# Non-commutative Khintchine Type Inequalities Associated with Free Groups

Gilles Pisier, Javier Parcet

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## To cite this version:

Gilles Pisier, Javier Parcet. Non-commutative Khintchine Type Inequalities Associated with Free Groups. Indiana University Mathematics Journal, 2005, 54 (2), pp.531-556. hal-05460871

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# Non-commutative Khintchine Type Inequalities Associated with Free Groups

JAVIER PARCET & GILLES PISIER

**ABSTRACT.** Let $\mathbb{F}_n$ denote the free group with $n$ generators $g_1, g_2, \dots, g_n$. Let $\lambda$ stand for the left regular representation of $\mathbb{F}_n$ and let $\tau$ be the standard trace associated to $\lambda$. Given any positive integer $d$, we study the operator space structure of the subspace $W_p(n, d)$ of $L_p(\tau)$ generated by the family of operators $\lambda(g_{i_1}g_{i_2}\cdots g_{i_d})$ with $1 \le i_k \le n$. Moreover, our description of this operator space holds up to a constant which does not depend on $n$ or $p$, so that our result remains valid for infinitely many generators. We also consider the subspace of $L_p(\tau)$ generated by the image under $\lambda$ of the set of reduced words of length $d$. Our result extends to any exponent $1 \le p \le \infty$ a previous result of Buchholz for the space $W_\infty(n, d)$. The main application is a certain interpolation theorem, valid for any degree $d$ (extending a result of the second author, restricted to $d=1$). In the simplest case $d=2$, our theorem can be stated as follows: consider the space $\mathcal{K}_p$ formed of all block matrices $a = (a_{ij})$ with entries in the Schatten class $S_p$, such that $a$ is in $S_p$ relative to $\ell_2 \otimes \ell_2$ and, moreover, such that $(\sum_{ij} a_{ij}^* a_{ij})^{1/2}$ and $(\sum_{ij} a_{ij} a_{ij}^*)^{1/2}$ both belong to $S_p$. We equip $\mathcal{K}_p$ with the maximum of the three corresponding norms. Then, for $2 \le p \le \infty$, we have $\mathcal{K}_p \simeq (\mathcal{K}_2, \mathcal{K}_\infty)_\theta$ with $1/p = (1-\theta)/2$.
INTRODUCTION

Let $\mathcal{R}_p(n)$ be the subspace of $L_p[0,1]$ generated by the classical Rademacher functions $r_1, r_2, \dots, r_n$. As is well-known, for any exponent $1 \le p < \infty$, the classical Khintchine inequalities provide a linear isomorphism between $\mathcal{R}_p(n)$ and $l_2(n)$ with constants independent of $n$. However, to describe the operator space structure of $\mathcal{R}_p(n)$ we need the so-called non-commutative Khintchine inequalities, introduced by F. Lust-Piquard in [6] and extended in [7] to the case $p = 1$, see also [2] for an analysis of the optimal constants.

To describe the non-commutative Khintchine inequalities, let us consider a family $x_1, x_2, \dots, x_n$ of elements in the Schatten class $S_p$. Then we have the following equivalences of norms for $1 \le p \le 2$

$$ \left\| \sum_{k=1}^{n} x_k r_k \right\|_{L_p([0,1];S_p)} \\ \approx \inf_{x_k = y_k + z_k} \left\{ \left\| \left( \sum_{k=1}^{n} y_k y_k^* \right)^{1/2} \right\|_{S_p} + \left\| \left( \sum_{k=1}^{n} z_k^* z_k \right)^{1/2} \right\|_{S_p} \right\} $$

while for $2 \le p < \infty$ we have

$$ \left\| \sum_{k=1}^{n} x_k r_k \right\|_{L_p([0,1];S_p)} \approx \max \left\{ \left\| \left( \sum_{k=1}^{n} x_k x_k^* \right)^{1/2} \right\|_{S_p}, \left\| \left( \sum_{k=1}^{n} x_k^* x_k \right)^{1/2} \right\|_{S_p} \right\}. $$

Again the constants do not depend on $n$. According to the purposes of this paper, it will be more convenient to rewrite these inequalities in terms of the spaces $R_p^n$ and $C_p^n$. Let us consider the Schatten class $S_p^n$ over the $n \times n$ matrices and let us denote its natural basis by

$$ \{e_{ij} | 1 \le i, j \le n\}. $$

We define $R_p^n$ to be the subspace of $S_p^n$ generated by $e_{11}, e_{12}, \dots, e_{1n}$, while $C_p^n$ will be the subspace generated by $e_{21}, \dots, e_{n1}$. That is, $R_p^n$ and $C_p^n$ can be regarded as the row and column subspaces of $S_p^n$ respectively. Note that both subspaces have a natural operator space structure inherited from $S_p^n$. In terms of the spaces $R_p^n$ and $C_p^n$, the non-commutative Khintchine inequalities can be rephrased by saying that $\mathcal{R}_p(n)$ is completely isomorphic to $R_p^n + C_p^n$ whenever $1 \le p \le 2$, and $\mathcal{R}_p(n)$ is completely isomorphic to $R_p^n \cap C_p^n$ for $2 \le p < \infty$. More concretely, when $1 \le p \le 2$ it follows that

$$ \left\| \sum_{k=1}^{n} x_k r_k \right\|_{L_p([0,1];S_p)} \\ \approx \inf_{x_k = y_k + z_k} \left\{ \left\| \sum_{k=1}^{n} y_k \otimes e_{1k} \right\|_{S_p(R_p^n)} + \left\| \sum_{k=1}^{n} z_k \otimes e_{k1} \right\|_{S_p(C_p^n)} \right\}, $$
while for $2 \le p \le \infty$ we have

$$ \left\| \sum_{k=1}^{n} x_k r_k \right\|_{L_p([0,1];S_p)} \approx \max \left\{ \left\| \sum_{k=1}^{n} x_k \otimes e_{1k} \right\|_{S_p(R_p^n)}, \left\| \sum_{k=1}^{n} x_k \otimes e_{k1} \right\|_{S_p(C_p^n)} \right\}. $$

Now let $\mathbf{F}_n$ be the free group with $n$ generators $g_1, g_2, \dots, g_n$. If $\lambda$ denotes the left regular representation of $\mathbf{F}_n$, the family of operators $\lambda(g_1), \lambda(g_2), \dots, \lambda(g_n)$ appear as the free analog of the sequence $r_1, r_2, \dots, r_n$ in this framework. Namely, let us consider the standard trace $\tau$ on $C_\lambda^*(\mathbf{F}_n)$. Then, by [3] it turns out that the subspace $W_p(n)$ of $L_p(\tau)$ generated by the operators $\lambda(g_1), \lambda(g_2), \dots, \lambda(g_n)$ is completely isomorphic to $\mathcal{R}_p(n)$ (with constants independent of $n$) for $1 \le p < \infty$. Moreover, for $p = \infty$ we have

$$ \left\| \sum_{k=1}^{n} a_k \otimes \lambda(g_k) \right\|_{L_{\infty}(\text{tr} \otimes \tau)} \approx \max \left\{ \left\| \sum_{k=1}^{n} a_k \otimes e_{1k} \right\|_{S_{\infty}(R_n)}, \left\| \sum_{k=1}^{n} a_k \otimes e_{k1} \right\|_{S_{\infty}(C_n)} \right\}, $$

where $R_n$ and $C_n$ are the usual expressions for $R_\infty^n$ and $C_\infty^n$. In other words, we also have $W_\infty(n) \approx R_n \cap C_n$ completely isomorphically. In this paper we shall generalize the mentioned complete isomorphism for $W_p(n)$, where the results above appear as the case of degree one. More concretely, given any positive integer $d$, we shall consider (operator-valued) homogeneous polynomials of degree $d$ in the variables $\lambda(g_1), \lambda(g_2), \dots, \lambda(g_n)$.

Let $W_p(n, d)$ be the subspace of $L_p(\tau)$ generated by the operators $\lambda(g_{i_1}g_{i_2}\cdots g_{i_d})$, where $1 \le i_k \le n$ for $1 \le k \le d$. The aim of this paper is to describe the operator space structure of $W_p(n, d)$ for any value of $d$. To that aim, we consider an auxiliary non-commutative Lebesgue space $L_p(\varphi)$ equipped with a faithful normal semi-finite trace $\varphi$. Then, describing the operator space structure of $W_p(n, d)$ becomes equivalent to describing the norm of

$$ \sum_{i_1, i_2, \dots, i_d = 1}^{n} a_{i_1 i_2 \dots i_d} \otimes \lambda(g_{i_1} g_{i_2} \dots g_{i_d}) $$

in $L_p(\varphi \otimes \tau)$, up to constants not depending on $n$ or $p$. Let $\mathcal{A}$ be the family of operators $\{a_{i_1 i_2 \dots i_d} \mid 1 \le i_k \le n\}$. This family can be regarded as an element of a matrix-valued $L_p(\varphi)$-space in several ways. Namely, given $0 \le k \le d$, we can construct a matrix $\mathcal{A}_k$ with entries in $\mathcal{A}$ by taking the first $k$ indices $i_1, i_2, \dots, i_k$ as the row index and the last $d-k$ indices $i_{k+1}, i_{k+2}, \dots, i_d$ as the column index. In other words, we consider the matrix

$$ \mathcal{A}_k = (a_{(i_1 \dots i_k), (i_{k+1} \dots i_d)}). $$
In particular, we see $\mathcal{A}_k$ as an element of the Haagerup tensor product $C_p^{n_k} \otimes_h R_p^{n_{d-k}}$ with values in $L_p(\varphi)$. This allows us to define the following family of spaces

$$ \mathcal{K}_p(n, d) = \sum_{k=0}^{d} C_p^{n_k} \otimes_h R_p^{n_{d-k}} \quad \text{for } 1 \le p \le 2, $$

$$ \mathcal{K}_p(n, d) = \bigcap_{k=0}^{d} C_p^{n_k} \otimes_h R_p^{n_{d-k}} \quad \text{for } 2 \le p \le \infty. $$

In this paper we shall prove that $W_p(n, d)$ and $\mathcal{K}_p(n, d)$ are completely isomorphic operator spaces with constants depending only on the degree $d$. More concretely, the following result holds.

**Theorem 0.1.** $\mathcal{K}_p(n, d)$ and $W_p(n, d)$ are completely isomorphic for $1 \le p \le \infty$ and there exists an absolute constant $c_d$, depending only on $d$, such that the following inequalities hold

$$ \frac{1}{c_d} \| \mathcal{A} \|_{L_p(\varphi; \mathcal{K}_p(n,d))} \le \left\| \sum_{i_1 \dots i_d = 1}^{n} a_{i_1 \dots i_d} \otimes \lambda(g_{i_1} \cdots g_{i_d}) \right\|_{L_p(\varphi \otimes \tau)} \\ \le c_d \| \mathcal{A} \|_{L_p(\varphi; \mathcal{K}_p(n,d))}. $$

Moreover, the natural projection $P: L_p(\tau) \to W_p(n, d)$ is c.b. with $\|P\|_{cb} \le c_d$.

The case $p = \infty$ of this result is the content of Buchholz's paper [1]. However, a slightly different definition of $W_\infty(n, d)$ was used there. Indeed, Buchholz takes the subspace of $L_\infty(\tau)$ generated by the image under $\lambda$ of the set of reduced words of length $d$. That is, inverses of generators are also allowed to appear. As we shall see, our arguments apply in this context for any $1 \le p \le \infty$, and we will also provide the proof. In Section 1 we set some preliminary results, while Section 2 is devoted to the proof of our main result and its analog for reduced words of length $d$.

# 1. PRELIMINARY RESULTS

One of the key points in the proof of our main result lies in the iteration of the non-commutative Khintchine inequality applied to the operator space $W_p(n)^{\otimes d}$. This process was already pointed in [11, Section 9.8], but we provide here an explicit description of the resulting inequalities. After that, we state an $L_p$-valued version of Fell's absorption principle that will be also needed in the proof.

**1.1. Iterations of the Khintchine inequality.** Let $F_n$ be the free group with $n$ generators $g_1, g_2, \dots, g_n$. If $(\delta_t)_{t \in F_n}$ denotes the natural basis of $\ell_2(F_n)$, the left regular representation $\lambda$ of $F_n$ is defined by the relation $\lambda(t_1)\delta_{t_2} = \delta_{t_1t_2}$. The reduced $C^*$-algebra $C_\lambda^*(F_n)$ is defined as the $C^*$-algebra generated in $B(\ell_2(F_n))$
by the operators $\lambda(t)$ when $t$ runs over $F_n$. Let us denote by $\tau$ the standard trace on $C_\lambda^*(F_n)$ defined by $\tau(x) = \langle x\delta_e, \delta_e \rangle$, where $e$ denotes the identity element of $F_n$. Then, we construct the non-commutative Lebesgue space $L_p(\tau)$ in the usual way and consider the subspace $W_p(n)$ of $L_p(\tau)$ generated by the operators $\lambda(g_1), \lambda(g_2), \dots, \lambda(g_n)$. The aim of this section is to describe the operator space structure of $W_p(n)^{\otimes d}$ as a subspace of $L_p(\tau^{\otimes d})$ for the exponents $2 \le p \le \infty$. As it was pointed out in [11], the case $1 \le p \le 2$ follows easily by duality. However, we shall not write the explicit inequalities in that case since we are not using them and the notation is considerably more complicated.

The following result can be regarded as a particular case of our main result for homogeneous polynomials of degree 1. It was proved in [3] for $p = \infty$, while the proof for $2 \le p < \infty$ can be found in Corollary 9.7.2 of [11]. We notice that its proof uses the fact that $R_p^n \cap C_p^n$ is an interpolation family for $2 \le p \le \infty$.

**Lemma 1.1.** The following equivalence of norms holds for $2 \le p \le \infty$,

$$ \left\| \sum_{k=1}^{n} a_k \otimes \lambda(g_k) \right\|_{L_p(\varphi \otimes \tau)} \\ \approx \max \left\{ \left\| \sum_{k=1}^{n} a_k \otimes e_{1k} \right\|_{L_p(\varphi; R_p^n)}, \left\| \sum_{k=1}^{n} a_k \otimes e_{k1} \right\|_{L_p(\varphi; C_p^n)} \right\}. $$

In fact, the linear map $u : R_p^n \cap C_p^n \to W_p(n)$ defined by

$$ u(e_{1k} \oplus e_{k1}) = \lambda(g_k), $$

is a complete isomorphism with $\|u\|_{cb} \le 2$ and completely contractive inverse. On the other hand, the canonical projection $P : L_p(\tau) \to W_p(n)$ satisfies $\|P\|_{cb} \le 2$.

Let us consider the group product $G_d = F_n \times F_n \times \dots \times F_n$ with $d$ factors. The left regular representation $\lambda_d$ of $G_d$ has the form

$$ \lambda_d(t_1, t_2, \dots, t_d) = \lambda(t_1) \otimes \lambda(t_2) \otimes \dots \otimes \lambda(t_d), $$

where $\lambda$ still denotes the left regular representation of $F_n$. In particular, the reduced $C^*$-algebra $C_{\lambda_d}^*(G_d)$ is endowed with the trace $\tau_d = \tau \otimes \tau \otimes \dots \otimes \tau$ with $d$ factors. This allows us to consider the non-commutative space $L_p(\tau_d)$ for any $1 \le p \le \infty$. Then we define the space $W_p(n)^{\otimes d}$ to be the subspace of $L_p(\tau_d)$ generated by the family of operators

$$ \lambda(g_{i_1}) \otimes \lambda(g_{i_2}) \otimes \dots \otimes \lambda(g_{i_d}). $$

If we apply repeatedly Lemma 1.1 to the sum

$$ S_d(a) = \sum_{i_1, \dots, i_d = 1}^{n} a_{i_1 i_2 \dots i_d} \otimes \lambda(g_{i_1}) \otimes \lambda(g_{i_2}) \otimes \dots \otimes \lambda(g_{i_d}) \in L_p(\varphi \otimes \tau_d), $$
then we easily get

$$ \|S_d(a)\|_{L_p(\varphi \otimes \tau_d)} \\ \le 2^d \max \left\{ \left\| \sum_{i_1, \dots, i_d=1}^n a_{i_1 \dots i_d} \otimes \xi_1(i_1) \otimes \dots \otimes \xi_d(i_d) \right\|_{S_p^{n_d}(L_p(\varphi))} \right\}, $$

where the maximum runs over all possible ways to choose the functions $\xi_1, \xi_2, \dots, \xi_d$ among $\xi_k(\cdot) = e_{1i}$ and $\xi_k(\cdot) = e_{1i}$. That is, each function $\xi_k$ can take values either in the space $R_p^n$ or in the space $C_p^n$. For a given selection of $\xi_1, \xi_2, \dots, \xi_d$ we split up these functions into two sets, one made up of the functions taking values in $R_p^n$ and the other taking values in $C_p^n$. More concretely, let us consider the sets

$$ R_{\xi} = \{k \mid \xi_k(i) = e_{1i}\}, $$

$$ C_{\xi} = \{k \mid \xi_k(i) = e_{i1}\}. $$

Then, if $C_{\xi}$ has $s$ elements, the sum

$$ \sum_{i_1, \dots, i_d=1}^{n} a_{i_1 \dots i_d} \otimes \xi_1(i_1) \otimes \dots \otimes \xi_d(i_d) $$

can be regarded as a $n^s \times n^{d-s}$ matrix with entries in $L_p(\varphi)$. Now we introduce a simpler notation already employed in [3]. Let $[m]$ be an abbreviation for the set $\{1, 2, \dots, m\}$. Then, if $\mathbb{P}_d(2)$ denotes the set of partitions $(\alpha, \beta)$ of $[d]$ into two disjoint subsets $\alpha$ and $\beta$, we denote by

$$ \pi_{\alpha} : [n]^d \to [n]^{|\alpha|} $$

the canonical projection given by $\pi_{\alpha}(I) = (i_k)_{k \in \alpha}$ for any $I = (i_1, \dots, i_d) \in [n]^d$. This notation allows us to express the inequality above in a much more understandable way. Namely, we have

$$ (1.1) \quad \|S_d(a)\|_{L_p(\varphi \otimes \tau_d)} \le 2^d \max_{(\alpha, \beta) \in \mathbb{P}_d(2)} \left\{ \left\| \sum_{I \in [n]^d} a_I \otimes e_{\pi_{\alpha}(I), \pi_{\beta}(I)} \right\|_{L_p(\varphi; S_{\beta}^{n_d})} \right\}. $$

**Remark 1.2.** By the same arguments, the converse of (1.1) holds with constant 1.

**1.2. Fell's absorption principle in $L_p$.** In the following, we shall use repeatedly the following $L_p$-valued version of the so-called Fell's absorption principle. This result might be known as folklore in the theory. However, we include the proof since we were not able to provide a reference.
**Absorption Principle in Lp.** Given a discrete group G, let us denote by $\lambda_G$ the
left regular representation of G and by $\tau_G$ the associated trace on the reduced C*-algebra of G.
Then, given any other unitary representation $\pi : G \to \pi(G)'',$ the
following representations are unitarily equivalent

$$
\lambda_G \otimes \pi \approx \lambda_G \otimes 1,
$$

where 1 stands for the trivial representation of G in π(G)''. Moreover, let us take any
faithful normalized trace ψ on π(G)''. Then, given any function a : G → Lp(φ)
finitely supported on G, the following equality holds for 1 ≤ p ≤ ∞

$$
(1.2) \quad \left\| \sum_{t \in G} a(t) \otimes \lambda_G(t) \otimes \pi(t) \right\|_{L_p(\varphi \otimes \tau_G \otimes \psi)} = \left\| \sum_{t \in G} a(t) \otimes \lambda_G(t) \right\|_{L_p(\varphi \otimes \tau_G)}.
$$

*Proof.* We refer the reader to Proposition 8.1 of [11] for a proof of the
claimed unitary equivalence. For the second assertion, it is easy to reduce to the
case when $\varphi$ is a tracial state. Then, we fix a pair of operators

$$
S = \sum_{t \in G} a(t) \otimes \lambda_G(t),
$$

$$
T = \sum_{t \in G} a(t) \otimes \lambda_G(t) \otimes \pi(t),
$$

with $a : G \to L_p(\varphi)$ finitely supported. Since $\psi$ is normalized, it is clear that

$$
(\varphi \otimes \tau_G)(S) = \varphi(a(e)) = \varphi(a(e))\psi(\pi(e)) = (\varphi \otimes \tau_G \otimes \psi)(T).
$$

Let us consider the operators $x = S*S$ and $z = T*T$. Recalling that the equality above holds for any pair of operators of the same kind, we deduce that

$$
(\varphi \otimes \tau_G)(x^n) = (\varphi \otimes \tau_G \otimes \psi)(z^n),
$$

for any integer $n \ge 0$. Now we let $A_x$ (resp. $A_z$) be the (commutative) algebra
generated by $x$ (resp. $z$) in $L_\infty(\varphi \otimes \tau_G)$ (resp. $L_\infty(\varphi \otimes \tau_G \otimes \psi)$). If $\mu_x$ (resp. $\mu_z$)
denotes the inherited probability measure on $A_x$ (resp. $A_z$), we have

$$
\int Q(x) d\mu_x = \int Q(z) d\mu_z,
$$

for any polynomial Q. By the Stone-Weierstrass theorem, we conclude that the
distribution of x with respect to μx coincides with the distribution of z with
respect to μz. Therefore, ||S||_{Lp}(φ⊗τG) = ||x||_{Lp/2}(μx) = ||z||_{Lp/2}(μz)
= ||T||_{Lp}(φ⊗τG⊗ψ). □
## 2. KHINTCHINE TYPE INEQUALITIES FOR $W_p(n, d)$

We first prove the case of degree 2, since the notation is simpler and it contains almost all the ingredients employed in the proof of the general case. This will simplify the reading of the paper. After the proof of the general case, we study the same problem when redefining the spaces $\mathcal{K}_p(n, d)$ and $W_p(n, d)$ so that we consider all the reduced words of length $d$.

**2.1. The case of degree 2.** As we did in the Introduction, we define $W_p(n, 2)$ to be the subspace of $L_p(\tau)$ generated by the operators $\lambda(g_ig_j)$ for $1 \le i, j \le n$. We shall also consider the spaces

$$ \mathcal{K}_p(n, 2) = \sum_{k=0}^{2} C_p^{nk} \otimes_h R_p^{n^{d-k}} \quad \text{for } 1 \le p \le 2, $$

$$ \mathcal{K}_p(n, 2) = \bigcap_{k=0}^{2} C_p^{nk} \otimes_h R_p^{n^{d-k}} \quad \text{for } 2 \le p \le \infty. $$

That is, if $\mathcal{A} = \{a_{ij} | 1 \le i, j \le n\} \subset L_p(\varphi)$, we consider the norms

$$ \left\| \sum_{i,j=1}^{n} a_{ij} \otimes e_{1,ij} \right\|_{L_p(\varphi; R_p^{n^2})}, $$

$$ \left\| \sum_{i,j=1}^{n} a_{ij} \otimes e_{ij} \right\|_{L_p(\varphi; S_p^n)}, $$

$$ \left\| \sum_{i,j=1}^{n} a_{ij} \otimes e_{ij,1} \right\|_{L_p(\varphi; C_p^{n^2})}. $$

We label them by $\|\mathcal{A}\|_0$, $\|\mathcal{A}\|_1$ and $\|\mathcal{A}\|_2$ respectively. Then, we have

$$ \|\mathcal{A}\|_{L_p(\varphi; \mathcal{K}_p(n,2))} = \max\{\|\mathcal{A}\|_k | 0 \le k \le 2\}, \quad \text{for } 2 \le p \le \infty. $$

This identity describes the operator space structure of $\mathcal{K}_p(n, 2)$ for $2 \le p \le \infty$. On the other hand, the obvious modifications lead to a description of the operator space structure of $\mathcal{K}_p(n, 2)$ for $1 \le p \le 2$. We shall prove the following result.

**Theorem 2.1.** $\mathcal{K}_p(n, 2)$ and $W_p(n, 2)$ are completely isomorphic for $1 \le p \le \infty$. More concretely, there exists an absolute constant $c$, independent of $n$ and $p$, such that the following inequalities hold

$$ \frac{1}{c} \|\mathcal{A}\|_{L_p(\varphi; \mathcal{K}_p(n,2))} \le \left\| \sum_{i,j=1}^{n} a_{ij} \otimes \lambda(g_i g_j) \right\|_{L_p(\varphi \otimes \tau)} \le c \|\mathcal{A}\|_{L_p(\varphi; \mathcal{K}_p(n,2))}. $$

Moreover, the natural projection $P: L_p(\tau) \to W_p(n, 2)$ is c.b. with $\|P\|_{cb} \le c$.
A statement similar to this result was proved by Haagerup (unpublished) and Buchholz [1] for $p = \infty$, see also Theorem 9.7.4 in [11]. Namely, the only difference is that Buchholz considered the whole set of words of length 2 instead of those words composed only of generators. Therefore, it is clear that Theorem 2.1 holds for $p = \infty$. By transposition, P also defines a completely bounded projection from $L_1(\tau)$ onto $W_1(n, 2)$. Hence, the last assertion of Theorem 2.1 follows by complex interpolation. In particular, if $p'$ stands for the conjugate exponent of $p$, it turns out that the dual $W_p(n, 2)^*$ is completely isomorphic to $W_{p'}(n, 2)$ for $1 \le p \le \infty$. On the other hand, it is obvious that $K_p(n, 2)^*$ is completely isometric to $K_{p'}(n, 2)$. In summary, it suffices to prove Theorem 2.1 for $2 \le p \le \infty$ since the case $1 \le p \le 2$ follows by duality.

**Remark 2.2.** From the previous considerations, it is clear that the space $W_p(n, 2)$ is completely isomorphic to $(W_\infty(n, 2), W_2(n, 2))_\theta$ for $\theta = 2/p$. Hence, if we knew a priori that $K_p(n, 2)$ is an interpolation family for $2 \le p \le \infty$, Theorem 2.1 would follow by complex interpolation between the obvious case for $p = 2$ and Buchholz's result. Conversely, Theorem 2.1 implies that $K_p(n, 2)$ is an interpolation family for $2 \le p \le \infty$. The fact that $K_p(n, 2)$ is an interpolation family was known to M. Junge [4] at the time of the preparation of this paper. He communicated to us the following sketch of his argument. Let $\tau_n$ stand for the normalized trace on the matrix algebra $M_n$ and let us write $\mathcal{N}$ for the free product of algebras

$$ \mathcal{N} = \mathcal{A}_k, \quad \text{with } \mathcal{A}_k = (M_n \oplus_{\infty} M_n, \varphi_n) \text{ and } \varphi_n = \frac{1}{2}(\tau_n \oplus \tau_n) $$

for $1 \le k \le n$. Then, if $\pi_k: \mathcal{A}_k \to \mathcal{N}$ denotes the natural inclusion, the map

$$ x \in R_p^{n^2} \cap S_p^n \cap C_p^{n^2} \to \sum_{k=1}^{n} \pi_k((x, -x)) \in L_p(\mathcal{N}) $$

is a complete isomorphism onto its image and the image is completely complemented in $L_p(\mathcal{N})$. Moreover, the constants appearing in the complete isomorphism and the projection considered above do not depend on $n$. This is based on the $L_p$ version of the operator-valued Voiculescu's inequality given in [5]. Here we shall give a different proof that will be useful in the proof of the general case of degree $d$.

Now we focus on the proof for the case $2 \le p \le \infty$. The lower estimate is much simpler and it even holds with $c = 1$. Namely, it suffices to check that

$$ \|\mathcal{A}\|_k \le \left\| \sum_{i,j=1}^{n} a_{ij} \otimes \lambda(g_i g_j) \right\|_{L_p(\varphi \otimes \tau)} $$

for any $0 \le k \le 2$. But we know that it holds trivially for $p = 2$ and also, by Buchholz's result, for $p = \infty$. Therefore, the lower estimate follows by complex
interpolation since $R_p^{n_1}$, $S_p^n$ and $C_p^{n_2}$ are interpolation families. Hence, we just need to prove the upper estimate. To that aim, we go back to Section 1, where we considered the group $G_2 = F_n \times F_n$ and the subspace $W_p(n) \otimes W_p(n)$ of $L_p(\tau_2)$ generated by the family of operators $\lambda(g_i) \otimes \lambda(g_j)$. We consider the subspace $V_p(n, 2)$ of $L_p(\tau_3)$ defined by

$$V_p(n, 2) = \left\{ \sum_{i,j=1}^{n} \alpha_{ij} \lambda(g_i) \otimes \lambda(g_j) \otimes \lambda(g_i g_j) \in L_p(\tau_3) \mid \alpha_{ij} \in \mathbb{C} \right\}.$$

**Lemma 2.3.** $V_p(n, 2)$ is a completely complemented subspace of $L_p(\tau_3)$.

*Proof.* We know that both $W_p(n)$ and $W_p(n, 2)$ are completely complemented in $L_p(\tau)$. In particular, the projection which maps

$$\sum_{u,v,w \in F_n} \alpha_{uvw} \lambda(u) \otimes \lambda(v) \otimes \lambda(w) \in L_p(\tau_3)$$

to the sum

$$\Sigma_2(\alpha) = \sum_{i,j,r,s=1}^{n} \alpha_{ijrs} \lambda(g_i) \otimes \lambda(g_j) \otimes \lambda(g_r g_s) \in W_p(n) \otimes W_p(n) \otimes W_p(n, 2),$$

is completely bounded with cb norm uniformly bounded in $n$ and $p$. This shows that $W_p(n) \otimes W_p(n) \otimes W_p(n, 2)$ is completely complemented in $L_p(\tau_3)$. After that, we project onto $V_p(n, 2)$ by using the standard diagonal projection

$$P(\Sigma_2(\alpha)) = \sum_{i,j,r,s=1}^{n} \iint \varepsilon_i \delta_j [\alpha_{ijrs} \lambda(g_i) \otimes \lambda(g_j) \otimes \lambda(g_r g_s)] \varepsilon_r \delta_s \, d\mu(\varepsilon) \, d\mu(\delta),$$

where $\mu$ is the normalized counting measure on $\{-1, 1\}^n$. Now, in the case $p = \infty$ it is not difficult to see that the norm of

$$\Sigma_2^{\pm}(\alpha) = \sum_{i,j,r,s=1}^{n} \varepsilon_i \delta_j [\alpha_{ijrs} \lambda(g_i) \otimes \lambda(g_j) \otimes \lambda(g_r g_s)] \varepsilon_r \delta_s$$

in $L_\infty(\tau_3)$ is completely equivalent to that of $\Sigma_2(\alpha)$ in $L_\infty(\tau_3)$ for any choice of signs $\varepsilon_i$, $\delta_j$, $\varepsilon_r$, $\delta_s$. Moreover, the constants do not depend on the signs taken. Indeed, by Buchholz's result the norm of $\Sigma_2^{\pm}(\alpha)$ is completely equivalent to the norm of an element in $L_\infty(\tau_2; \mathcal{K}_\infty(n, d))$. In that case, the signs $\varepsilon_r \delta_s$ can be regarded as a Schur multiplier. Hence, $\varepsilon_r$ and $\delta_s$ can be dropped by means of [11, Exercise 1.5]. On the other hand, the signs $\varepsilon_i$ and $\delta_j$ disappear by applying Fell's absorption principle. Moreover, the norms of $\Sigma_2(\alpha)$ and $\Sigma_2^{\pm}(\alpha)$ clearly coincide for $p = 2$. Therefore, since $W_p(n) \otimes W_p(n) \otimes W_p(n, 2)$ is an interpolation
family, both norms are completely equivalent for any $2 \le p \le \infty$. In particular, by Jensen's inequality we have

$$ \|P(\Sigma_2(\alpha))\|_{L_p(\tau_3)} \le c \|\Sigma_2(\alpha)\|_{L_p(\tau_3)}, $$

for some absolute constant c. Since the same holds taking values in $S_p$, it turns out that P is a completely bounded projection with constants independent of $n$ and $p$. In summary, putting all together, the result follows. This completes the proof. □

The next step in the proof of Theorem 2.1 is to show that $W_p(n, 2)$ and $V_p(n, 2)$ are completely isomorphic operator spaces for any exponent $2 \le p \le \infty$. Namely, given a family $\mathcal{A} = \{a_{ij} | 1 \le i, j \le n\}$ in $L_p(\varphi)$, we consider the sum

$$ \Sigma_2(a) = \sum_{i,j=1}^{n} \lambda(g_i) \otimes \lambda(g_j) \otimes a_{ij} \otimes \lambda(g_i g_j) \in L_p(\tau_2 \otimes \varphi \otimes \tau). $$

Applying Buchholz's result to $\Sigma_2(a)$, we get the equivalence of norms

$$ \|\Sigma_2(a)\|_{L_\infty(\tau_2 \otimes \varphi \otimes \tau)} \approx \max\{\|\mathcal{A}'\|_k \mid 0 \le k \le 2\}, $$

where $\mathcal{A}' = \{a'_{ij} | 1 \le i, j \le n\}$, with $a'_{ij} = \lambda(g_i) \otimes \lambda(g_j) \otimes a_{ij}$.

**Remark 2.4.** Note that

$$ \sum_{i,j=1}^{n} a'_{ij} \otimes e_{ij} = \Phi_1 \cdot \left[ \sum_{i,j=1}^{n} 1 \otimes a_{ij} \otimes e_{ij} \right] \cdot \Phi_2, $$

where

$$ \Phi_1 = \sum_{i=1}^{n} \lambda(g_i) \otimes 1 \otimes 1 \otimes e_{ii}, $$

$$ \Phi_2 = \sum_{j=1}^{n} 1 \otimes \lambda(g_j) \otimes 1 \otimes e_{jj}. $$

Therefore, according to Remark 2.4 and since $\Phi_1$ and $\Phi_2$ are unitary, we obtain $\|\mathcal{A}\|_1 = \|\mathcal{A}'\|_1$. The obvious modifications lead to $\|\mathcal{A}\|_k = \|\mathcal{A}'\|_k$ for $k = 0$ and $k = 2$. In summary, if

$$ S_2(a) = \sum_{i,j=1}^{n} a_{ij} \otimes \lambda(g_i g_j), $$

we conclude that the norm of $S_2(a)$ in $L_\infty(\varphi \otimes \tau)$ is completely equivalent to the norm of $\Sigma_2(a)$ in $L_\infty(\tau_2 \otimes \varphi \otimes \tau)$. Now recall that by Lemma 2.3, $V_p(n, 2)$
is an interpolation family for $2 \le p \le \infty$. Then, since the norm of these sums obviously coincide when $p = 2$, we get by complex interpolation

$$ (2.1) \quad \left\| \sum_{i,j=1}^{n} a_{ij} \otimes \lambda(g_i g_j) \right\|_{p} \le c \left\| \sum_{i,j=1}^{n} \lambda(g_i) \otimes \lambda(g_j) \otimes a_{ij} \otimes \lambda(g_i g_j) \right\|_{p}, $$

for any $2 \le p \le \infty$. Here $c$ denotes an absolute constant, independent of $n$ and $p$. In what follows, the value of $c$ might change from one instance to another. Now we apply the iteration of the Khintchine inequality (1.1) to inequality (2.1) to obtain

$$ (2.2) \quad \left\| \sum_{i,j=1}^{n} a_{ij} \otimes \lambda(g_i g_j) \right\|_{p} \le c \max_{(\alpha, \beta) \in \mathbb{P}_2(2)} \left\{ \left\| \sum_{I \in [n]^2} \tilde{a}_I \otimes e_{\pi_\alpha(I), \pi_\beta(I)} \right\|_{L_p(\varphi \otimes \tau; S_p^{n^2})} \right\}, $$

with $\tilde{a}_{ij} = a_{ij} \otimes \lambda(g_ig_j)$. Hence, we have four terms on the right

$$ \left\| \sum_{i,j=1}^{n} \tilde{a}_{ij} \otimes e_{1,ij} \right\|_{L_p(\varphi \otimes \tau; R_p^{n^2})}, $$

$$ \left\| \sum_{i,j=1}^{n} \tilde{a}_{ij} \otimes e_{ij} \right\|_{L_p(\varphi \otimes \tau; S_p^{n^2})}, $$

$$ \left\| \sum_{i,j=1}^{n} \tilde{a}_{ij} \otimes e_{ij,1} \right\|_{L_p(\varphi \otimes \tau; C_p^{n^2})}, $$

$$ \left\| \sum_{i,j=1}^{n} \tilde{a}_{ij} \otimes e_{ji} \right\|_{L_p(\varphi \otimes \tau; S_p^{n^2})}. $$

If $\tilde{\mathcal{A}} = \{\tilde{a}_{ij} | 1 \le i, j \le n\}$, the first three terms are nothing but $\|\tilde{\mathcal{A}}\|_0$, $\|\tilde{\mathcal{A}}\|_1$, $\|\tilde{\mathcal{A}}\|_2$. Arguing as above we have

$$ \max\{\|\mathcal{A}\|_k | 0 \le k \le 2\} = \max\{\|\tilde{\mathcal{A}}\|_k | 0 \le k \le 2\}. $$

In particular, the proof will be completed if we see that

$$ \left\| \sum_{i,j=1}^{n} \tilde{a}_{ij} \otimes e_{ji} \right\|_{L_p(\varphi \otimes \tau; S_p^{n^2})} \le c \max\{\|\mathcal{A}\|_k | 0 \le k \le 2\}. $$

This is the content of the following lemma. The proof is not complicated but, as we shall see in the next paragraph, it constitutes one of the key points in the proof of the general case.
**Lemma 2.5.** The following inequality holds

$$ \left\| \sum_{i,j=1}^{n} a_{ij} \otimes \lambda(g_i g_j) \otimes e_{ji} \right\|_{L_p(\varphi \otimes \tau; S_p^n)} \\ \leq \operatorname{cmax} \left\{ \left\| \sum_{i,j=1}^{n} a_{ij} \otimes e_{1,ij} \right\|_{L_p(\varphi; R_p^{n^2})}, \left\| \sum_{i,j=1}^{n} a_{ij} \otimes e_{ij,1} \right\|_{L_p(\varphi; C_p^{n^2})} \right\}. $$

*Proof.* When $p = \infty$ we can apply Buchholz's result to obtain

$$ \left\| \sum_{i,j=1}^{n} a_{ij} \otimes \lambda(g_i g_j) \otimes e_{ji} \right\|_{L_{\infty}(\varphi \otimes \tau; S_{\infty}^n)} \leq \operatorname{cmax}\{A, B, C\}, $$

where the terms A and C are given by

$$ A = \left\| \left( \sum_{i,j=1}^{n} (a_{ij} \otimes e_{ji})(a_{ij} \otimes e_{ji})^* \right)^{1/2} \right\|_{\infty} = \sup_{1 \le j \le n} \left\| \left( \sum_{i=1}^{n} a_{ij} a_{ij}^* \right)^{1/2} \right\|_{\infty}, $$

$$ C = \left\| \left( \sum_{i,j=1}^{n} (a_{ij} \otimes e_{ji})^* (a_{ij} \otimes e_{ji}) \right)^{1/2} \right\|_{\infty} = \sup_{1 \le i, j \le n} \left\| \left( \sum_{j=1}^{n} a_{ij}^* a_{ij} \right)^{1/2} \right\|_{\infty}. $$

In particular, we have the following estimates

$$ A \le \left\| \left( \sum_{i,j=1}^{n} a_{ij} a_{ij}^* \right)^{1/2} \right\|_{\infty} = \left\| \sum_{i,j=1}^{n} a_{ij} \otimes e_{1,ij} \right\|_{L_{\infty}(\varphi; R_{\infty}^{n^2})}, $$

$$ C \le \left\| \left( \sum_{i,j=1}^{n} a_{ij}^* a_{ij} \right)^{1/2} \right\|_{\infty} = \left\| \sum_{i,j=1}^{n} a_{ij} \otimes e_{ij,1} \right\|_{L_{\infty}(\varphi; C_{\infty}^{n^2})}. $$

It remains to estimate the middle term B. We have

$$ \begin{aligned} B &= \left\| \sum_{i,j=1}^{n} a_{ij} \otimes e_{ji} \otimes e_{ij} \right\|_{L_{\infty}(\varphi; S_{\infty}^n \otimes_{\min} S_{\infty}^n)} \\ &= \left\| \left( \sum_{i,j=1}^{n} a_{ij}^* a_{ij} \otimes e_{ii} \otimes e_{jj} \right)^{1/2} \right\|_{L_{\infty}(\varphi; S_{\infty}^n \otimes_{\min} S_{\infty}^n)} \\ &= \sup_{1 \le i,j \le n} \|a_{ij}\|_{L_{\infty}(\varphi)}. \end{aligned} $$

Therefore, the term B is even smaller than A, C. This completes the proof for the case $p = \infty$. On the other hand, for the case $p = 2$ we clearly have an equality. Finally, we recall that

$$ L_p(\varphi; S_p^n (W_p(n, 2))^{op}) \quad \text{and} \quad L_p(\varphi; R_p^{n^2} \cap C_p^{n^2}) $$
are interpolation families, see Section 9.5 in [11] for the details. Therefore, the result follows for $2 \le p \le \infty$ by complex interpolation. This completes the proof. □

**2.2. The general case.** Now we prove the analog of Theorem 2.1 for any positive integer $d$. As we shall see, there exist a lot of similarities with the proof for degree 2. Therefore, we shall not repeat in detail those arguments which already appeared above. The statement of this result is the following.

**Theorem 2.6.** $\mathcal{K}_p(n, d)$ and $W_p(n, d)$ are completely isomorphic for $1 \le p \le \infty$. More concretely, there exists an absolute constant $c_d$, depending only on $d$, such that the following inequalities hold

$$ \frac{1}{c_d} \|\mathcal{A}\|_{L_p(\varphi; \mathcal{K}_p(n,d))} \le \left\| \sum_{i_1 \dots i_d = 1}^{n} a_{i_1 \dots i_d} \otimes \lambda(g_{i_1} \dots g_{i_d}) \right\|_{L_p(\varphi \otimes \tau)} \\ \le c_d \|\mathcal{A}\|_{L_p(\varphi; \mathcal{K}_p(n,d))}. $$

Moreover, the natural projection $P: L_p(\tau) \to W_p(n, d)$ is c.b. with $\|P\|_{cb} \le c_d$.

Before starting the proof of Theorem 2.6, we point out some remarks analogous to those given for Theorem 2.1. The arguments needed to prove the assertions given below are the same as the ones we used for the case of degree 2.

* Again, a statement similar to this result was proved by Buchholz [1] for $p = \infty$. Buchholz's considered the whole set of words of length $d$. Therefore, Theorem 2.6 holds for $p = \infty$ by Buchholz's more general statement.

* By transposition and complex interpolation, the last assertion of Theorem 2.6 follows. Hence, $W_p(n, d)$ interpolates well up to complete isomorphism for $1 \le p \le \infty$. Moreover, $W_{p'}(n, d)$ is completely isomorphic to $W_p(n, d)^*$. In particular, since $\mathcal{K}_p(n, d)$ behaves well with respect to duality, it suffices to prove Theorem 2.6 for $2 \le p \le \infty$.

* Given a family of operators $\mathcal{A} = \{a_{i_1 \dots i_d} | 1 \le i_k \le n\}$ in $L_p(\varphi)$, we define

$$ \|\mathcal{A}\|_k = \left\| \sum_{i_1, \dots, i_d = 1}^{n} a_{i_1 \dots i_d} \otimes e_{(i_1 \dots i_k), (i_{k+1} \dots i_d)} \right\|_{L_p(\varphi; C_p^{n_k} \otimes_h R_p^{n^{d-k}})} $$

* If $2 \le p \le \infty$, the lower estimate holds with $c_d = 1$. Namely, it follows by Buchholz's result and complex interpolation since we are allowed to look separately at the inequalities

$$ \|\mathcal{A}\|_k \le \left\| \sum_{i_1, \dots, i_d = 1}^{n} a_{i_1 \dots i_d} \otimes \lambda(g_{i_1} \dots g_{i_d}) \right\|_{L_p(\varphi \otimes \tau)}, \quad \text{for } 0 \le k \le d. $$

Recall that $C_p^{n_k} \otimes_h R_p^{n^{d-k}}$ is a rectangular Schatten $p$-class of size $n^k \times n^{d-k}$. In particular, it follows that it is an interpolation family.
In what follows, $c_d$ will denote a constant depending only on $d$ and whose value might change from one instance to another. Now we start the proof of Theorem 2.6. By the considerations above, it suffices to prove the upper estimate for $2 \le p \le \infty$. Let $G$ stand for the free group $F_{nd}$ with $nd$ generators and let $\psi_{nd}$ be the natural trace on the reduced $C^*$-algebra of $G$. We label the generators by $g_{1k}, g_{2k}, \dots, g_{dk}$ with $1 \le k \le n$. If $\lambda_G$ denotes the left regular representation of $G$, we consider the family of operators

$$ \mathcal{A}^{\dagger} = \{ a_{i_1 i_2 \dots i_d} \otimes \lambda_G(g_{1i_1} g_{2i_2} \dots g_{di_d}) \mid 1 \le i_k \le n \}. $$

That is, we take the image under $\lambda_G$ of the set of reduced words of length $d$ where the first letter is one of the first $n$ generators, the second letter is one of the second $n$ generators and so on. Let us write $W_p(n, d)$ to denote the subspace of $L_p(\psi_{nd})$ generated by the operators $\lambda_G(g_{1i_1} \dots g_{di_d})$. Then, we have

$$ (2.3) \qquad \left\| \sum_{i_1, \dots, i_d = 1}^{n} a_{i_1 \dots i_d} \otimes \lambda(g_{i_1} \dots g_{i_d}) \right\|_p \\ \le c_d \left\| \sum_{i_1, \dots, i_d = 1}^{n} a_{i_1 \dots i_d} \otimes \lambda_G(g_{1i_1} \dots g_{di_d}) \right\|_p. $$

Namely, for $p = \infty$ this follows by Buchholz's result. Then, since both $W_p(n, d)$ and $W_p(n, n, d)$ are interpolation families, inequality (2.3) holds for any $2 \le p \le \infty$ by complex interpolation. Now, proceeding as in the previous paragraph, we consider the group $G_d = F_n \times F_n \times \dots \times F_n$ and the subspace $W_p(n)^{\otimes d}$ of $L_p(\tau_d)$ generated by the family of operators

$$ \lambda(g_{i_1}) \otimes \lambda(g_{i_2}) \otimes \dots \otimes \lambda(g_{i_d}). $$

Then we define $\mathcal{V}_p(n, d)$ as the subspace of $L_p(\tau_d \otimes \psi_{nd})$ defined by

$$ \mathcal{V}_p(n, d) = \left\{ \sum_{i_1, \dots, i_d = 1}^{n} \alpha_{i_1 \dots i_d} \lambda(g_{i_1}) \otimes \dots \otimes \lambda(g_{i_d}) \otimes \lambda_G(g_{1i_1} \dots g_{di_d}) \mid \alpha_{i_1 \dots i_d} \in \mathbb{C} \right\}. $$

Recalling that both $W_p(n)$ and $W_p(n, n, d)$ are completely complemented in their respective $L_p$ spaces, it can be showed just like in Lemma 2.3 that $\mathcal{V}_p(n, d)$ is completely complemented in $L_p(\tau_d \otimes \psi_{nd})$, with constants depending only on the degree $d$. The next step in the proof is to obtain the analog of inequality (2.1).
Namely, the inequality

$$
(2.4) \quad \left\| \sum_{i_1, \dots, i_d = 1}^{n} a_{i_1 \dots i_d} \otimes \lambda_G(g_{1i_1} \cdots g_{di_d}) \right\|_p \\
\le c_d \left\| \sum_{i_1, \dots, i_d = 1}^{n} \lambda(g_{i_1}) \otimes \cdots \otimes \lambda(g_{i_d}) \otimes a_{i_1 \dots i_d} \otimes \lambda_G(g_{1i_1} \cdots g_{di_d}) \right\|_p,
$$

for $2 \le p \le \infty$. The proof of this inequality is identical to the one given for inequality (2.1). Indeed, we have just showed that both $W_p(n,d,n,d)$ and $V_p(n,d)$ are interpolation families. Therefore, the proof of (2.4) works by complex interpolation between the obvious case $p=2$ and the case $p=\infty$. When $p=\infty$, the idea consists in applying to both terms in (2.4) Buchholz's result for degree $d$. Then, inequality (2.4) becomes equivalent to

$$
\max\{\|\mathcal{A}\|_k \mid 0 \le k \le d\} \le c_d \max\{\|\mathcal{A}'\|_k \mid 0 \le k \le d\},
$$

where $\mathcal{A}'$ is given by

$$
\mathcal{A}' = \{ a_{i_1 i_2 \dots i_d} \otimes \lambda(g_{i_1}) \otimes \dots \otimes \lambda(g_{i_d}) \mid 1 \le i_k \le n \}.
$$

**Remark 2.7.** Again it is clear that $\|\mathcal{A}\|_k = \|\mathcal{A}'\|_k$ for any $0 \le k \le d$. Namely, as we pointed out in Remark 2.4, the sum

$$
\sum_{i_1, \dots, i_d = 1}^{n} \lambda(g_{i_1}) \otimes \dots \otimes \lambda(g_{i_d}) \otimes a_{i_1 i_2 \dots i_d} \otimes e_{(i_1 \dots i_k), (i_{k+1} \dots i_d)}
$$

factorizes as

$$
\Phi_1 \cdot \left[ \sum_{i_1, \dots, i_d = 1}^{n} a_{i_1 i_2 \dots i_d} \otimes e_{(i_1 \dots i_k), (i_{k+1} \dots i_d)} \right] \cdot \Phi_2,
$$

where $\Phi_1$ (resp. $\Phi_2$) is a $n^k \times n^k$ (resp. $n^{d-k} \times n^{d-k}$) unitary mapping.

This completes the proof of inequality (2.4). Another possible approach to (2.4) is given by iterating Fell's absorption principle *d* times, with the suitable choice for $\pi$ each time. We leave the details to the reader. Notice that Fell's absorption principle shows that (2.4) is in fact an equality with $c_d = 1$. Then we apply the iteration of Khintchine inequality (1.1) to inequality (2.4). This gives

$$
(2.5) \quad \left\| \sum_{i,j=1}^{n} a_{i_1 \dots i_d}^{\dagger} \right\|_{p} \\
\le c_d \max_{(\alpha, \beta) \in \mathbb{P}_d(2)} \left\{ \left\| \sum_{I \in [n]^d} a_I^{\dagger} \otimes e_{\pi_{\alpha}(I), \pi_{\beta}(I)} \right\|_{L_p(\varphi \otimes \psi_{nd}; S_p^{nd})} \right\},
$$
with $a_{i_1 \dots i_d}^\dagger = a_{i_1 \dots i_d} \otimes \lambda_G(g_{1i_1} \dots g_{di_d})$. Now we have $2^d$ terms on the right. Before going on, let us look for a moment at the norm of the space $\mathcal{K}_p(n, d)$. Concretely, if we rewrite the definition of $\mathcal{K}_p(n, d)$ for $2 \le p \le \infty$ with the notation employed at the end of Section 1, we obtain

$$ (2.6) \quad \begin{aligned} & \|\mathcal{A}\|_{L_p(\phi; \mathcal{K}_p(n, d))} \\ &= \max_{0 \le k \le d} \left\{ \left\| \sum_{I \in [n]^k} \sum_{J \in [n]^{d-k}} a_{IJ} \otimes e_{I,J} \right\|_{L_p(\phi; C_p^{nk} \otimes_h R_p^{d-k})} \right\}. \end{aligned} $$

To complete the proof of Theorem 2.6 it remains to see that the right side of (2.5) is controlled by the right side of (2.6). Here is where the proof of the general case differs from that of degree 2. Let us sketch briefly how we shall conclude the proof. If $\text{RHS}_{(2.5)}$ stands for the right hand side of (2.5), we shall prove that

$$ (2.7) \quad \text{RHS}_{(2.5)} \le c_d \max_{0 \le k \le d} \left\{ \left\| \sum_{I \in [n]^k} \sum_{J \in [n]^{d-k}} a_{IJ}^\dagger \otimes e_{I,J} \right\|_{L_p(\phi \otimes \psi_{nd}; C_p^{nk} \otimes_h R_p^{d-k})} \right\}. $$

Assuming we have (2.7), the proof is completed since the right side of (2.7) coincides with the right side of (2.6). Namely, it clearly follows by the same factorization argument as above. That is,

$$ \sum_{I \in [n]^k} \sum_{J \in [n]^{d-k}} a_{IJ}^\dagger \otimes e_{I,J} = \Phi_1 \cdot \left[ \sum_{I \in [n]^k} \sum_{J \in [n]^{d-k}} a_{IJ} \otimes e_{I,J} \right] \cdot \Phi_2, $$

with $\Phi_1$ and $\Phi_2$ suitably chosen unitary mappings. In summary, it remains to prove inequality (2.7). Recall that the $d+1$ terms which appear on the right hand side of (2.7) also appear on its left hand side. They correspond to $\alpha = \emptyset$, [1], [2], ..., [d]. The remaining terms correspond to certain transpositions just like the term we bounded with the aid of Lemma 2.5. Thus, we just need to show that the transposed terms are controlled by the non-transposed ones. To that aim, we need to introduce some notation. Given $(\alpha, \beta) \in \mathbb{P}_d(2)$, we define

$$ a = \max\{k \mid k \in \alpha\}, \quad b = \min\{k \mid k \in \beta\}. $$

We define $a = 0$ for $\alpha = \emptyset$ and $b = d+1$ for $\beta = \emptyset$. We shall say that $(\alpha, \beta)$ is *non-transposed* whenever $a < b$ and $(\alpha, \beta)$ will be called *transposed* otherwise. Let us introduce the number $T(\alpha, \beta) = a - b$, so that $(\alpha, \beta)$ is transposed whenever $T(\alpha, \beta) > 0$. The proof of the remaining inequality lies on the following claim.

**Claim 2.8.** Let $(\alpha, \beta)$ be a transposed element of $\mathbb{P}_d(2)$. Then, we have

$$ \begin{aligned} & \left\| \sum_{I \in [n]^d} a_I^\dagger \otimes e_{\pi_\alpha(I), \pi_\beta(I)} \right\|_p \\ & \le c_d \max \left\{ \left\| \sum_{I \in [n]^d} a_I^\dagger \otimes e_{\pi_{\alpha_1}(I), \pi_{\beta_1}(I)} \right\|_p, \left\| \sum_{I \in [n]^d} a_I^\dagger \otimes e_{\pi_{\alpha_2}(I), \pi_{\beta_2}(I)} \right\|_p \right\}. \end{aligned} $$
for some $(\alpha_1, \beta_1)$ and $(\alpha_2, \beta_2)$ in $\mathbb{P}_d(2)$ satisfying

$$T(\alpha_1, \beta_1) < T(\alpha, \beta), \quad T(\alpha_2, \beta_2) < T(\alpha, \beta).$$

**Remark 2.9.** Clearly, iteration of Claim 2.8 concludes the proof of Theorem 2.6.

*Proof.* If $G = F_{nd}$, we define $\pi : G \to \mathcal{B}(\ell_2(G))$ by

$$\pi(g_{rs}) = \begin{cases} \lambda_G(g_{rs}) & \text{if } r = a, b, \\ 1 & \text{otherwise,} \end{cases}$$

where $\lambda_G$ denotes the left regular representation of $G$ and $1 \le s \le n$. Clearly, the mapping $\pi$ extends to a unitary representation of $G$. Therefore, by Fell's absorption principle we have

$$\left\| \sum_{I \in [n]^d} a_I^\dagger \otimes e_{\pi_\alpha(I), \pi_\beta(I)} \right\|_p = \left\| \sum_{I \in [n]^d} a_I^\dagger \otimes \lambda_G(g_{bi_b} g_{ai_a}) \otimes e_{\pi_\alpha(I), \pi_\beta(I)} \right\|_p.$$

Recall that $\alpha, \beta \ne \emptyset$ since otherwise $(\alpha, \beta)$ would be non-transposed. Hence we can assume that $1 \le b < a \le d$. Then we define

$$\alpha_1 = \alpha \setminus \{a\}, \quad \alpha_2 = \alpha \cup \{b\},$$

$$\beta_1 = \beta \cup \{a\}, \quad \beta_2 = \beta \setminus \{b\}.$$

In particular, we can write

$$(2.8) \qquad \sum_{I \in [n]^d} a_I^\dagger \otimes \lambda_G(g_{bi_b} g_{ai_a}) \otimes e_{\pi_\alpha(I), \pi_\beta(I)} \\ = \sum_{i_a, i_b = 1}^n x_{i_b i_a} \otimes \lambda_G(g_{bi_b} g_{ai_a}) \otimes e_{i_a, i_b},$$

where $x_{i_b i_a}$ has the following form

$$x_{i_b i_a} = \sum_{i_1, \dots, i_{b-1} = 1}^{n} \sum_{i_{b+1}, \dots, i_{a-1} = 1}^{n} \sum_{i_{a+1}, \dots, i_d = 1}^{n} a_{i_1 \dots i_d}^{\dagger} \otimes e_{\pi_{\alpha_1}(i_1 \dots i_d), \pi_{\beta_2}(i_1 \dots i_d)},$$

with the obvious modifications on the sum indices if $a = d$ or $b = 1$ or $a = b + 1$. On the other hand, since $x_{i_b i_a}$ lives in some non-commutative Lebesgue space $L_p(\varphi)$, we can apply Lemma 2.5 to the right hand side of (2.8) to obtain

$$\left\| \sum_{i_a, i_b = 1}^{n} x_{i_b i_a} \otimes \lambda_G(g_{bi_b} g_{ai_a}) \otimes e_{i_a, i_b} \right\|_p \\ \le c_d \max \left\{ \left\| \sum_{i_a, i_b = 1}^{n} x_{i_b i_a} \otimes e_{1, i_b i_a} \right\|_{L_p(\varphi; R_p^{n_2})}, \left\| \sum_{i_a, i_b = 1}^{n} x_{i_b i_a} \otimes e_{i_b i_a, 1} \right\|_{L_p(\varphi; C_p^{n_2})} \right\}.$$
Finally, we observe that

$$ \sum_{i_a, i_b = 1}^{n} x_{i_b i_a} \otimes e_{1, i_b i_a} = \sum_{I \in [n]^d} a_I^\dagger \otimes e_{\pi_{\alpha_1}(I), \pi_{\beta_1}(I)}, $$

$$ \sum_{i_a, i_b = 1}^{n} x_{i_b i_a} \otimes e_{i_b i_a, 1} = \sum_{I \in [n]^d} a_I^\dagger \otimes e_{\pi_{\alpha_2}(I), \pi_{\beta_2}(I)}. $$

This completes the proof since it is clear that $T(\alpha_1, \beta_1)$, $T(\alpha_2, \beta_2) < T(\alpha, \beta)$. □

**Remark 2.10.** A consequence of our proof is the following equivalence of norms

$$ \begin{aligned} & \max_{0 \le k \le d} \left\{ \left\| \sum_{I \in [n]^k} \sum_{J \in [n]^{d-k}} a_{IJ}^\dagger \otimes e_{I,J} \right\|_p \right\} \\ & \approx \max_{(\alpha, \beta) \in \mathbb{P}_d(2)} \left\{ \left\| \sum_{I \in [n]^d} a_I^\dagger \otimes e_{\pi_\alpha(I), \pi_\beta(I)} \right\|_p \right\}. \end{aligned} $$

This means that, for operators of the form $a_{i_1 \dots i_d} \otimes \lambda(g_{1i_1} \cdots g_{di_d})$, the transposed terms are controlled by the non-transposed ones. The presence of $\lambda(g_{1i_1} \cdots g_{di_d})$ is essential to apply Fell's absorption principle. Moreover, this equivalence is no longer true for arbitrary families of operators. A simple counterexample is given by the 2-indexed family $a_{ij} = e_{ji} \in S_p^n$. Namely, it is easy to check that

$$ \left\| \sum_{i,j=1}^{n} e_{ji} \otimes e_{1,ij} \right\|_{S_p^n (R_p^{n^2})} = n^{1/2+1/p}, $$

$$ \left\| \sum_{i,j=1}^{n} e_{ji} \otimes e_{ij} \right\|_{S_p^n (S_p^n)} = n^{2/p}, $$

$$ \left\| \sum_{i,j=1}^{n} e_{ji} \otimes e_{ij,1} \right\|_{S_p^n (C_p^{n^2})} = n^{1/2+1/p}, $$

$$ \left\| \sum_{i,j=1}^{n} e_{ji} \otimes e_{ji} \right\|_{S_p^n (S_p^n)} = n. $$

In other words, the non-transposed term is not controlled by the transposed ones. In particular, we conclude that the estimation given in Section 1 for the iteration of Khintchine inequality is not equivalent to that provided by Theorem 2.6.

**2.3. The main result for words of length d.** As we have pointed out several times in this paper, Buchholz's result also holds for the whole set of reduced words of length d. Therefore, it is natural to seek for the analog of Theorem 2.6 in this case. We shall need the following modified version of Lemma 2.5.
**Lemma 2.11.** The following inequality holds for any exponent $2 \le p \le \infty$

$$ \left\| \sum_{i,j=1}^{n} a_{ij} \otimes \lambda(g_i g_j^{-1}) \otimes e_{ji} \right\|_{L_p(\varphi \otimes \tau; S_p^n)} \\ \le c \max \left\{ \left\| \sum_{i,j=1}^{n} a_{ij} \otimes e_{1,ij} \right\|_{L_p(\varphi; R_p^{n^2})}, \left\| \sum_{i,j=1}^{n} a_{ij} \otimes e_{ij,1} \right\|_{L_p(\varphi; C_p^{n^2})} \right\}. $$

*Proof.* We can split the sum on the left hand side as follows

$$ \sum_{i,j=1}^{n} a_{ij} \otimes \lambda(g_i g_j^{-1}) \otimes e_{ji} = \sum_{k=1}^{n} a_{kk} \otimes 1 \otimes e_{kk} + \sum_{1 \le i \ne j \le n} a_{ij} \otimes \lambda(g_i g_j^{-1}) \otimes e_{ji}. $$

Since $g_i g_j^{-1}$ is a reduced word of length 2 whenever $i \ne j$, the arguments employed in the proof of Lemma 2.5 apply to estimate the norm of the second sum on the right in $L_p(\varphi \otimes \tau; S_p^n)$. For the first sum, the estimation is obvious since it follows by complex interpolation when we replace max by min above. □

Before stating the announced result, we redefine the analogs of the operator spaces $K_p(n, d)$ and $W_p(n, d)$ in this new framework. To that aim, let us define the elements $h_1, h_2, \dots, h_{2n}$ of $F_n$ as follows

$$ h_k = \begin{cases} g_k & \text{if } 1 \le k \le n, \\ g_{k-n}^{-1} & \text{otherwise,} \end{cases} $$

where $g_1, g_2, \dots, g_n$ are the generators of $F_n$. In this paragraph, $W_p(n, d)$ will denote the subspace of $L_p(\tau)$ generated by the image under $\lambda$ of the set of reduced words of length $d$. In other words, an element of $W_p(n, d)$ has the form

$$ \sum_{|t|=d} \alpha_t \lambda(t) = \sum_{i_1, \dots, i_d=1}^{2n} \alpha_{i_1 i_2 \dots i_d} \lambda(h_{i_1} h_{i_2} \dots h_{i_d}) \in L_p(\tau), $$

where the family of scalars $\mathcal{A} = \{\alpha_{i_1 i_2 \dots i_d} \mid 1 \le i_k \le 2n\}$, satisfies the following cancellation property

$$ (2.9) \qquad \alpha_{i_1 i_2 \dots i_d} = 0 \quad \text{if } i_s \equiv n + i_{s+1} \pmod{2n}, $$

for some $1 \le s < d$. Note that the cancellation property (2.9) is taken so that we only consider reduced words of length $d$. This notation will allow us to handle the space $W_p(n, d)$ just like $W_p(n, d)$ in the previous paragraph. On the other hand, as pointed out in the Introduction, we can regard the family $\mathcal{A}$ as a $(2n)^k \times (2n)^{d-k}$ matrix as follows

$$ \mathcal{A}_k = (\alpha_{(i_1 \dots i_k), (i_{k+1} \dots i_d)}) \in C_p^{(2n)^k} \otimes_h R_p^{(2n)^{d-k}}. $$
Then, we define $K_p(n, d)$ as the subspace of

$$J_p(n, d) = \sum_{k=0}^{d} C_p^{(2n)^k} \otimes_h R_p^{(2n)^{d-k}} \quad \text{if } 1 \le p \le 2,$$

$$J_p(n, d) = \bigcap_{k=0}^{d} C_p^{(2n)^k} \otimes_h R_p^{(2n)^{d-k}} \quad \text{if } 2 \le p \le \infty,$$

where certain entries are zero according to (2.9). We shall prove the following result.

**Corollary 2.12.** $K_p(n, d)$ and $W_p(n, d)$ are completely isomorphic for $1 \le p \le \infty$. More concretely, there exists an absolute constant $c_d$, depending only on $d$, such that the following inequalities hold for any family $\mathcal{A}$ of operators in $L_p(\varphi)$ satisfying the cancellation property (2.9)

$$ \frac{1}{c_d} \|\mathcal{A}\|_{L_p(\varphi; K_p(n,d))} \le \left\| \sum_{i_1 \dots i_d = 1}^{2n} a_{i_1 \dots i_d} \otimes \lambda(h_{i_1} \dots h_{i_d}) \right\|_{L_p(\varphi \otimes \tau)} \\ \le c_d \|\mathcal{A}\|_{L_p(\varphi; K_p(n,d))}. $$

Moreover, the natural projection $P: L_p(\tau) \to W_p(n, d)$ is c.b. with $\|P\|_{cb} \le c_d$.

The proof we are giving is quite similar to that of Theorem 2.6. In particular, we shall skip those arguments which already appeared above. The first remark is that the case $p = \infty$ is exactly the content of Buchholz's result in [1]. Therefore, arguing as we did after the statement of Theorem 2.6, we have:

* The last assertion of Corollary 2.12 holds.

* The spaces $W_p(n, d)$ are an interpolation family for $1 \le p \le \infty$.

* The dual of the space $W_p(n, d)$ is completely isomorphic to $W_{p'}(n, d)$.

* The case $1 \le p \le 2$ in Corollary 2.12 follows from the case $2 \le p \le \infty$.

* The lower estimate for the case $2 \le p \le \infty$ holds with some constant $c_d$.

The last two points use that $K_p(n, d)$ is completely complemented in $J_p(n, d)$, with constants independent on $n$ and $p$. The proof of this fact is simple. Indeed, by transposition and complex interpolation it suffices to prove it for $p = \infty$. Now, since $J_\infty(n, d)$ is an intersection space, we just need to see it for each space appearing in the intersection. Let $\mathcal{I}$ be the set of indices in $[2n]^d$ satisfying the cancellation property (2.9), and let $H_\infty(n, d)$ be the subspace of elements of $J_\infty(n, d)$ supported in $\mathcal{I}$. Then, it is clear that the projection $Q$ onto $H_\infty(n, d)$ is completely bounded since it decomposes as a sum of $d-1$ diagonal projections. In particular, the projection $P$ onto $K_p(n, d)$ is also completely bounded.

*Sketch of the proof.* We only prove the upper estimate for $2 \le p \le \infty$. As above, let us write $G$ for the free group $F_{nd}$ and $\psi_{nd}$ for the standard trace on its reduced $C^*$-algebra. Now, following the notation just introduced, we label the
set of generators and its inverses by $h_{1k}$, $h_{2k}$, ..., $h_{dk}$ with $1 \le k \le 2n$. If $\lambda_G$ denotes the left regular representation of G, we consider the family of operators

$$ \mathcal{A}^{\dagger} = \{ a_{i_1 i_2 \dots i_d} \otimes \lambda_G (h_{1i_1} h_{2i_2} \dots h_{di_d}) \mid 1 \le i_k \le 2n \}. $$

The following chain of inequalities can be proved applying the same arguments as for the proof of Theorem 2.6. Namely, essentially we use Buchholz's result, complex interpolation, and the iteration of Khintchine inequality described in Section 1.

$$
\begin{align*}
(2.10) \quad & \left\| \sum_{i_1, \dots, i_d = 1}^{2n} a_{i_1 \dots i_d} \otimes \lambda(h_{i_1} \dots h_{i_d}) \right\|_p \\
& \le c_d \left\| \sum_{i_1, \dots, i_d = 1}^{2n} a_{i_1 \dots i_d} \otimes \lambda_G(h_{1i_1} \dots h_{di_d}) \right\|_p \\
& \le c_d \left\| \sum_{i_1, \dots, i_d = 1}^{2n} \lambda(h_{i_1}) \otimes \dots \otimes \lambda(h_{i_d}) \otimes a_{i_1 \dots i_d} \otimes \lambda_G(h_{1i_1} \dots h_{di_d}) \right\|_p \\
& \le c_d \max_{(\alpha, \beta) \in \mathbb{P}_d(2)} \left\{ \left\| \sum_{I \in [2n]^d} a_I \otimes \lambda_G(h_{1i_I} \dots h_{di_I}) \otimes e_{\pi_\alpha(1), \pi_\beta(1)} \right\|_p \right\}.
\end{align*}
$$

Then, the proof reduces again to the proof of

$$ (2.11) \quad \text{RHS}_{(2.10)} \le c_d \max_{0 \le k \le d} \left\{ \left\| \sum_{I \in [2n]^k} \sum_{J \in [2n]^{d-k}} a_{IJ} \otimes \lambda_G(h_{1i_I} \cdots h_{di_I}) \otimes e_{I,J} \right\|_p \right\}. $$

By Fell's absorption principle, we have

$$ \left\| \sum_{I \in [2n]^d} a_I^\dagger \otimes e_{\pi_\alpha(1), \pi_\beta(1)} \right\|_p = \left\| \sum_{I \in [2n]^d} a_I^\dagger \otimes \lambda_G(h_{bi_I} h_{ai_I}) \otimes e_{\pi_\alpha(1), \pi_\beta(1)} \right\|_p, $$

with $a_{i_1 \dots i_d}^\dagger = a_{i_1 \dots i_d} \otimes \lambda_G(h_{1i_1} \dots h_{di_1})$. Moreover, we can write

$$
\begin{align*}
(2.12) \quad & \sum_{I \in [2n]^d} a_I^\dagger \otimes \lambda_G(h_{bi_I} h_{ai_I}) \otimes e_{\pi_\alpha(1), \pi_\beta(1)} \\
& = \sum_{i_a, i_b = 1}^{2n} x_{i_b i_a} \otimes \lambda_G(h_{bi_b} h_{ai_b}) \otimes e_{i_a, i_b},
\end{align*}
$$

where $x_{i_b i_a}$ has the form

$$ x_{i_b i_a} = \sum_{i_1, \dots, i_{b-1} = 1}^{2n} \sum_{i_{b+1}, \dots, i_{a-1} = 1}^{2n} \sum_{i_{a+1}, \dots, i_d = 1}^{2n} a_{i_1 \dots i_d}^{\dagger} \otimes e_{\pi_{\alpha_1}(i_1 \dots i_d), \pi_{\beta_2}(i_1 \dots i_d)}, $$
with the obvious modifications if $a = d$ or $b = 1$ or $a = b + 1$. Recall that the operators $x_{i_b, i_a}$ do not necessarily satisfy the cancellation property (2.9). However, we can decompose the sum in (2.12) as follows

$$
\begin{align*}
\sum_{i_a, i_b = 1}^{2n} x_{i_b i_a} \otimes \lambda_G(h_{b i_b} h_{a i_a}) \otimes e_{i_a, i_b} \\
&= \sum_{i_a, i_b = 1}^{n} x_{i_b i_a} \otimes \lambda_G(h_{b i_b} h_{a i_a}) \otimes e_{i_a, i_b} \\
&\quad + \sum_{i_a = 1}^{n} \sum_{i_b = n+1}^{2n} x_{i_b i_a} \otimes \lambda_G(h_{b i_b} h_{a i_a}) \otimes e_{i_a, i_b} \\
&\quad + \sum_{i_b = 1}^{n} \sum_{i_a = n+1}^{2n} x_{i_b i_a} \otimes \lambda_G(h_{b i_b} h_{a i_a}) \otimes e_{i_a, i_b} \\
&\quad + \sum_{i_a, i_b = n+1}^{2n} x_{i_b i_a} \otimes \lambda_G(h_{b i_b} h_{a i_a}) \otimes e_{i_a, i_b}.
\end{align*}
$$

Then it is clear that Lemma 2.5 applies to the first and the fourth sums, while
Lemma 2.11 applies to the second and third sums. In summary, we have

$$
\left\| \sum_{i_a, i_b=1}^{2n} x_{i_b i_a} \otimes \lambda_G(h_{b i_b} h_{a i_a}) \otimes e_{i_a, i_b} \right\|_p \\
\le c_d \max \left\{ \left\| \sum_{i_a, i_b=1}^{2n} x_{i_b i_a} \otimes e_{1, i_b i_a} \right\|_{L_p(\phi; R_p^{n^2})}, \left\| \sum_{i_a, i_b=1}^{2n} x_{i_b i_a} \otimes e_{i_b i_a, 1} \right\|_{L_p(\phi; C_p^{n^2})} \right\}.
$$

Finally, we conclude as in Claim 2.8. This completes the proof of (2.11). □

<figure><img src="image_3.png" /></figure>

**Remark 2.13.** In Voiculescu’s free probability theory, stochastic independence of random variables is replaced by freeness of non-commutative random variables. In this setting, the Wigner’s probability distribution

$$
d\mu_W(t) = 1_{[-2,2]} \frac{\sqrt{4-t^2}}{2\pi} dt
$$

plays a crucial role. Namely, given a free family $x_1, x_2, \dots, x_n$ of self-adjoint
random variables in a non-commutative probability space $(\mathcal{M}, \tau)$, we say that
$x_1, x_2, \dots, x_n$ is a free semi-circular system if each $x_k$ is equipped with Wigner's
distribution. This family is the free analog of a system of $n$ independent standard
real-valued gaussian random variables. Explicit constructions of free semi-circular
systems are available by means of the creation and annihilation operators on the
full Fock space, see [11, 12] for more on this. The free analog of $n$ independent complex-valued gaussians is now given by taking

$$z_k = \frac{1}{\sqrt{2}}(x'_k + ix''_k),$$

with $x'_1, x''_1, x'_2, x''_2, \dots, x'_n, x''_n$ being a free semi-circular system. This new system is called a *free circular system*. At this point, it is natural to guess that the analog of Theorem 2.6 should hold when we replace free generators by free circular random variables. Indeed, as it was pointed out in the Introduction, the family of operators $\lambda(g_1), \lambda(g_2), \dots, \lambda(g_n)$ is the free analog of the sequence of Rademacher functions $r_1, r_2, \dots, r_n$. Therefore, a free version of the central limit theorem is exactly what is needed here. A precise statement of this result can be found in [12] and supports the previous identification between real-valued gaussians and semi-circular random variables. Although we are not giving the details, it can be checked that the central limit theorem for free random variables provides the analog of Theorem 2.6 for free circular variables. In other words, if we replace the operators $\lambda(g_{i_1}g_{i_2}\cdots g_{i_d})$ by the products $z_{i_1}z_{i_2}\cdots z_{i_d}$ in Theorem 2.6, then the same conclusions hold. In passing, we also refer the interested reader to Nou's paper [8], which contains the analog of Buchholz's result for q-gaussian random variables.

**Remark 2.14.** The paper [10] deals with the notion of $p$-orthogonal sums in non commutative $L_p$ spaces. Applying some combinatorial techniques, it is shown that the Khintchine type inequality that applies for $W_p(n)$ majorizes the behaviour of a much larger class of operators, the so-called $p$-orthogonal sums, for any even integer $p$. On the other hand, the bounds given in Section 1 for $W(n)^{\otimes d}$ constitute an upper bound of a more general family of operators. Namely, let $(M, \tau)$ be a von Neumann algebra endowed with a standard trace satisfying $\tau(1) = 1$ and let $L_p(\tau)$ be the associated non-commutative $L_p$ space. Let $\Gamma$ stand for the product set $[n] \times \cdots \times [n]$ with $d$ factors. Then, given an even integer $p$ and a family $f = (f_y)_{y \in \Gamma}$ of operators in $L_p(\tau)$ indexed by $\Gamma$, we shall say that $f$ is $p$-orthogonal with $d$ indices if

$$\tau(f_{h(1)}^*f_{h(2)}f_{h(3)}^*f_{h(4)}\cdots f_{h(p-1)}^*f_{h(p)}) = 0$$

whenever the function $h: \{1, 2, \dots, p\} \to \Gamma$ has an injective projection. In other words, whenever the coordinate function $\pi_k \circ h: \{1, 2, \dots, p\} \to [n]$ is an injective function for some index $1 \le k \le d$. The paper [9] extends the results in [10] to this more general setting by studying the norm in $L_p(\tau)$ of the sum

$$\sum_{y \in \Gamma} f_y.$$

More concretely, the norm of this sum in $L_p(\tau)$ is bounded above by the expressions given in Section 1, see [9] for a precise statement. Moreover, we should point
out that, in contrast with the image under $\lambda$ of the words of length $d$, the family $\lambda(g_{i_1}g_{i_2}\cdots g_{i_d})$ is also a $p$-orthogonal family with $d$ indices.

**Acknowledgements.** The first author was partially supported by the Project BFM 2001/0189, Spain. The second author was partially supported by the NSF and by the Texas Advanced Research Program 010366-163.

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JAVIER PARCET:
Universidad Autónoma de Madrid
SPAIN
and Texas A&M University U. S. A..
E-MAIL: javier.parcet@uam.es

GILLES PISIER:
Université Paris VI, France
and
Texas A&M University
U. S. A. .
KEY WORDS AND PHRASES: Author: Please supply.

2000 MATHEMATICS SUBJECT CLASSIFICATION: Author: Please supply.

Received: March 11th, 2004; revised: July 8th, 2004.