1 Introduction

The plan of our note is the following:

  1. 1.

    q-CCR(CAR) relations for \(|q|>1\), and q-continuous Hermite polynomials.

  2. 2.

    Combinatorial results on 2-partitions of \(\{1,2,\ldots ,2n\}\) – \(P_2(2n)\).

  3. 3.

    q-discrete Hermite polynomials of type I, II.

  4. 4.

    q-analogue of classical convolutions of Carnovale and Koornwinder [15] for \(0 \le q \le 1\), (\(q=0\), Boolean convolution, \(q=1\) classical convolution).

  5. 5.

    Braided Hopf algebras of Kempf and Majid.

  6. 6.

    The construction of q-Discrete Fock space \(\mathcal {F}_q^{disc}(\mathcal {H})\) and q-Discrete Brownian motions corresponding to q-Discrete Hermite polynomials of type I (\(0 \le q \le 1\)).

  7. 7.

    Matrix version of Khintchine inequalities.

2 q-CCR(CAR) for \(|q|>1\), and q-Continuous Hermite Polynomials

Continuous q-Hermite are defined as:

$$\begin{aligned} x H_n^{q}(\infty )= & {} H_{n+1}^{q}(x) + \frac{q^n-1}{q-1} H_{n-1}^{q}(x), H_0=1, H_1=x \\ {[}n]_q! \delta _{n,m}= & {} \int _{-\frac{!}{\sqrt{1-q}}}^{\frac{!}{\sqrt{1-q}}} H_n^{(q)}(x) H_m^{(q)}(x) d \mu _q^{c} (x), \end{aligned}$$

where

$$\begin{aligned} d \mu _q^{c} (x)= & {} \frac{1}{2\pi } q^{-\frac{1}{8}} \theta _1 \left( \frac{\theta }{\pi },\frac{1}{2\pi i} \log q \right) dx \\= & {} \frac{1}{\pi }\sqrt{1-q}\sin (\theta ) \prod _{n=1}^\infty (1-q^n)| 1-q^n \exp (2\pi \theta )|^2 dx \end{aligned}$$

for \(0 \le q < 1\), \(\theta _1\) – Jacobi theta one function.

\(2\cos v = x \sqrt{1-q}\), \({{\,\textrm{supp}\,}}\mu _q^c = [-\frac{2}{\sqrt{1-q}},\frac{2}{\sqrt{1-q}}]\) (see [3] and [9] for more details).

Theorem 1

([10]) If \(-1 \le q \le 1\), \(s>0\), then there exist operators \(A^\pm (f)=A^\pm _{q,s}(f)\), \(g,f \in \mathbb {R}^N\), \(N=\infty ,1,2,\ldots \):

$$\begin{aligned}{} & {} A(f)A^+(g)-(s q)A^+(g)A(f)=s^N<f,g>I. \\{} & {} A(f) \Omega = 0 \end{aligned}$$

where \(\mathcal {H}^{\otimes 0} = \mathbb {C}\Omega \).

Construction 1

Take q-CCR operators: \(a^\pm _q(f)=a(f)\).

$$\begin{aligned} a(f) a^+(g) - q a^+(g) a(f) = <f,g> I, \end{aligned}$$

on \(\mathcal {F}_q(\mathcal {H})=\bigoplus _{n=0}^{\infty } \mathcal {H}^{\otimes n}\), as in Bożejko-Speicher [13] with scalar product

$$\begin{aligned}<f_1 \otimes \ldots \otimes f_n | g_1 \otimes \ldots \otimes g_n>_q = <P_q^{(n)}(f_1 \otimes \ldots \otimes f_n) | g_1 \otimes \ldots \otimes g_n>. \end{aligned}$$

where \(P_q^{(n)}=\sum _{\sigma \in S(n)} q^{inv(\sigma )} \sigma \). Here inverse of permutation \(\sigma \) is defined as \(inv(\sigma )=\#\{\pi \in S(n): i<j\;\textrm{and}\; \pi (i)>\pi (j)\}\) and S(n) is a permutation group on n letters. Now we define annihilation operator \(A^\pm (f)=A^\pm _{q,s}(f)\) as

$$\begin{aligned} A^\pm _{q,s}(f)=s^{N-1} a^\pm _q(f),\; s>0 \end{aligned}$$

where N on \(\mathcal {H}^{\otimes n}\) is defined as:

$$\begin{aligned} N(x_1 \otimes \ldots \otimes x_n)=n(x_1\otimes \ldots \otimes x_n) \end{aligned}$$

where \(a^+_q(f) = f \otimes \xi \) (this is q-creation) and \(a_q(f) = [a^+_q(f)]^*\), \(f \in \mathcal {H}_\mathbb {R}\) (this is q-annihilation). This conjugation for vectors \(\xi ,\eta \in \mathcal {H}^{\otimes n}\) is defined in the new scalar product \(<\xi |\eta>_{q,s} = s^{\left( {\begin{array}{c}n\\ 2\end{array}}\right) } <\xi |\eta >_q\).

3 Combinatorial Results on \(P_2(2n)\)

Definition 1

(q-conditions cummulants - Ph.Biane, M.Anshelevich) If \(\mu \) – probability measure on \(\mathbb {R}\) with all moments, then the q-continuous cummulants are defined as follows:

$$\begin{aligned} \mu \rightarrow \left( R^{(q)}_\mu (n)\right) _{n=1}^\infty \end{aligned}$$

in such a way that:

$$\begin{aligned} \int _{-\infty }^\infty x^n d \mu (x)=\sum _{\mathcal {V} \in P(n)} q^{cr(V)} R_\mu ^{(q)}(\mathcal {V}), \end{aligned}$$
(1)

where P(n) is the set of all set-partitions on \(\{1,2,\ldots ,n\}\), and

$$\begin{aligned} R_\mu ^{(q)}(\mathcal {V})=\prod _{B \in \mathcal {V}} R_\mu ^{(q)}(|B|), \end{aligned}$$

where \(\mathcal {V}\) – partition of \(\{1,2,\ldots ,n\}\), and \(cr(\mathcal {V})\) is a number of of hyperbolic (restricted) crossings defined by Ph. Biane [14] (Fig. 1).

Fig. 1
figure 1

Crossings

Full size image

Remark 1

A.Nica defined left-reduced number of crossing \(c_0(\mathcal {V})\) as: \( c_0(\mathcal {V}) = \#\{(m_1,m_2,m_3,m_4):1 \le m_1 \le m_2 \le m_3 \le m_4 \le n: \) \((m_1,m_3) \in \mathcal {V}, (m_2,m_4) \in \mathcal {V}, (m_2,m_3) \notin \mathcal {V}, \mathrm{\;each\;} m_1, m_2 \mathrm{\;minimal\;in\;the} \) \( \mathrm{\;class\;of\;} \mathcal {V} \mathrm{\;containing\;it}\}, \) then Nica’s q-cummulants \(\widetilde{R}_\mu ^{(q)}(n)\) come from (1), where \(cr(\mathcal {V})\) is replaced by \(c_0(\mathcal {V})\).

F.Oravecz [24] showed that Nica’s q-cummulants are not positivity preserving, i.e.

If we define a,,q-convolututon”: \(\mu =\mu _1 *_q \mu _2\): (Ph. Biane idea) is done as:

$$\begin{aligned} R^{(q)}_{\mu _1}(n) + R^{(q)}_{\mu _2}(n) = R^{(q)}_{\mu }(n), n=1,2,3,\ldots , \end{aligned}$$
(2)

then we have the following open problem:

Problem 1

(open) Is Ph. Biane,,\( *_q\)-convolutions” positivity preserving?

Recently Carnovole and Koornwinder defined q-Discrete version of (2):

\(\mu =\mu _1 *_q^{disc} \mu _2\), \(d \mu _i(x)=f_i(x) dx\), \(f_1 *_q^{disc} f_2 = f\).

$$\begin{aligned} m_n^{disc}(f)=q^{\left( {\begin{array}{c}n\\ 2\end{array}}\right) } \int _{-\infty }^\infty x^n f(x) d_q(x) = \sum _{k=0}^n \genfrac[]{0.0pt}{}{n}{k}_q m_k^{disc} (f_1) m_{n-k}^{disc} (f_2), \end{aligned}$$
(3)

where \(d_q(x)\) is the Jackson integral.

For \(q=1\) we have classical convolutions.

For \(q=0\) we have Boolean convolutions.

Now, we are describing the new q-convolution corresponding to q-Discrete Hermite polynomials of the type I. We give also Wick formula for that case.

Theorem 2

(Bożejko–Yoshida[10] (Wick formula)) If \(G(f)=A(f)+A^+(f)\), then

$$\begin{aligned}<G(f_1)\ldots G(f_{2n}) \Omega | \Omega>=\sum _{\mathcal {V} \in P_2(2n)} s^{\frac{1}{2}ip(\mathcal {V}} \cdot q^{cr(\mathcal {V})} \prod _{(i,j) \in \mathcal {V}} <f_i | f_j> \end{aligned}$$

where \(ip(\mathcal {V})=\sum _{(i,j) \in \mathcal {V}} inpt(i,j)\), \(inpt(i,j) = \#\{\)of k with \(i<k<j\}\)

$$\begin{aligned} =\sum _{k=1}^n (j_k - i_k -1), \textrm{if}\; \mathcal {V}=\{(i_1,j_1),\ldots ,(i_n,j_n)\} \in P_2(2n). \end{aligned}$$

For the proof see [10].

We are recalling the crossing number definition for 2-partitions \(\mathcal {V}\) (see [7]):

Theorem 3

([5]) If \(\mathcal {V} \in P_2(2n)\), then

$$\begin{aligned} cr(\mathcal {V}) + pbr(\mathcal {V})=\frac{1}{2}ip(\mathcal {V}) \end{aligned}$$

where

figure a

The \(pbr(\mathcal {V})\), also denoted as \(nest(\mathcal {V})\), was introduced by A.Nica [22], de Médicis and Viennot [18]. For more detailes on nesting, see papers: N.Blitvic [8], Bożejko and Ejsmont [11]. For the proof, we need the following Lemma, more details, see Bożejko [5].

Lemma 1

If

$$\begin{aligned} \sum _{\mathcal {V} \in P_2(2n)} t_1(\mathcal {V}) \prod _{(i,j) \in \mathcal {V}}<f_i | f_j> = \sum _{\mathcal {V} \in P_2(2n)} t_2(\mathcal {V}) \prod _{(i,j) \in \mathcal {V}} <f_i | f_j>, \end{aligned}$$

for all vectors \(f_i \in \)Hilbert space, than

$$\begin{aligned} t_1(\mathcal {V}) = t_2(\mathcal {V}) \end{aligned}$$

for all 2-partitions \(\mathcal {V} \in P_2(2n)\).

From this general construction we obtain q-CCR relation for \(|q| \ge 1\).

Theorem 4

([5]) If \(q \ge 1\), then there exist operators on a proper Fock space satisfying the (q-CCR):

$$\begin{aligned} B(f) B^+(g) - B^+(g) B(f) = q^N <f,g> I, f,g, \in \mathcal {H}\;{\mathrm{(Hilbert~space),}} \end{aligned}$$

where \(N(x_1 \otimes \ldots \otimes x_n) = n(x_1 \otimes \ldots \otimes x_n)\).

Moreover \(\widetilde{G}(f) = B(f) + B^+(f)\):

$$\begin{aligned}<\widetilde{G}(f_1) \ldots \widetilde{G}(f_{2n}) \Omega | \Omega> = \sum _{\mathcal {V} \in P_2(2n)} q^{pbr(\mathcal {V})} \prod _{(i,j) \in \mathcal {V} <f_i | f_j>}. \end{aligned}$$

Proof’s idea: Consider \(A_{1/q,q}(f)\), submitting 1/q instead of q, \(s=q\), where \(B^\pm (f)=A^\pm _{1/q,q}(f)\), \(f \in \mathcal {H}\) were constructed in Theorem 1.

4 q-Discrete Hermite Polynomials, \(0 \le q \le 1\)

We recall the definition of q-Discrete Hermite polynomial of type I and type II for \(0 \le q \le 1\) as:

I type: \(h_0=1\), \(h_1(x)=x\), \(x h_n(x) = h_{n+1}(x) + q^{n-1}[n]_q h_{n-1}(x)\), \([n]_q=\frac{q^n-1}{q-1}=1+q+\ldots +q^{n-1}\). Later we will denote \({h_n(x;q)}= h_n(x)\).

II type: \(\widetilde{h}_n(x;q)=i^{-n} h_n(ix;q^{-1})\), where \(i=\sqrt{-1}\).

Now we recall the definition of two exponential functions

$$\begin{aligned} e_q(x) = \sum _{k=0}^\infty \frac{x^k}{(q:q)_k}, E_q(x) = \sum _{k=0}^\infty \frac{q^{\left( {\begin{array}{c}n\\ 2\end{array}}\right) } x^k}{(q:q)_k} \end{aligned}$$

where \((a:q)_k = (1-a)(1-aq)\ldots (1-aq^{k-1})\).

$$\begin{aligned} {[}k]_q!=\frac{(q:q)_k}{(1-q)^k}, \genfrac[]{0.0pt}{}{n}{j}_q = \frac{[n]_q!}{[j]_q! [n-j]_q!} \mathrm{(Gauss~symbol)}. \end{aligned}$$

Facts (see Andrews et al. [3]):

  1. 1.

    \(E_q(z)=\prod _{n=0}^\infty (1+q^n z)\), \(z \in \mathbb {C}\),

  2. 2.

    \(e_q(z)E_{q}(-z)=1\), \(z \in \mathbb {C}\),

  3. 3.

    (I type) \(\int _{-1}^1 h_m(x:q)h_n(x:q) E_{q^2}(-q^2x^2)d_qx = b_q \cdot q^{\left( {\begin{array}{c}n\\ 2\end{array}}\right) }(q:q)_n \delta _{n,m}\),

  4. 4.

    (II type) \(\int _{-\infty }^\infty \widetilde{h}_m(x:q) \widetilde{h}_n(x:q) e_{q^2} (-x^2) d_q x = c_q \cdot q^{-n^2}(q:q)_n \delta _{n,m}\), where

    $$\begin{aligned} \int _0^x f(x) d_q(x) = (1-q) \sum _{k=0}^\infty f(q^k x) q^k x \end{aligned}$$

    is well known Jackson integral for functions with support \({{\,\textrm{supp}\,}}(f) \subset \mathbb {R}^+\), and for arbitrary \(f:\mathbb {R} \rightarrow \mathbb {C}\) we define

    $$\begin{aligned} \int _{-\infty }^\infty f(x) d_q(x) = (1-q) \sum _{k=-\infty }^\infty \sum _{\varepsilon = \pm 1} q^k f(\varepsilon q^k); {{\,\textrm{supp}\,}}(f) \subset \mathbb {R}. \end{aligned}$$

Commutation relation to the Fock representation of type I discrete Hermite polynomials.

In Theoreom 1 put \(s=q\), \(q=q\), \(0 \le 1 \le 1\), then operators

$$\begin{aligned} C_q^\pm (f)=A_{q,q}^\pm (f), \widehat{G}(f)=C(f)+C^+(f) \end{aligned}$$

satisfy the following theorem:

Theorem 5

  1. 1.

    If \(||f_i||=1\), \(i=1,2,\ldots ,2n\), then

    $$\begin{aligned}<\widehat{G}(f_1)\ldots \widehat{G}(f_{2n}) \Omega | \Omega>=\sum _{\mathcal {V} \in P_2(2n)} q^{\frac{1}{2} ip(\mathcal {V})+cr(\mathcal {V})} \prod _{(i,j) \in \mathcal {V}} <f_i | f_j> \end{aligned}$$
  2. 2.
    $$\begin{aligned} \int x^{2n} d \mu _q^I(x)=<\widehat{G}(f_1)^{2n} \Omega | \Omega >=[1]_q [3]_q\ldots [2n-1]_q = \sum _{\mathcal {V} \in P_2(2n)} q^{e_0}(\mathcal {V}), \end{aligned}$$

    where \(e_0(\mathcal {V})\) was introduced by [18], where

    $$\begin{aligned} e_0(\mathcal {V})=pbr(\mathcal {V})+ 2cr(\mathcal {V})=\frac{1}{2}ip(\mathcal {V})+cr(\mathcal {V}). \end{aligned}$$
  3. 3.

    Moreover

    $$\begin{aligned} A_q(f) A_q^+(g) - q^2 A_q^+(g) A_q(f) = q^N <f,g>I. \end{aligned}$$

    for \(f,g \in \mathcal {H}\).

Problems:

  1. 1.

    Prove positivity of q-Discrete (continuous) convolutions for \(0< q < 1\)?

  2. 2.

    Describe q-Discrete Poisson measure (process)?

  3. 3.

    Calculate the operator norm of \(||\widehat{G}(f_i)||=?\), \(i=1,2,\ldots \)

  4. 4.

    If \(\Omega \) is faithful state in the corresponding Fock space?

5 Another q-Analogues of Classical Convolutions

Let us define Jackson,,q-moments” for,,good” function \(f:\mathbb {R}\rightarrow \mathbb {R}\) as follows:

$$\begin{aligned} m_n^{disc}(f)=q^{\left( {\begin{array}{c}n\\ 2\end{array}}\right) } \int _{-\infty }^\infty f(x) x^n d_q(x), \end{aligned}$$

and,,q-Discrete convolutions” of Carnowale and Koornwinder [15]

$$\begin{aligned} (f \otimes _q g)(x) = \sum _{n=0}^\infty \frac{(-1)^n m_n^{disc}(f)}{[n]_q!} (\delta _q^n g) (x) \end{aligned}$$

where

$$\begin{aligned} \delta _q f(x) = \left\{ \begin{array}{ll} \frac{f(x)-f(qx)}{x-qx},&{} x \ne 0,\quad \lim _{q\rightarrow 1} \delta _q f(x) = f'(x),\\ f'(0),&{} x=0. \end{array} \right. \end{aligned}$$

Note that if \(q=1\), we have

$$\begin{aligned}{} & {} \left( \int _{-\infty }^\infty dt f(t) \frac{(-1)^n t^n}{n!} \right) g^{(n)}(x)= \int _{-\infty }^\infty dt f(t) \left( \sum _{n=0}^\infty \frac{(-t)^n}{n!} g^{(n)}(x) \right) \\{} & {} \quad =\int _{-\infty }^\infty dt f(t)\; g(x-t) = (f * g)(x). \end{aligned}$$

which is the classical convolution.

Theorem 6

(Carnovale, Koornwinder [15]) For,,good” functions \(f,g: \mathbb {R} \rightarrow \mathbb {R}\) q-Discrete convolution is associative and commutative. Moreover

$$\begin{aligned} m_n^{disc} (f \otimes _q g) = \sum _{n=0}^n \genfrac[]{0.0pt}{}{n}{j}_q m_k^{disc}(f)\; m_{n-k}^{disc}(g). \end{aligned}$$

If \(q=0\), we get Boolean convolution.

If \(q=1\), we get classical convolution on \(\mathbb {R}\).

Problem: Find characterization q-Discrete moments sequence \(m_n^{disc}(f)\), i.e. for \(f \ge 0\)

$$\begin{aligned} m_n^{disc}(f) = \int _{-\infty }^\infty f(x) x^n d_q(x) ? \end{aligned}$$

6 Braided Hopf Algebras of Kempf and Majid

Definition 2

Braided line is a braided algebra \(\mathcal {A}=\mathbb {C}[[x]]\) formal power series in variable x which has braiding

$$\begin{aligned} \Phi (x^k \otimes x^l) = q^{kl} x^l \otimes x^k, \end{aligned}$$

comultiplication:

$$\begin{aligned} \Delta (x^k)=\sum _{j=0}^k \genfrac[]{0.0pt}{}{k}{j}_q x^{k-j} \otimes x^j \end{aligned}$$

co-unit

$$\begin{aligned} \varepsilon (x^k)=\delta _{k,0} \end{aligned}$$

braided antipode

$$\begin{aligned} S(x^k)=(-1)^k q^{\left( {\begin{array}{c}k\\ 2\end{array}}\right) } x^k = (-1)^k q^{\frac{k(k-1)}{2}} x^k, \end{aligned}$$

and then we get the q-analogue of of Taylor’s formula:

$$\begin{aligned} \Delta (f(x))=\sum _{j=0}^\infty \frac{x^j}{[j]_q!} \otimes \delta _q(f(x)). \end{aligned}$$

Theorem 7

(Kemp, Majid [16]) If \(Q f(x) = f(qx)\), then we have

$$\begin{aligned} (f*_q g)(x)= (f \otimes id)(m \otimes id)[id \otimes Q \otimes id](id \otimes S \otimes id)(id \otimes \Delta )(f \otimes g)(x) \end{aligned}$$

Moreover as observed by Koornwinder we have

$$\begin{aligned} \Delta (e_q(x))= & {} e_q(x) \otimes e_q(x),\\ S(e_q(x))= & {} E_q(-x),\\ \varepsilon (e_q(x))= & {} 1. \end{aligned}$$

7 The Construction of q-Discrete Fock Space \(\mathcal {F}_q^{disc}(\mathcal {H})\) and q-Discrete Brownian Motions

Now we present, for , the construction of q-Discrete Fock space \(\mathcal {F}_q^{disc}(\mathcal {H})\) for q-Discrete Hermite of Type I, which is the completion of the full Fock space \(\mathcal {F}(\mathcal {H})=\bigoplus _{n=0}^\infty \mathcal {H}^{\oplus n}=\mathbb {C} \Omega \oplus \mathcal {H} \oplus (\mathcal {H} \otimes \mathcal {H}) \ldots \) under the positive inner product on \(\mathcal {H}^{\oplus n}\) done by:

$$\begin{aligned}{} & {} <x_1 \otimes \ldots \otimes x_n | y_1 \otimes \ldots \otimes y_m>_q \\{} & {} \qquad =\delta _{n,m} q^{\left( {\begin{array}{c}n\\ 2\end{array}}\right) } \sum _{\pi \in S(n)} q^{inv(\pi )}<x_1|y_{\pi (1)}>\cdot \ldots \cdot <x_n|y_{\pi (n)}>. \end{aligned}$$

We define creation operator \(A_q^+(f)\xi _n=f \otimes \xi _n\), \(f \in \mathcal {H}\), \(\xi _n \in \mathcal {H}^{\otimes n}\) and the annihilation operator

$$\begin{aligned} A_q(f)x_1 \otimes \ldots \otimes x_n= q^{n-1} \sum _{k=1}^n q^{k-1} <x_k | f> x_1 \otimes \ldots \otimes \check{x}_k \otimes \ldots \otimes x_n \end{aligned}$$

In the paper [B-Y] we have more general construction

$$\begin{aligned} \widetilde{A}_{q,s}(f)x_1 \otimes \ldots \otimes x_n= s^{2(n-1)} \sum _{k=1}^n q^{k-1} <x_k | f> x_1 \otimes \ldots \otimes \check{x}_k \otimes \ldots \otimes x_n \end{aligned}$$

If we put \(s^2=q\) we get our q-Discrete Fock space \(\mathcal {F}_q^{disc}(\mathcal {H})\).

Remark 2

For \(f,g \in \mathcal {H}\) we have the following q-Discrete Commutation Relation:

$$\begin{aligned} A(f)A^+(g)-q^2 A^+(g)A(f)=q^N<f,g>I. \end{aligned}$$

See more details in [8, 10] for a more general case.

Moreover, we have q-Discrete Gaussian random variables \(\widehat{G}_q(f)=A_q(f)+A_q^+(f)\). We get q-version of Wick formula

$$\begin{aligned}<\widehat{G}_q(f_1)\ldots \widehat{G}_q(f_{2n}) \Omega | \Omega>=\sum _{\mathcal {V} \in P_2(2n)} q^{\frac{1}{2}ip(\mathcal {V})} \cdot q^{cr(\mathcal {V})} \prod _{(i,j) \in \mathcal {V}} <f_i | f_j>. \end{aligned}$$

Our Gaussian \(\widehat{G}_q(f)\), at the vacuum state \(\Omega \), has the spectral measure \(\mu ^{disc}_q\) corresponding to q-Discrete Hermite polynomials of type I which were defined by following recurrence:

$$\begin{aligned} x h_n(x) = h_{n+1}(x) + q^{n-1}[n]_q h_{n-1}(x),[n]_q=\frac{q^n-1}{q-1}=1+q+\ldots +q^{n-1}, \end{aligned}$$

\(h_0(x)=1\), \(h_1(x)=x\), in the Sect. 4.

Now we define q-Discrete Brownian motion \(BM_t\) as follows.

Take \( \mathcal {H}=L^2(\mathbb {R}^+,dx) \) and \(f=\chi _{[0,t)}\),

$$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1&{}\textrm{for}\; x \in [0,t),\\ 0&{}\mathrm{otherwise.} \end{array}\right. \end{aligned}$$

Then

$$\begin{aligned} BM_t=\widehat{G}_q(\chi _{[0,t)}) \end{aligned}$$

is our q-Discrete Brownian motion.

Remark 3

Case \(q=1\) is the classical Brownian motion, and \(q=0\) is the Boolean Brownian motion.

Problem: Is the von Neumann algebra \(BB_q =\) WO-closure of \(\{BM_t: t \ge 0\}\) is factorial, that means it has one-dimensional center?

This \(BB_q\) algebra corresponds to q-Discrete Hermite polynomials of type I, but the corresponding problem for continuous q-Discrete Hermite polynomials was solved by Bożejko, Kümmerer and Speicher [9].

8 Matrix Version of Khintchine Inequalities

We are looking for matricial version Khintchine inequalities for random variables \(X_1,X_2,\ldots ,X_n\),

$$\begin{aligned} \left\| \sum _{j=1}^n \alpha _j \otimes X_j\right\| \cong \max \left\{ \left\| \sum _{j=1}^n \alpha _j \alpha _j^*\right\| ^{\frac{1}{2}},\left\| \sum _{j=1}^n \alpha _j^* \alpha _j\right\| ^{\frac{1}{2}}\right\} \end{aligned}$$
(4)

for \(n=1,2,\ldots \) and \(\alpha _j\) are complex matrices of arbitrary sizes and the norms are operator norms, where \(a \cong b\) iff \(K_1b \le a \le K_2 b\) for some \(K_1,K_2 > 0\).

Theorem 8

Inequality (4) holds for q-continuous, q-discrete Gaussian, Kesten Gaussian, Booleand Gaussian and many others examples.

We give only proof for Boolean Gaussian (\(0=q\)-discrete Gaussian) which is much more, since we have isometrical-isomorphism, that is for \(\Vert f_i\Vert =1\), \(f_i \in \mathcal {H}\), \(a_i = a(f_i)\), \(G^B(f_i) = a_i + a^{+}_i\)

$$\begin{aligned} \left\| \sum _{i=1}^N \alpha _i \otimes G^B(f_i)\right\| = \max \left\{ \left\| \sum _{j=1}^n \alpha _j \alpha _j^*\right\| ^{\frac{1}{2}},\left\| \sum _{j=1}^n \alpha _j^* \alpha _j\right\| ^{\frac{1}{2}}\right\} . \end{aligned}$$
(5)

Proof

Our Boolean Fock space \(\mathcal {F}_0(\mathcal {H}=\mathbb {C}\Omega \oplus \mathcal {H}=\mathbb {C}\Omega \oplus lin\{e_1,e_2,\ldots ,e_n\}\). Let \(G^B(f_i)(\Omega )=\omega _i(\Omega )=(a_i+a^+_i)(\Omega )=e_i\). Since by definition \(a_i(\Omega )=0\) and \(a^+_i(\Omega )=e_i\) so

$$\begin{aligned} \omega _i(e_k)=a_i(e_k)+a^+_i(e_k)=\delta _{ik}\Omega . \end{aligned}$$

Then

$$\begin{aligned} \omega _i^2(\Omega )=\omega _i(e_i)=\Omega . \end{aligned}$$

Therefore

$$\begin{aligned} \omega _i^2(e_k)=\omega _i(\omega _i(e_k))=\omega _i(\delta _{ik}\Omega )=\delta _{ik} e_i = e_k. \end{aligned}$$

So

$$\begin{aligned} \omega _i^2=\mathbb {I}. \end{aligned}$$

From that consideration we get

$$\begin{aligned} \omega _1= & {} \left( \begin{array}{c|c|c} 0&{}1&{}0 \\ \hline 1&{}0&{}0 \\ \hline 0&{}0&{}0 \end{array}~~~~\right) ,\\ \omega _2= & {} \left( \begin{array}{c|c|c} 0&{}0&{}1 \\ \hline 0&{}0&{}0 \\ \hline 1&{}0&{}0 \end{array}~~~~\right) . \end{aligned}$$

Let

$$\begin{aligned} T= & {} \sum _{i=1}^N \alpha _i \otimes \omega _i=\left( \begin{array}{c|c|c|c|c} 0&{}\alpha _1&{}\alpha _2&{}\alpha _3&{}\ldots \alpha _N \\ \hline \alpha _1&{}0&{}0&{}0&{}0 \\ \hline \alpha _2&{}0&{}0&{}0&{}0 \\ \hline \vdots &{}&{}&{}&{} \\ \hline \alpha _N&{}0&{}0&{}0&{}0 \\ \end{array}~~~~\right) ,\\ T^*= & {} \left( \begin{array}{c|c|c|c|c} 0&{}\alpha _1^*&{}\alpha _2^*&{}\alpha _3^*&{}\ldots \alpha _N^* \\ \hline \alpha _1^*&{}0&{}0&{}0&{}0 \\ \hline \alpha _2^*&{}0&{}0&{}0&{}0 \\ \hline \vdots &{}&{}&{}&{} \\ \hline \alpha _N^*&{}0&{}0&{}0&{}0 \\ \end{array}~~~~\right) . \end{aligned}$$

So we get

$$\begin{aligned} T T^*=\left( \begin{array}{c|c|c|c|c} \sum _{i=1}^N \alpha _i \alpha _i^*&{}0&{}0&{}\ldots &{}0 \\ \hline 0&{}\alpha _1 \alpha _1^*&{}\alpha _1 \alpha _2^*&{}\ldots &{}\alpha _1 \alpha _N^* \\ \hline 0&{}\alpha _2 \alpha _1^*&{}\alpha _2 \alpha _2^*&{}\ldots &{}\alpha _2 \alpha _N^* \\ \hline \vdots &{}&{}&{}&{} \\ \hline 0&{}\alpha _N \alpha _1^*&{}\alpha _N \alpha _2^*&{}\ldots &{}\alpha _N \alpha _N^* \end{array}~~~~\right) =\left( \begin{array}{c|c} A&{}0 \\ \hline 0&{}B \end{array}\right) , \end{aligned}$$

\(A=(\sum _{i=1}^N \alpha _i \alpha _i^*)\), \(B=(\alpha _i \alpha _j^*)_{i,j=1,2,\ldots ,N}\).

From this form we obtain \(\Vert T T^* \Vert = \max \{\Vert A\Vert ,\Vert B\Vert \}\). It is easy to see that \(\Vert A\Vert =\Vert \sum _{i=1}^N \alpha _i \alpha _i^*\Vert \). To calculate \(\Vert B\Vert \) we observe, that \(\Vert (\alpha _i \alpha _j^*)\Vert =\)

$$\begin{aligned}= & {} \left\| \left( \begin{array}{cccc} \alpha _1&{} 0 &{} \ldots &{} 0 \\ \alpha _2&{} 0 &{} \ldots &{} 0\\ \ldots &{} 0 &{} \ldots &{} 0\\ \alpha _N &{} 0 &{} \ldots &{} 0\\ \end{array}\right) \left( \begin{array}{cccc} \alpha _1&{} 0 &{} \ldots &{} 0 \\ \alpha _2&{} 0 &{} \ldots &{} 0\\ \ldots &{} 0 &{} \ldots &{} 0\\ \alpha _N &{} 0 &{} \ldots &{} 0\\ \end{array}\right) ^* \right\| \\= & {} \left\| \left( \begin{array}{cccc} \alpha _1^*&{} \alpha _2^* &{} \ldots &{} \alpha _N^* \\ 0 &{} 0 &{} \ldots &{} 0\\ \ldots &{} 0 &{} \ldots &{} 0\\ 0 &{} 0 &{} \ldots &{} 0\\ \end{array}\right) \left( \begin{array}{cccc} \alpha _1&{} 0 &{} \ldots &{} 0 \\ \alpha _2&{} 0 &{} \ldots &{} 0\\ \ldots &{} 0 &{} \ldots &{} 0\\ \alpha _N &{} 0 &{} \ldots &{} 0\\ \end{array}\right) \right\| \\= & {} \left\| \sum _{i=1}^N \alpha _i^* \alpha _i \right\| = \Vert B\Vert \end{aligned}$$

so we get our theorem for Boolean Gaussian. For other cases proofs are a little bit more complicated, see [6, 12]. \(\square \)

Corollary 1

If \(VN(X_1,X_2,\ldots ,X_n)\) has trace, then for some \(q=q(N)\), \(VN(X_1,X_2,\) \(\ldots ,X_n)\) is NOT injective and also it is a FACTOR.