Using exactly the same method but replacing stochastic independence by free-
ness in the sense of [35] and Gaussian by semi-circular (or “free-Gaussian”), it
is easy to extend the preceding Corollary to the free case. More generally, we
can use the q-Fock space (−1 ≤ q ≤ 1) and the associated q-Gaussian variables
described in [3, 4, 2].

Fix $q$ with $-1 \le q < 1$. Given a complex Hilbert space $\mathcal{H}$, we denote by $\mathcal{F}_q(\mathcal{H})$ the $q$-Fock space associated to $\mathcal{H}$. Let us assume that $\mathcal{H}$ is the complexification of a real Hilbert space $H$ so that $\mathcal{H} = H + iH$. For simplicity we assume $H = \ell_2(\mathbb{R})$. To any real Hilbert subspace $K \subset H$ we can associate (following [2]) a von Neumann algebra $\Gamma_q(K)$, so that we have natural embeddings $\Gamma_q(K_1) \subset \Gamma_q(K_2)$ when $K_1 \subset K_2$. Moreover $\Gamma_q(H)$ is equipped with a normalized trace (faithful and normal) denoted by $\tau_q$, that we may restrict to $\Gamma_q(K)$ to view the latter as a non-commutative probability space.

For any $h \in H$ we denote by $a^*(h)$ (resp. $a(h)$) the $q$-creation (resp. $q$-annihilation) operator on $\mathcal{F}_q(\mathcal{H})$ and we let $g_q(h) = a(h) + a^*(h)$. By definition, the von Neumann algebras $\Gamma_q(K)$ is generated by $\{g_q(h) \mid h \in K\}$. We will say that a family $X_1, \dots, X_n$ in $\Gamma_q(H)$ is $q$-independent if there are mutually orthogonal real subspaces $K_j \subset H$ such that $X_j \in \Gamma_q(K_j)$.

Let $B = B(\ell_2)$. More generally, consider $x_1, \dots, x_n$ in $B \otimes \Gamma_q(H)$. We will say that $x_1, \dots, x_n$ are $q$-independent if there are $K_j$ as above such that $x_j \in B \otimes \Gamma_q(K_j)$ for all $j = 1, \dots, n$. The elements of $g_q(H) = \{g_q(h) \mid h \in H\}$ will be called $q$-Gaussian.

More generally, an element $x \in B \otimes \Gamma_q(H)$ will be called $q$-Gaussian if $x \in B \otimes g_q(H)$. The $q$-Gaussian elements satisfy an analogue (called “second quantization”) of the rotational invariance of Gaussian distributions: For any $\mathbb{R}$-isometry $T: K \to H$ the families $\{g_q(t) \mid t \in K\}$ and $\{g_q(Tt) \mid t \in K\}$ have identical distributions. Here “same distribution” means equality of the moments of all non-commutative monomials. We will denote by $\tilde{T}: B \otimes g_q(K) \to B \otimes g_q(H)$ the linear map taking $b \otimes g_q(t)$ to $b \otimes g_q(Tt)$ ($b \in B, t \in K$). In particular, if $x \in g_q(K)$ and if $K_1 = K_2 = \dots = K_s = K$ we may use the isometry $u_s: K \to K_1 \oplus \dots \oplus K_s \subset H$ defined by $u_s(x) = s^{-1/2}(x \oplus \dots \oplus x)$ in order to define elements $x_j$ in $g_q(K_j)$ each with the same distribution as $x$ such that $x \stackrel{\text{dist}}{=} s^{-1/2}(x_1 + \dots + x_s)$.

Let $x_1, \dots, x_n$ be any sequence in $B \otimes g_q(H)$ and let $\pi$ be a partition. For any block $\alpha$ of $\pi$ we give ourselves an isometry $u_\alpha: H \to H_\alpha \subset H$ where $H_\alpha$ are mutually orthogonal (real) Hilbert subspaces. Then we define a sequence $(y_1, \dots, y_n)$ in $B \otimes g_q(H)$ by setting

$$
y_j = \tilde{u}_{\alpha} x_j, \qquad \forall j \in \alpha
$$

It is not hard to check that $\hat{\tau}(y_1 \dot{\otimes} \dots \dot{\otimes} y_n)$ depends only on $x = (x_1, \dots, x_n)$ and $\pi$ (and not on the $u_\alpha$'s). Therefore we may set

$$
\hat{\tau}(x^{\pi}) \stackrel{\text{def}}{=} \hat{\tau}(y_1 \dot{\otimes} \cdots \dot{\otimes} y_n).
$$

As before, by symmetry $\hat{\tau}_q(x_1 \dot{\otimes} \cdots \dot{\otimes} x_n) = 0$ for all odd $n$.
The $q$-Gaussian analogue of (15.1) is as follows:
PROPOSITION 15.6. For any even integer $n$ and any $n$-tuple $x_1, \dots, x_n$ in $B \otimes g_q(H)$ ($-1 \le q < 1$), we have

$$ \hat{\tau}_q(x_1 \dot{\otimes} \dots \dot{\otimes} x_n) = \sum_{\nu \in P_2(n)} \hat{\tau}_q(x^\nu). $$

*Proof.* With the preceding ingredients, this can be proved exactly as Proposition 15.1 above. □

In particular, replacing $B$ by $C$, we find

COROLLARY 15.7. Any $q$-Gaussian sequence $(x_j)$ in the sense of [2, Def. 3.3] with covariance equal to the identity matrix has $p$-th moment, defined by parings for any even integer $p$.

ACKNOWLEDGMENT. I am extremely grateful to Quanhua Xu for a careful reading that led to numerous corrections and improvements.

REFERENCES

[1] G. Andrews, *The theory of partitions*, Cambridge Univ. Press, 1984.

[2] M. Bożejko, B. Kmmerer, and R. Speicher, Roland. $q$-Gaussian processes: non-commutative and classical aspects. *Comm. Math. Phys.* 185 (1997), 129-154.

[3] M. Bożejko, R. Speicher, An example of a generalized Brownian motion. *Comm. Math. Phys.* 137 (1991), 519-531.

[4] M. Bożejko, R. Speicher, Interpolations between bosonic and fermionic relations given by generalized Brownian motions. *Math. Z.* 222 (1996), 135-159.

[5] A. Buchholz, Norm of convolution by operator-valued functions on free groups. Proc. Amer. Math. Soc. 127 (1999), no. 6, 1671-1682.

[6] A. Buchholz, Operator Khintchine inequality in non-commutative probability. *Math. Ann.* 319 (2001), 1-16.

[7] A. Buchholz, Optimal Constants in Khintchine Type Inequalities for Fermions, Rademachers and $q$-Gaussian Operators. *Bull. Polish Acad. Sci. Math.* 53 (2005), 315-321.

[8] D. Burkholder, Martingales and singular integrals in Banach spaces. Handbook of the geometry of Banach spaces, Vol. I, 233-269, North-Holland, Amsterdam, 2001.

[9] B. Collins and C. Male, The strong asymptotic freeness of Haar and deterministic matrices, To appear.
[10] M. Cotlar, A unified theory of Hilbert transforms and ergodic theorems. *Rev. Mat. Cuyana* 1 (1955), 105-167 (1956).

[11] E. G. Effros and Z-J.Ruan, *Operator spaces*, Oxford University Press, New York, 2000. xvi+363 pp.

[12] U. Haagerup and S. Thorbjørnsen, Random matrices and K-theory for exact $C^*$-algebras, *Doc. Math.* 4 (1999), 341-450 (electronic).

[13] A. Harcharras, Fourier analysis, Schur multipliers on $S^p$ and non-commutative $\Lambda(p)$-sets. *Studia Math.* 137 (1999), no. 3, 203-260.

[14] A. Harcharras, S. Neuwirth and K. Oleszkiewicz Lacunary matrices. *Indi-ana Univ. Math. J.* 50 (2001), no. 4, 1675-1689.

[15] M. Junge, Doob's inequality for non-commutative martingales. *J. Reine Angew. Math.* 549 (2002), 149-190.

[16] M. Junge, J. Parcet and Q. Xu, Rosenthal type inequalities for free chaos. *Ann. Probab.* 35 (2007), no. 4, 1374-1437.

[17] M. Junge and Q. Xu, Noncommutative Burkholder/Rosenthal inequalities. *Ann. Probab.* 31 (2003), no. 2, 948-995.

[18] M. Junge and Q. Xu, On the best constants in some non-commutative martingale inequalities. *Bull. London Math. Soc.* 37 (2005), no. 2, 243-253.

[19] M. Junge and Q. Xu, Noncommutative Burkholder/Rosenthal inequalities. II. Applications. *Israel J. Math.* 167 (2008), 227-282.

[20] F. Lust-Piquard, Inégalités de Khintchine daus $C_p$ ($1 < p < \infty$), *C.R. Acad. Sci. Paris* 303 (1986), 289-292.

[21] J. Parcet and N. Randrianantoanina, Gundy's decomposition for non-commutative martingales and applications. *Proc. London Math. Soc.* 93 (2006), no. 1, 227-252.

[22] G. Pisier, The operator Hilbert space OH, complex interpolation and tensor norms, *Mem. Amer. Math. Soc.* 122 (1996), no. 585.

[23] G. Pisier, Noncommutative vector valued $L_p$-spaces and completely $p$-summing maps, *Soc. Math. France. Astérisque* 237 (1998), vi+131 pp.

[24] G. Pisier, An inequality for $p$-orthogonal sums in non-commutative $L_p$, *Illinois J. Math.* 44 (2000), 901-923.

[25] G. Pisier, *Introduction to operator space theory*, Cambridge University Press, Cambridge, 2003.
[26] G. Pisier and Q. Xu, Non-commutative martingale inequalities, *Comm. Math. Phys.* 189 (1997), no. 3, 667-698.

[27] G. Pisier and Q. Xu, Non-commutative $L_p$-spaces, *Handbook of the Geometry of Banach Spaces, Vol. II*, North-Holland, Amsterdam, 2003.

[28] N. Randrianantoanina, Non-commutative martingale transforms. *J. Funct. Anal.* 194 (2002), no. 1, 181-212.

[29] N. Randrianantoanina, Square function inequalities for non-commutative martingales. *Israel J. Math.* 140 (2004), 333-365.

[30] N. Randrianantoanina, A weak type inequality for non-commutative martingales and applications. *Proc. London Math. Soc.* 91 (2005), no. 2, 509-542.

[31] N. Randrianantoanina, Conditioned square functions for noncommutative martingales. *Ann. Probab.* 35 (2007), no. 3, 1039-1070.

[32] É. Ricard and Q. Xu, Khintchine type inequalities for reduced free products and applications. *J. Reine Angew. Math.* 599 (2006), 27-59.

[33] W. Rudin, Trigonometric series with gaps. *J. Math. Mech.* 9 (1960) 203-228.

[34] A. Sinclair and R.R. Smith, *Hochschild cohomology of von Neumann algebras*, Cambridge University Press, Cambridge, 1995.

[35] D. Voiculescu, K. Dykema and A. Nica, *Free random variables*, American Mathematical Society, Providence, RI, 1992.

[36] Q. Xu, Embedding of Cq and Rq into noncommutative $L_p$-spaces, $1 \le p < q \le 2$. *Math. Ann.* 335 (2006) 109-131.

[37] Q. Xu, Operator-space Grothendieck inequalities for noncommutative $L_p$-spaces. *Duke Math. J.* 131 (2006), 525-574.

Gilles Pisier
Texas A&M University
College Station
TX 77843
U. S. A.
and
Université Paris VI
Equipe d'Analyse
Case 186
75252
Paris Cedex 05
France