 3.1: the semigroup $\{e^{-tG_1}:t\ge 0\}$ is dominated by $\{e^{-tG_0}:t\ge 0\}$ if and only if each one of the following properties is verified:

• (a) $D(q_1)\subseteq D(q_0)$,

• (b) $q_0(\eta |\zeta )\le q_1(\eta |\zeta )$ for all $\eta ,\zeta \in D(q_1)\cap L^2_+(M)$,

• (c) if $\eta \in D(q_0)\cap L^2_+(M)$, $\zeta \in D(q_1)\cap L^2_+(M)$ and $\eta \le \zeta$, then $\eta \in D(q_1)$. Condition a) holds true since $D(q_1):={{\tilde{{\mathcal {F}}}}}_{X_m}\subseteq {\mathcal {F}}^{\lambda _m}_{X_m}=:D(q_0)$. To prove b), consider the set $C(e)\subseteq L^2(h)$ of all Hilbert-Schmidt operators which are finite linear combination of the partial isometries $\{e_j\otimes e_k^*:j,k\in \mathbb {N}\}$ of the natural basis $e:=\{e_k\in h:k\in \mathbb {N}\}$ and set $C_+(e):=C(e)\cap L^2_+(h)$, $C_{\mathbb {R}}(e):=C(e)\cap L^2_{\mathbb {R}}(h)$, where $L^2_{\mathbb {R}}(h)=L^2_+(h)-L^2_+(h)$ is the self-adjoint part of $L^2(h)$. Since $\{e_j\otimes e_k^*:j,k\in \mathbb {N}\}$ is a Hilbert basis for $L^2(h)$, C(e) is dense in $L^2(h)$. For $A\in L^2_{\mathbb {R}}(h)$ and $B\in C(e)$ we have $(B^*+B)/2\in C_{\mathbb {R}}(e)$ and $\Vert A-(B^*+B)/2\Vert _2\le \Vert A-B\Vert _2$ so that $C_{\mathbb {R}}(e)$ is dense in $L^2_{\mathbb {R}}(h)$. For any $B\in C_{\mathbb {R}}(e)$ we have $B_+\in C_+(e)$ since $C_{\mathbb {R}}(e)=\bigcup _{j\in \mathbb {N}}L^2_{\mathbb {R}}(h_j)$, where $h_j:=\textrm{Lin}\{e_k\in h:k=0,\ldots ,j\}$, and if $B\in L^2_{\mathbb {R}}(h_j)$ for some $j\in \mathbb {N}$, then $B_+\in L^2_{\mathbb {R}}(h_j)$. Since the Hilbertian projection of $L^2_{\mathbb {R}}(h)$ onto $L^2_+(h)$ is a contraction, for any $A\in L^2_+(h)$ and $B\in C_{\mathbb {R}}(e)$ we have $\Vert A-B_+\Vert _2=\Vert A_+ -B_+\Vert _2\le \Vert A-B\Vert _2$ showing that the cone $C_+(e)$ is dense in the positive cone $L^2_+(h)$. It follows from Lemma 5.1 that C(e) is a J-invariant core for $(X_m,D(X_m))$ and $(X_m^*,D(X_m^*))$ which is left globally invariant by both operators: $X_m(C(e))\subseteq C(e)$, $X_m^*(C(e))\subseteq C(e)$. Let $P_j$ the finite rank projection on h with range $h_j$, for any $j\in \mathbb {N}$. Then if $\eta ,\zeta \in C_+(e)$ then $X_m\zeta =P_jX_mP_k\zeta$ and $X_m^*\eta =P_jX_m^*P_k\eta$ for sufficiently large $j,k\in \mathbb {N}$. Since $P_{j+m}X_m P_j\in B(h)$, $(P_j X_m P_{j+m})J(P_j X_m P_{j+m})J$ is positivity preserving and we have

  $$\begin{aligned} \left( X_m^*\eta |JX_mJ\zeta \right) =\left( \eta |(P_j X_m P_{j+m}\right) J\left( P_j X_m P_{j+m})J\zeta \right) \ge 0. \end{aligned}$$

  By the core property, the positivity of $(X_m^*\eta |JX_mJ\zeta )$ extends to any $\eta ,\zeta \in D(X_m^*)\cap JD(X_m)$ and an analogous reasoning shows that $(X_m\eta |JX_m^*J\zeta )\ge 0$ is true for any $\eta ,\zeta \in D(X_m)\cap JD(X_m^*)$. Since $D(q_1)={\tilde{{\mathcal {F}}}}_{X_m}$, altogether these properties allows to check b) as follows for $\eta ,\zeta \in D(q_1)\cap L^2_+(M)$

  $$\begin{aligned} q_1(\eta |\zeta )-q_0(\eta |\zeta )=w(\eta |\zeta )=2\left[ \left( X_m\eta |JX_m^*J\zeta \right) +\left( X_m^*\eta |JX_mJ\zeta \right) \right] . \end{aligned}$$

  To check c), since $D(q_1):={\tilde{{\mathcal {F}}}}_{X_m}$ is core for $({\mathcal {E}}^{\lambda _m}_{X_m},{\mathcal {F}}^{\lambda _m}_{X_m})$, let $\eta _n\in D(q_1)$ be a sequence such that

  $$\begin{aligned} \lim _n \left( q_0[\eta _n-\eta ]+\Vert \eta _n-\eta \Vert _2^2]\right) =0. \end{aligned}$$

  Let $\eta _n\wedge \zeta :=\textrm{Proj}(\eta _n,\zeta -L^2_+(M))$ be the Hilbert projection of $\eta _n\in L^2_+(M)$ onto the closed and convex set $\zeta -L^2_+(M)\subset L^2_{\mathbb {R}}(M)$. Since, by Lemma 4.4 in [8], we have $\eta _n\wedge \zeta =\zeta \wedge \eta _n=\eta _n-(\zeta -\eta _n)_-$, the continuity of the Hilbert projections and the fact that $\eta \le \zeta$, imply

  $$\begin{aligned} \lim _n\Vert \eta -\eta _n\wedge \zeta \Vert _2=\lim _n\Vert \eta -\eta _n+(\zeta -\eta _n)_-\Vert _2=\Vert (\zeta -\eta )_-\Vert _2=0. \end{aligned}$$

  Since $\{e^{-tG_0}:t\ge 0\}$ and $\{e^{-tG_1}:t\ge 0\}$ are positivity preserving, by Proposition 4.5 iii) in [8] we have

  $$\begin{aligned} \eta _n\wedge \zeta \in D(q_1),\qquad q_0[\eta _n\wedge \zeta ]\le q_0[\eta _n\wedge \zeta ]+q_0[\eta _n\vee \zeta ]\le q_0[\eta _n]+q_0[\zeta ]. \end{aligned}$$

  Since $\eta _n\wedge \zeta ,\zeta \in D(q_1)$ and, by definition, $\eta _n\wedge \zeta \le \zeta$, we have also (using the property of the quadratic form w established in the proof of b)) $w[\eta _n\wedge \zeta ]\le w[\zeta ]$ so that

  $$\begin{aligned} q_1[\eta _n\wedge \zeta ]=q_0[\eta _n\wedge \zeta ]+w[\eta _n\wedge \zeta ]\le q_0[\eta _n]+q_0[\zeta ]+w[\zeta ]=q_0[\eta _n]+q_1[\zeta ]. \end{aligned}$$

  Since the quadratic form $(q_1,D(q_1))$ is closed on $L^2(M)$, it is lower semicontinuous when considered as a functional on $L^2(M)$ taking values in the extended positive half-line $[0,+\infty ]$ and it is finite exactly on $D(q_1)$. We then have

  $$\begin{aligned} q_1[\eta ]\le \liminf _n q_1[\eta _n\wedge \zeta ]\le \liminf _n\bigl (q_0[\eta _n]+q_1[\zeta ]\bigr )=q_0[\eta ]+q_1[\zeta ]<+\infty \end{aligned}$$

  so that $\eta \in D(q_1)$. By [3] Theorem 3.1, $\{e^{-tG_1}:t\ge 0\}$ is dominated by $\{e^{-tG_0}:t\ge 0\}$: $e^{-tG_1}\eta \le e^{-tH^{\lambda _m}_{X_m}}\eta$ for all $\eta \in L^2_+(M)$ and $t\ge 0$. Choosing $\eta :=\xi _0$ one has $e^{-tG_1}\xi _0\le e^{-tH^{\lambda _m}_{X_m}}\xi _0\le \xi _0$ for all $t\ge 0$ so that $\{e^{-tG_1}:t\ge 0\}$ is Markovian.

$\square$

6 Dirichlet forms associated to deformations of the CCR relations

In this section we outline the construction of Dirichlet forms associated to deformations of annihilation and creation operators in the framework and notations of Sect. 5. To use the tools of Sect. 2 to this end, we need to represent eigenvectors of (isolated) eigenvalues of the Araki Hamiltonian as in Lemma 2.1.

6.1 Deformation of the CCR relations

Let $g:{\mathbb {R}}\rightarrow {\mathbb {R}}$ be a function vanishing on $(-\infty ,0]$, strictly increasing on $[0,+\infty )$ and satisfying, for $\beta >0$ and $\ell \in \mathbb {N}$ to be fixed later,

$$\begin{aligned} \sum _{n=0}^\infty n^{\ell } e^{-\beta g(n)}<+\infty . \end{aligned}$$

(6.1)

Consider the automorphisms group of the $C^*$-algebra of compact operators

$$\begin{aligned} \alpha _t(B):=e^{itg(N)}Be^{-itg(N)}\qquad t\in {\mathbb {R}},\quad B\in {\mathcal {K}}(h) \end{aligned}$$

whose Gibbs equilibrium state $\omega _\beta (\cdot )=\textrm{Tr}(\cdot \rho _\beta )$ is represented by the density matrix $\rho _\beta :=e^{-\beta g(N)}/Z(\beta )$ with partition function $Z(\beta ):=\textrm{Tr}(e^{-\beta g(N)})$. Let $\xi _0:=\rho _\beta ^{1/2}\in L^2_+(h)$ be the cyclic vector giving rise to the modular group of the normal extension of $\omega _\beta$ to B(h)

$$\begin{aligned} \sigma _t^{\omega _\beta }(B)=\alpha _{-t\beta }(B)=e^{-it\beta g(N)}Be^{it\beta g(N)}\qquad B\in B(h),\quad t\in {\mathbb {R}}. \end{aligned}$$

Then $\Delta _0^{it}(\eta )=\rho _\beta ^{it}\eta \rho _\beta ^{-it}$ for all $\eta \in L^2(h)$ and the Araki Hamiltonian is the strong sum

$$\begin{aligned} \ln \Delta _0=-\beta \overline{g(N)-j(g(N))}. \end{aligned}$$

Since for each $m,n\in \mathbb {N}$, $\nu _{m,n}:=\beta (g(m)-g(n))$ is an eigenvalue of $\ln \Delta _0$ with eigenvector $e_m\otimes e_n^*\in L^2(h)$ and $\{e_m\otimes e_n^*:m,n\in \mathbb {N}\}$ is a Hilbert basis, the spectrum of $\ln \Delta _0$ is

$$\begin{aligned} sp(\ln \Delta _0)=\overline{\{\nu _{m,n}:m,n\in \mathbb {N}\}}. \end{aligned}$$

All eigenvalues are isolated if, for example,

$$\begin{aligned} \liminf _{m>n\ge 0}\frac{g(m)-g(n)}{m-n}>0. \end{aligned}$$

Proposition 6.1

Suppose $\nu :=\nu _{m,n}\ge 0$ to be an isolated eigenvalue of the Araki Hamiltonian with $m\ge n$, set $\ell :=m-n\in \mathbb {N}$ and let $f\in C^\infty _0({\mathbb {R}})$ be a Schwartz function whose Fourier Transform²${\hat{f}}\in C^\infty _0({\mathbb {R}})$ is supported by $[\nu -\varepsilon ,\nu +\varepsilon ]$ and is strictly positive on $(\nu -\varepsilon ,\nu +\varepsilon )$, with ${{\hat{f}}}(\nu )=1$, for

$$\begin{aligned} 0<\varepsilon < \textrm{dist}(\nu ,sp(\ln \Delta _0)\setminus \{\nu \}). \end{aligned}$$

Then, setting $k(t):=\beta (g(t+\ell ))-g(t))$ and $p(t):={{\hat{f}}}(k(t))$ for $t\in {\mathbb {R}}$, we have

• (i) $sp(k(N))\subset sp(\ln \Delta _0)$ and $\nu \in sp(k(N))$ is an isolated eigenvalue of k(N) acting on h;

• (ii) p(N) is the spectral projection of N corresponding to the Borel set

  $$\begin{aligned} B:=\left\{ n'\in \mathbb {N}: g(m)-g(n)=g(n'+\ell )-g(n')\right\} \subseteq sp(N) \end{aligned}$$

  and $p(N-\ell \cdot I)$ is the spectral projection of N corresponding to $B+\ell \subseteq \mathbb {N}$;

• iii) the densely defined, closed operator (X, D(X)) on h, given by

  $$\begin{aligned} D(X):=D\left( N^{\ell /2}\right) \qquad X:=p(N)\circ A^\ell , \end{aligned}$$

  (6.2)

  where A is the annihilation operator defined in Sect. 5, satisfies the relations

  $$\begin{aligned} \begin{aligned} XX^*&=(N+1)\cdots (N+\ell \cdot I)p(N)\\ X^*X&=N(N-I)\cdots (N-(\ell -1)\cdot I)p(N-\ell \cdot I)\\ [X,X^* ]&=(N+1)\cdots (N+\ell \cdot I)p(N)- \\&N(N-I)\cdots (N-(\ell -1)\cdot I)p(N-\ell \cdot I); \end{aligned} \end{aligned}$$

  (6.3)

• iv) if B is unbounded, $(X^*X,D(N^\ell ))$ and $(XX^*,D(N^\ell ))$ are unbounded with discrete spectra;

• v) if B and $B+\ell$ differ by a finite set, then $([X,X^*],D(N^\ell ))$ is infinitesimally small with respect to $(N^\ell ,D(N^\ell ))$;

• vi) $\xi :=X\xi _0\in L^2(h)$ is an eigenvector of $\ln \Delta _0$ with eigenvalue $\nu$:

  $$\begin{aligned} (\ln \Delta _0)\xi =\nu \cdot \xi . \end{aligned}$$

Proof

i) follows from $sp(k(N))=\overline{\{\nu _{n'+m-n,n'}:n'\in \mathbb {N}\}}\subset \overline{\{\nu _{m,n}:m,n\in \mathbb {N}\}}= sp(\ln \Delta _0)$; ii) follows from i), the assumption on $\varepsilon$ and the Spectral Theorem; iii) by the CCR we have

$$\begin{aligned} NA=A(N-I),\qquad A^* N=(N-I)A^* \end{aligned}$$

(6.4)

as identities among closed operators on their common domain $D(N^{3/2})$. By induction

$$\begin{aligned} (A^*)^\ell A^\ell =N(N-I)\cdots (N-(\ell -1)\cdot I), \qquad A^\ell (A^*)^\ell =(N+I)\cdots (N+\ell \cdot I) \end{aligned}$$

on the domain $D(N^\ell )$ so that, by (6.2), one gets the first relation (6.3)

$$\begin{aligned} XX^*=p(N)A^\ell (A^*)^\ell p(N)=(N+I)\cdots (N+\ell \cdot I)p(N). \end{aligned}$$

Since, by (6.4), $p(N)A=Ap(N-I)$, by induction one obtains the second relation (6.3) $X^*X=(A^*)^\ell p(N)A^\ell =(A^*)^\ell A^\ell p(N-\ell \cdot I)=N(N-I)\cdots (N-(\ell -1)\cdot I)p(N-\ell \cdot I)$; the last relation (6.3) follows by difference; iv) follows from (6.3) and the fact that $N(N-I)\cdots (N-(\ell -1)\cdot I)$ and $(N+I)\cdots (N+\ell \cdot I)$ are polynomials; v) in this case $p(N)-p(N-\ell \cdot I)$ has finite rank and $(N+1)\cdots (N+\ell \cdot I)-N(N-I)\cdots (N-(\ell -1)\cdot I)$ is polynomial of degree at most $\ell -1$; vi) since $sp(\ln \Delta _0)\cap [\nu -\varepsilon ,\nu +\varepsilon ]=\{\nu \}$ and ${{\hat{f}}}(\nu )=1$, by the Spectral Theorem, the spectral projection P of $\ln \Delta _0$, corresponding to $\{\nu \}$, can be represented as

$$\begin{aligned} P={{\hat{f}}}(\ln \Delta _0)=\int _{\mathbb {R}}\textrm{d}t f(t)e^{it\ln \Delta _0}=\int _{\mathbb {R}}\textrm{d}t f(t)\Delta _0^{it}; \end{aligned}$$

since $\xi _0\in D(A^\ell )=D(N^{\ell /2})$ by (6.1), by (6.4) we have

$$\begin{aligned} \begin{aligned} \Delta _0^{it}\left( A^\ell (\xi _0)\right)&=\rho _\beta ^{it}\circ \left( A^\ell (\xi _0)\right) \circ \rho _\beta ^{-it}=\rho _\beta ^{it} A^\ell \rho _\beta ^{1/2}\rho _\beta ^{-it}\\&=e^{-it\beta g(N)}e^{it\beta g\left( N+\ell \cdot I\right) }A^\ell \rho _\beta ^{1/2}=e^{itk(N)}\left( A^\ell (\xi _0)\right) . \end{aligned} \end{aligned}$$

Hence, $P(A^\ell \xi _0)=\int _{\mathbb {R}}dt f(t) e^{itk(N)}(A^\ell (\xi _0))=({{\hat{f}}}(k(N))A^\ell )(\xi _0)=X(\xi _0)=:\xi$ does not vanish and it is an eigenvector of $\ln \Delta _0$ corresponding to the eigenvalue $\nu$. $\square$

Example

• (1) If $g(t)=t$ for any $t\in {\mathbb {R}}$, $B=\mathbb {N}$, $p(N)=I$, $X=A^\ell$ and we reproduce the "unperturbed" case treated in Theorem 5.2.

• (2) If $g(t):=t+[t/2]$ for $t\ge 0$, $n\in 2\mathbb {N}$ is even and $m\in 1+2\mathbb {N}$ is odd, then $\ell \in 1+2\mathbb {N}$ is odd, $g(m)-g(n)=3\ell /2-1/2$ and $B=\{n'\in \mathbb {N}:g(m)-g(n)=g(n'+\ell )-g(n')\}=2\mathbb {N}$.

Remark 6.2

The canonical commutation relations CCR arise in the spectral analysis of the quantum harmonic oscillator, which can be considered the canonical quantization of the classical harmonic oscillator whose phase space is the plane ${\mathbb {R}}^2$. D. Shale and W. F. Stinespring constructed in [32] a quantum system which can be regarded as the quantization of a harmonic oscillator whose phase space is the hyperbolic plane ${\mathbb {H}}^2$ with a fixed negative constant curvature $k<0$. It can be also considered as a quantum harmonic oscillator with self-interaction, the coupling constant being proportional to the curvature. In their work the authors found that the dynamics is generated by an Hamiltonian $H=\hbar \omega N$ proportional to the Number Operator and that annihilation and creation operators are replaced by operators X and $X^*$ satisfying a deformed CCR

$$\begin{aligned}{}[X,X^*]=\hbar \cdot I-k\hbar ^2\cdot N. \end{aligned}$$

A similar commutation relation is satisfied by $X:=A^2$ where A is the annihilation operator

$$\begin{aligned} \begin{aligned}{}[X,X^*]&=\left[ A^2,(A^*)^2\right] =A\left[ A,(A^*)^2\right] +\left[ A,(A^*)^2\right] A\\&=A[A,A^*]A^*+AA^*[A,A^*]+[A,A^*]A^* A+A^*[A,A^*]A\\&=AA^*+AA^*+A^* A+A^*A=2I+4N. \end{aligned} \end{aligned}$$

In reference to Sect. 5, $e^{-tH_{X_2}^{\lambda _2}}$, compared with the quantum Ornstein–Uhlenbeck semigroup $e^{-tH_{X_1}^{\lambda _1}}$ (see [12]), could be called quantum Ornstein–Uhlenbeck hyperbolic semigroup.

Appendix

1.1 Generators of a class of positivity preserving semigroups

Let (A, D(A)) be a lower bounded, self-adjoint operator affiliated to a von Neumann algebra M and consider the $C_0$-continuous, self-adjoint, positivity preserving semigroup on $L^2(M)$, defined by

$$\begin{aligned} T^A_t:=e^{-tA}j\left( e^{-tA}\right) =e^{-tA}Je^{-tA}J\qquad t\ge 0. \end{aligned}$$

If $(q_A,D(q_A))$ is the lower bounded, closed quadratic form of (A, D(A)), then the lower bounded, closed quadratic form of (j(A)), JD(A)) is given by $JD(q_A)\ni \eta \mapsto q_A[J\eta ]$ and the quadratic form $(t_A,D(q_A)\cap JD(q_A))$ given by $t_A[\eta ]:=q_A[\eta ]+q_A[J\eta ]$ is lower bounded and closed as a sum of forms sharing these same properties.

Lemma 7.1

The lower bounded, closed, quadratic form of the $C_0$-continuous, self-adjoint semigroup $\{T^A_t:t\ge 0\}$ is given by $(t_A,D(q_A)\cap JD(q_A))$ and the associated self-adjoint generator, i.e. the generalized sum $A\dot{+}j(A)$ (see [19]), is given by the closure $\overline{A+j(A)}$

$$\begin{aligned} T^A_t=e^{-tA}j\left( e^{-tA}\right) =e^{-t\left( A\dot{+}j(A)\right) }=e^{-t\overline{A+j(A)}}\qquad t\ge 0. \end{aligned}$$

Proof

Since $(q_A,D(q_A))$ is lower bounded, for $\eta \in L^2(M)$ the limit

$$\begin{aligned}{} & {} \lim _{t\rightarrow 0^+}t^{-1}\left[ \left( \eta |(I-T^A_t)\eta \right) \right] \\ {}{} & {} \quad =\lim _{t\rightarrow 0^+}t^{-1}\left[ \left( \eta |(I-e^{-tA}\eta )\right) +\left( e^{-tA}\eta |J(I-e^{-tA}J\eta )\right) \right] \end{aligned}$$

exists in ${\mathbb {R}}$ if and only if both limits on the right-hand side exist in ${\mathbb {R}}$, i.e. if and only if $\eta \in D(q_A)\cap JD(q_A)$ and in this case $\lim _{t\rightarrow 0^+}t^{-1}[(\eta |(I-T^A_t)\eta )]=q_A[\eta ]+q_A[J\eta ]=:t_A[\eta ]$. Hence the lower bounded, closed quadratic form of $\{T^A_t:t\ge 0\}$ is $(t_A,D(q_A)\cap J D(q_A))$ and this form is densely defined. As (A, D(A)) and (j(A), JD(A)) are affiliated to commuting von Neumann algebras, they strongly commute and the sum $(A+J(A),D(A)\cap J D(A))$ is densely defined, lower bounded, symmetric and essentially self-adjoint so that $A\dot{+}j(A)=\overline{A+j(A)}$. $\square$

1.2 Superbounded semigroups on abelian atomic von Neumann algebras.

The von Neumann algebra B(h) is atomic and this suggests to have a look at the superboundedness property in the abelian situation of atomic measured spaces.

Let (X, m) be a locally compact, second countable, Hausdorff space, endowed with a fully supported Borel measure. Consider a real valued function U such that $e^{-U}\in L^1 (X,m)$ and define a probability measure by

$$\begin{aligned} m_U:=e^{-U}m\Big /\int _X e^{- U}\textrm{d}m. \end{aligned}$$

By the unit norm function $u_0:=e^{- U/2}/\Vert e^{-U/2}\Vert _{L^2(X,m)}\in L^2(X,m)$, one recovers the integral with respect to $m_U$ by

$$\begin{aligned} \int _Xvdm_U=(u_0|vu_0)_{L^2(X,m)}, \end{aligned}$$

one has the embedding $i_0:L^\infty (X,m)\rightarrow L^2(X,m)$ $i_0(v):=vu_0$ with $\Vert i_0(w)\Vert _{L^2(X,m)}=\Vert w\Vert _{L^2(X,m_U)}$.

A $C_0$-continuous semigroup $T_t:L^2(X,m)\rightarrow L^2(X,m)$ is Markovian with respect to $m_U$ (in the sense we are discussing in this work, i.e. the one introduced in [8]), if

$$\begin{aligned} 0\le v\le u_0\quad \Rightarrow \quad 0\le T_tv\le u_0\qquad t\ge 0. \end{aligned}$$

Such a semigroup induces a semigroup on the abelian von Neumann algebra $L^\infty (X,m)$ by

$$\begin{aligned} S_t:L^\infty (X,m)\rightarrow L^\infty (X,m)\qquad i_0(S_t u)=T_t(i_0(u))\qquad u\in L^\infty (X,m), \end{aligned}$$

which is Markovian in the usual sense

$$\begin{aligned} 0\le u\le 1\quad \Rightarrow \quad 0\le S_tu\le 1\qquad t\ge 0. \end{aligned}$$

The definition of superboundedness considered above on von Neumann algebras, in the commutative setting reduces to say that $T_t$ is superbounded with respect to $m_U$ if

$$\begin{aligned} T_t\left( L^2(X,m)\right) \subset i_0\left( L^\infty (X,m)\right) \qquad t>t_0 \end{aligned}$$

for some $t_0\ge 0$ and

$$\begin{aligned} \Vert u\Vert _{L^\infty (X,m)}\le \Vert v\Vert _{L^2(X,m)} \end{aligned}$$

whenever $T_t v=i_0(u)$ for $v\in L^2(X,m)$, $u\in L^\infty (X,m)$ and $t>t_0$. In other words, $T_t$ is superbounded with respect to $m_U$, if the induced Markovian semigroup satisfies

$$\begin{aligned} \Vert S_t u\Vert _{L^\infty (X,m)}\le \Vert i_0(u)\Vert _{L^2(X,m)}=\Vert u\Vert _{L^2(X,m_U)}\qquad u\in L^\infty (X,m),\qquad t>t_0. \end{aligned}$$

In case (X, m) is an atomic measured space, the classical definition of super or ultracontractivity typically trivializes (see [15] Section 2.1): this happens, for example, if m is the counting measure because of the contractive embedding $L^2(X,m)\subseteq L^\infty (X,m)$. Superboundedness however may still be non trivial.

Let (X, m) be a countable, atomic measured space and let $m=e^{-h}m_0$ for some function h and the counting measure $m_0$. To simplify notations, we assume that $\Vert e^{-U}\Vert _{L^1(X,m)}=1$.

For a fixed nonnegative measurable function $V:X\rightarrow [0,+\infty )$ let us consider the semigroup

$$\begin{aligned} T_t:L^2(X,m)\rightarrow L^2(X,m)\qquad T_t v:=e^{-tV}v\qquad t\ge 0 \end{aligned}$$

which is clearly Markovian with respect to the probability measure $m_U$.

Lemma 7.2

The semigroup $T_t$ is supercontractive with respect to $m_U$ if and only if

$$\begin{aligned} (U+h)_+/V\in L^\infty (X,m). \end{aligned}$$

More precisely, $T_t$ extends to a contraction from $L^2(X,m_U)$ to $L^2(X,m)$ if and only if

$$\begin{aligned} t\ge t_0:=\frac{1}{2}\Vert (U+h)_+/V\Vert _\infty . \end{aligned}$$

In case m is the counting measure we have $t_0=\Vert U/V\Vert _\infty /2$.

Proof

On one hand, if $t_0$ is finite and $t\ge t_0$, the result follows from $\Vert S_t v\Vert _\infty = \Vert ve^{-tV}\Vert _\infty \le \Vert ve^{-tV}\Vert _{L^2(X,m_0)}= \Vert ve^{(U+h)/2-tV}\Vert _{L^2(X,m_U)}\le \Vert v\Vert _{L^2(X,m_U)}$. On the other hand, if $\Vert S_tv\Vert _\infty \le \Vert v\Vert _{L^2(X,m_U)}$ for some $t_0\ge 0$ and all $t\ge t_0$, choosing $v:=1_{\{x\}}$ for any $x\in X$, we have $t_0\ge \Vert (U+h)_+/V\Vert _\infty /2$. $\square$