Completely Sidon sets in $C^*$-algebras (New title)

TL;DR

This paper introduces the concept of completely Sidon sets in $C^*$-algebras, generalizing classical Sidon set stability under unions to a non-commutative operator space context, with applications to free group $C^*$-algebras.

Contribution

It extends Drury's theorem to completely Sidon sets in $C^*$-algebras, including non-commutative generalizations related to free groups and von Neumann algebras.

Findings

Complete Sidon sets are stable under finite unions in $C^*$-algebras.

The non-commutative version of Drury's theorem is established.

Extensions to von Neumann algebras with tracial states are provided.

Abstract

A sequence in a $C^*$-algebra $A$ is called completely Sidon if its span in $A$ is completely isomorphic to the operator space version of the space $\ell_1$ (i.e. $\ell_1$ equipped with its maximal operator space structure). The latter can also be described as the span of the free unitary generators in the (full) $C^*$-algebra of the free group $\F_\infty$ with countably infinitely many generators. Our main result is a generalization to this context of Drury's classical theorem stating that Sidon sets are stable under finite unions. In the particular case when $A=C^*(G)$ the (maximal) $C^*$-algebra of a discrete group $G$, we recover the non-commutative (operator space) version of Drury's theorem that we recently proved. We also give several non-commutative generalizations of our recent work on uniformly bounded orthonormal systems to the case of von Neumann algebras equipped with normal faithful tracial states.

Equations

$$\Vert \sum a_n\otimes U_n{\Vert }_{\min }\le C\Vert \sum a_n\otimes {\psi }_n{\Vert }_{\min }.$$

$$\Vert {\sum }_1^K{a}_n\otimes u_n\Vert \le C\Vert {\sum }_1^K{a}_n\otimes {\psi }_n\Vert .$$

$$\Vert {\sum }_1^K{a}_n\otimes u_n{\Vert }_{B{\otimes }_{\min }D}\le C\Vert {\sum }_1^K{a}_n\otimes {\psi }_n{\Vert }_{B{\otimes }_{\min }A}.$$

$${\sup }_{(iii)}\Vert {\sum }_1^{2m}{a}_n\otimes u_n\Vert \ge \max \{\Vert {\sum }_1^m{a}_n\otimes u_n\Vert ,\Vert {\sum }_{m+1}^{2m}{a}_n\otimes u_n\Vert \}.$$

$${\sup }_{(iii)}\Vert {\sum }_1^{2m}{a}_n\otimes u_n\Vert \ge (1/2){\sup }_{(ii)}\Vert {\sum }_1^{2m}{a}_n\otimes u_n\Vert ,$$

$$u=u_1-u_2+i(u_3-u_4).$$

$$\Vert u{\Vert }_{dec}=\inf \{\max \{\Vert S_1\Vert ,\Vert S_2\Vert \}\}$$

$$V:x\to \left(\begin{array}{cc}{S}_1(x)& u(x)\\ u(x^{\ast }{)}^{\ast }& S_2(x)\end{array}\right)$$

$$\Vert v{\Vert }_{dec}=\Vert v^{\ast }{\Vert }_{dec}.$$

$$\Vert i{d}_C\otimes v:C{\otimes }_{\max }A\to C{\otimes }_{\max }B\Vert \le \Vert v{\Vert }_{dec}.$$

$$\Vert v_1\otimes v_2:A_1{\otimes }_{\max }{A}_2\to B_1{\otimes }_{\max }{B}_2{\Vert }_{cb}\le \Vert v_1\otimes v_2:A_1{\otimes }_{\max }{A}_2\to B_1{\otimes }_{\max }{B}_2{\Vert }_{dec}$$

$$\le \Vert v_1{\Vert }_{dec}\Vert v_2{\Vert }_{dec}.$$

$$\forall a\in M_k(Y^{\ast }{)}_{+}{\sum }_{ij}{a}_{ij}(y_{ij})\ge 0.$$

$$\forall n\ge 1{\phi }_{-n}={\phi }_n^{-1}.$$

$$\forall a\in M{y}_n(a)=\tau ({\phi }_n^{\ast }a).$$

$$\forall a\in M{y}_n^{\ast }(a)=\tau ({\phi }_{n}a).$$

$$y_n=J({\phi }_n^{\ast })\text{ and }{y}_n^{\ast }=J({\phi }_n).$$

$$\forall a\in \mathcal{A}{P}_1(a)=\sum \tau ({\phi }_n^{\ast }a){\phi }_n.$$

$$\forall n\in {\mathbb{Z}}_{\ast }\pi (U_n)={\phi }_n.$$

$$\mathcal{E}\frac{\text{buildrel}{\pi }_1}{⟶}B(H)\frac{\text{buildrel}{\pi }_2}{⟶}M$$

$$\forall n\in {\mathbb{Z}}_{\ast }{\pi }_2{\pi }_1(U_n)=\pi (U_n)={\phi }_n$$

$$\tau (z_n{\phi }_m^{\ast })=0\text{ if }n\ne m\text{ and }\tau (z_n{\phi }_n^{\ast })=1,$$

$$\Vert \sum u_n\otimes a_n{\Vert }_{M_{k^{\prime }}{\otimes }_{\min }{M}_k}\le \alpha \Vert \sum a_n\otimes z_n^1\otimes {z_n^2}^{\ast }{\Vert }_{M_k(M{\otimes }_{\max }{M}^{op})}.$$

$$f(x\otimes y)=T(x)(y).$$

$$⟨{\Phi }_T,x\otimes y⟩=\tau (T(x)y).$$

$$\Vert T\otimes i{d}_{M^{op}}:M{\otimes }_{\max }{M}^{op}\to M{\otimes }_{\max }{M}^{op}\Vert \le \Vert T{\Vert }_{dec}.$$

$$T_{\epsilon }=({\theta }_{\epsilon }-{\theta }_0)/\epsilon \text{ and }{R}_{\epsilon }=-\sum _{\ell \ge 2}{\epsilon }^{\ell -1}{P}_{\ell },$$

$$P_1=T_{\epsilon }+R_{\epsilon }.$$

$$\Vert T_{\epsilon }{\Vert }_{dec}\le 2/\epsilon$$

$$\Vert R_{\epsilon }:M\to L_2(\tau )\Vert \le \Vert R_{\epsilon }:L_2(\tau )\to L_2(\tau )\Vert \le \epsilon .$$

Full text

MSC Classif. 43A46, 46L06

Recently, following the impulse of Bourgain and Lewko [1], we studied in [17] the uniformly bounded orthonormal systems that span in $L_{\infty}$ a subspace isomorphic to $\ell_{1}$ by the basis to basis equivalence, and we called them Sidon sequences in analogy with the case of characters on compact abelian groups. One of the main results in [17] says that if a uniformly bounded orthonormal system $\{\psi_{n}\}$ in $L_{2}$ of a probability space $(\Omega,\mathbb{P})$ is the union $\{\psi^{1}_{n}\}\cup\{\psi^{2}_{n}\}$ of two Sidon sequences, then the sequence $\{\psi_{n}\otimes\psi_{n}\otimes\psi_{n}\otimes\psi_{n}\}$ or simply $\{\psi_{n}^{\otimes^{4}}\}$ is Sidon in $L_{\infty}(\Omega^{4},\mathbb{P}^{4})$. Our goal in this paper is to generalize this result to sequences in a non-commutative $C^{*}$-algebra. The central ingredient of our method in [17] is the spectral decomposition of the Ornstein-Uhlenbeck semigroup for a Gaussian measure on $\mathbb{R}^{n}$. Since this has all sorts of non-commutative analogues, it is natural to try to extend the results of [17] to non-commutative von Neumann or $C^{*}$-algebras in place of $L_{\infty}$. In [17] “subgaussian” and “randomly Sidon” sequences play an important role. Although the non-commutative analogue of a subgaussian system is not clear (see however Remark 4.10), and that of “randomly Sidon set” eludes us for the moment, we are able in the present paper to extend several of the main results of [17], in particular we recover an analogue of Drury’s famous union theorem for Sidon sets in groups. In the commutative case the fundamental example of Sidon set is the set formed of the canonical generators in the group $\mathbb{Z}_{\infty}$ formed of all the finitely supported functions $f:\mathbb{N}\to\mathbb{Z}$. This is sometimes referred to as the free Abelian group with countably infinitely many generators. The dual of the discrete group $\mathbb{Z}_{\infty}$ is the compact group $\mathbb{T}^{\mathbb{N}}$, and the von Neumann algebra of $\mathbb{Z}_{\infty}$ can be identified with $L_{\infty}(\mathbb{T}^{\mathbb{N}})$. The analogue of this for our work is the free group $\mathbb{F}_{\infty}$ with countably infinitely many generators, and its von Neumann algebra $M_{\mathbb{F}_{\infty}}$. In the commutative case the generators of $\mathbb{Z}_{\infty}$ correspond in $L_{\infty}(\mathbb{T}^{\mathbb{N}})$ to independent random variables uniformly distributed over $\mathbb{T}$. In classical Sidon set theory, the associated Riesz product plays a crucial role, because of its special interpolation property derived from its spectral decomposition. In more modern approaches, these variables are replaced by standard i.i.d. gaussian random variables, and in [17] the “spectral/interpolation property” of Riesz products is replaced by the spectral expansion of the Ornstein-Uhlenbeck semigroup $T_{\varepsilon}$, i.e. the one that multiplies a multivariate Hermite function of degree $d$ by $\varepsilon^{d}$ (here $0\leq\varepsilon\leq 1$). Equivalently this is obtained by second quantization applied to $\varepsilon\times\text{Identity}$ on the symmetric Fock space. In our new setting, the proper analogue comes from Voiculescu’s free probability theory (cf. [21]), where the analogues of gaussian variables are operators on the full Fock space. Not surprisingly, the techniques we use come from non-commutative probability, in connection with operator space theory for which we refer to [16].

By definition an operator space is a subspace $E\subset A$ of a $C^{*}$-algebra, and the operator space structure (o.s.s. in short) on $E$ consists of the sequence of norms induced on $M_{n}(E)$ by $M_{n}(A)$, where $M_{n}(E)$ denotes the space of $n\times n$-matrices with entries in $E$, and $M_{n}(A)$ is equipped with its natural norm as a $C^{*}$-algebra.

Let $(e_{n})$ denote the canonical basis of the Banach space $\ell_{1}$ of absolutely summable complex sequences. The space $\ell_{1}$ is equipped with a special o.s.s. called the maximal one. The latter operator space structure is induced by the isometric embedding $\ell_{1}\subset C^{*}(\mathbb{F}_{\infty})$ taking $e_{n}$ to the $n$-th free unitary generator in the maximal $C^{*}$-algebra $C^{*}(\mathbb{F}_{\infty})$ (see [16, p. 183]). See [16, §3] for more information and references.

We recall that the algebraic tensor product $A\otimes B$ of two $C^{*}$-algebras can be equipped with a minimal and a maximal $C^{*}$-norm, which after completion produce the $C^{*}$-algebras $A\otimes_{\min}B$ and $A\otimes_{\max}B$. If either $A$ or $B$ is commutative (or nuclear) then $A\otimes_{\min}B=A\otimes_{\max}B$ isometrically.

It is well known that the linear maps between $C^{*}$-algebras that are compatible with the minimal tensor products are the completely bounded (c.b. in short) ones, see [8, 13, 16]. We should emphasize that the analogous maps for the maximal tensor products are the decomposable ones (see (1.4) and (1.5) below) for which we extensively use Haagerup’s results in [11]. See [16, §11] or [7] for more background.

The natural non-commutative generalizations of the notions in [19] are like this:

The sequence $(\psi_{n})$ is called completely $\otimes_{\max}^{k}$-Sidon in $A$ if the sequence $(\psi_{n}\otimes\cdots\otimes\psi_{n})$ ($k$-times) is completely Sidon when viewed as sitting in $A\otimes_{\max}\cdots\otimes_{\max}A$ ($k$-times).

When $A$ is commutative, we simply say that $(\psi_{n})$ is $\otimes^{k}$-Sidon (and when $k=1$ we just call it Sidon), thus recovering the terminology in [17].

Let $({\mathcal{M}},\tau)$ be a non-commutative tracial probability space, i.e. a von Neumann algebra equipped with a faithful normal tracial state. Toward the end of this paper we reach our (already announced) goal: we show that if $\{\psi_{n}\}\subset L_{2}({\mathcal{M}},\tau)$ is an orthonormal system that is uniformly bounded in ${\mathcal{M}}$ and is the union $\{\psi^{1}_{n}\}\cup\{\psi^{2}_{n}\}$ of two completely Sidon sequences then $\{\psi_{n}\}$ is completely $\otimes_{\max}^{4}$-Sidon.

One difficulty is the apparent lack of a suitable non-commutative analogue of the notion of subgaussian sequence, that is crucially used in [17], as well as that of a sequence “dominated” by gaussians. We say that sequence $(x^{1}_{n})$ in an $L_{1}$-space is dominated by another one $(x^{2}_{n})$ if there is a bounded linear map $v:L_{1}\to L_{1}$ taking $x^{2}_{n}$ to $x^{1}_{n}$ for all $n$. In the non-commutative case, we use the same definition but c.b. maps are not enough, we must consider decomposable maps $v:{\mathcal{M}}_{*}\to{\mathcal{N}}_{*}$ between non-commutative $L_{1}$-space, i.e. preduals of von Neumann algebras ${\mathcal{M},\mathcal{N}}$. By $v$ decomposable we just mean that the adjoint $v^{*}:\mathcal{N}\to\mathcal{M}$ is decomposable as a linear combination of c.p. maps. When ${\mathcal{M},\mathcal{N}}$ are commutative any bounded linear map between them is decomposable, but in general it is not so.

A key point in the commutative setting of [17] is that if a uniformly bounded orthonormal sequence $\{\psi_{n}\}$ in $A=L_{\infty}(\mathbb{P})$ is Sidon with constant $C$, then there is a biorthogonal sequence $(y_{n})$ in $A^{*}$ that is dominated by an i.i.d. bounded sequence $(x_{n})$ (and a fortiori dominated by i.i.d. gaussians). The proof is very simple: the mapping $u:\text{span}\{\psi_{n}\}\to L_{\infty}(\mathbb{T}^{\mathbb{N}})$ taking $\psi_{n}$ to the $n$-coordinate $z_{n}$ on $\mathbb{T}^{\mathbb{N}}$ has norm $\leq C$, it extends to a mapping $\widetilde{u}:A\to L_{\infty}(\mathbb{T}^{\mathbb{N}})$ with the same norm, then if we set $x_{n}=\bar{z}_{n}$ viewed as an element of $L_{1}(\mathbb{T}^{\mathbb{N}})$, $(y_{n})$ defined by $y_{n}=(\widetilde{u})^{*}(x_{n})$ does the job.

A second point in [17] is that if a sequence in $L_{1}(\mathbb{P})$ is dominated by a sequence such as $(x_{n})$ then any bounded sequence in $L_{\infty}(\mathbb{P})$ that is biorthogonal to it must be $\otimes^{2}$-Sidon.

In the non-commutative setting, $L_{\infty}(\mathbb{P})$ is replaced by $({\mathcal{M}},\tau)$ and $L_{\infty}(\mathbb{T}^{\mathbb{N}})$ by $M=M_{\mathbb{F}_{\infty}}$ equipped with its usual trace $\tau_{\mathbb{F}_{\infty}}$. The non-commutative version of $(x_{n})$ is the sequence $(y_{n})$ formed of variables each one having the same distribution as $\bar{z}_{n}$ or $z_{n}$ (i.e. normalized Haar measure on $\mathbb{T}$) but instead of stochastic independence we require freeness. More formally we take for $y_{n}$ the element of $M_{*}$ associated to the $n$-th free generator of ${\mathbb{F}_{\infty}}$. Equivalently $(y_{n})$ can be any free Haar unitary sequence in the sense of [21]. We could use just as well any free semicircular (also called “free gaussian”) sequence in Voiculescu’s sense. More generally we call “pseudo-free” (see Remark 3.3) any sequence that is equivalent in a suitable sense (see Definition 1.5) to such a sequence $(y_{n})$. Surprisingly, in this setting there is no need to distinguish between the gaussian and i.i.d. unimodular case, because free gaussian variables, unlike the gaussian ones, are uniformy bounded.

The non-commutative analogue of the preceding two points can be described schematically like this:

See §3 for the proof. This is particularly useful in the case $A=C^{*}(G)$ ($G$ a discrete group) when $\psi_{n}=U_{G}(t_{n})$, the $t_{n}$’s being distinct elements of $G$, and $U_{G}$ being the universal unitary representation of $G$. We say that the set $\Lambda=\{t_{n}\}$ is completely Sidon in $G$ when $(U_{G}(t_{n}))$ is so in $A=C^{*}(G)$. In this case completely $\otimes_{\max}^{k}$-Sidon in $A$ automatically implies completely Sidon, therefore “completely Sidon” is equivalent to having a biothogonal sequence dominated by $(x_{n})$.

We deduce from this in Corollary 4.9 that the union of two completely Sidon subsets of $G$ is completely Sidon. This reduces to show that the union $\Lambda=\{t_{n}\}$ of the two completely Sidon sets is such that, in the group $G^{2}$, $\{(t_{n},t_{n})\}$ has a biothogonal sequence dominated by $(y_{n})$. Indeed, the preceding Theorem then tells us that $\{(t_{n},t_{n},t_{n},t_{n})\}$ is completely Sidon in $G^{4}$ and this clearly is the same as saying $\Lambda=\{t_{n}\}$ is completely Sidon. Note that while the property “dominated by $(x_{n})$” is preserved by the union of two sequences with it (see Remark 4.6), when dealing with a disjoint union $\Lambda_{1}\cup\Lambda_{2}\subset A$ we have to find a way around the following difficulty : the union of a system biorthogonal to $\Lambda_{1}$ with one biorthogonal to $\Lambda_{2}$ is not necessarily biorthogonal to $\Lambda_{1}\cup\Lambda_{2}$. This explains why we pass to $G^{2}$. A similar difficulty arises for a general $A$. This point leads us to conclude that the union is completely $\otimes_{\max}^{k}$-Sidon only for $k=4$ when we would hope to find $k=2$ (see the proof of Theorem 4.7).

The proof of Theorem 0.2 reduces to a special case that we prove in Corollary 2.6, namely the case when $A=M$ and $(\psi_{n})=(y_{n})$. This is analogous to the commutative result proved in [17]: any subgaussian sequence in $L_{1}(\mathbb{P})$ (in particular the above sequence $(\bar{z}_{n})$ in $L_{1}(\mathbb{T}^{\mathbb{N}})$) is such that any bounded biorthogonal sequence in $L_{\infty}(\mathbb{P})$ is $\otimes^{2}$-Sidon. See Remark 4.10 for a further discussion of possible generalization of the “subgaussian” property.

There is an extensive literature on Sidon sets in commutative discrete groups or in duals (dual objects) of compact groups, see e.g. [9], but not much seems to be available on Sidon sets in non-abelian discrete groups or a fortiori in $C^{*}$-algebras. Bożejko and Picardello investigated several closely connected notions of Sidon set, those that span $\ell_{1}$ isomorphically but only as a Banach space and not an operator space, see [2, 3, 14]. Apparently no version of Drury’s theorem is known for these notions in non-abelian groups. We refer to Bożejko and Speicher’s [6] and also the recent work [5] for some results on completely positive functions on Coxeter groups that may be related to our own. See also [4].

See also [22] for a study of Sidon sets in compact quantum groups.

We refer to [16] for background on completely bounded (c.b. in short), completely positive (c.p. in short), and decomposable maps. See also [11]. Some of the connections of the latter notions with the harmonic analysis of the present paper are described in chapter 8 and §9.6 and §9.7 in [16].

1 Completely Sidon sets

Let $U_{n}$ be the unitary generators in $C^{*}(\mathbb{F}_{\infty})$.

Let $A$ be a $C^{*}$-algebra. Let $(\psi_{n})$ be a bounded sequence in $A$.

Definition 1.1**.**

We say that $(\psi_{n})$ is completely Sidon if there is $C$ such that for any matricial coefficients $(a_{n})$

[TABLE]

Equivalently, the operator space spanned by $(\psi_{n})$ in $A$ is completely isomorphic to $\ell_{1}$ equipped with the maximal operator space structure.

Fix an integer $k\geq 1$. We say that $(\psi_{n})$ is completely $\otimes_{\max}^{k}$-Sidon in $A$ if the sequence $(\psi_{n}\otimes\cdots\otimes\psi_{n})$ ($k$-times) is completely Sidon in $A\otimes_{\max}\cdots\otimes_{\max}A$ ($k$-times).

Remark 1.2*.*

It is important to note that for $k>1$ the notion of completely $\otimes_{\max}^{k}$-Sidon is relative to the ambient $C^{*}$-algebra $A$. If $A$ is a $C^{*}$-subalgebra of a $C^{*}$-algebra $B$, and $(\psi_{n})$ is completely $\otimes_{\max}^{k}$-Sidon in $A$, it does not follow in general that $(\psi_{n})$ is completely $\otimes_{\max}^{k}$-Sidon in $B$. This does hold nevertheless if there is a c.p. or decomposable projection from $B$ to $A$. It obviously holds without restriction if $k=1$, but for $k>1$ the precision $\otimes_{\max}^{k}$-Sidon “in $A$” is important. However, when there is no risk of confusion we will omit “in $A$”.

Proposition 1.3**.**

The following are equivalent:

(i) The sequence $(\psi_{n})$ is completely Sidon.

(ii) There is $C$ such that for any ${K},m,k$, any $(a_{n})$ in $M_{m}$, and any $(u_{n})$ in $U(M_{k})$ we have

[TABLE]

(ii)’ Same as (ii) but for $(u_{n})$ in the unit ball of $M_{k}$.

(ii)” There is $C$ such that for any $C^{*}$-algebras $B$ and $D$, any $(a_{n})$ in $B$, and any $(u_{n})$ in the unit ball of $D$ we have

[TABLE]

(iii) Same as (ii) but with ${K}$ even, say ${K}=2m$ and the $u_{n}$’s restricted to be such that $u_{m+j}=u_{j}^{-1}$ for $1\leq j\leq m$.

(iv) Same as (ii) but with the $u_{n}$’s restricted to be selfadjoint unitaries.

Sketch.

The equivalence (i) $\Leftrightarrow$ (ii) is just an explicit reformulation of the preceding definition. To justify (iii) $\Rightarrow$ (ii) we can use $(zu_{j},\bar{z}u_{m+j}^{-1})_{1\leq j\leq m}$. Then after integrating in $z\in\mathbb{T}$, we can separate the two parts of the sum appearing in (ii). This gives us for the sup over all the $u_{n}$’s as in (iii)

[TABLE]

and hence (triangle inequality)

[TABLE]

where the last sup runs over all $(u_{n})$ as in (ii). We then deduce (ii) from (iii) possibly with a different constant.

To justify (iv) $\Rightarrow$ (ii) we can use a $2\times 2$-matrix trick: if $(u_{n})$ is an arbitrary sequence in $U(k)$, $(\begin{matrix}0&u_{n}\\ u_{n}^{*}&0\end{matrix})$ are selfadjoint in $U(2k)$. We then deduce (ii) from (iv) with the same constant.

Lastly the equivalence (ii) $\Leftrightarrow$ (ii)’ is obvious by an extreme point argument, and (ii)’ $\Leftrightarrow$ (ii)” (which reduces to $B=B(H)$ and hence to the matricial case) follows by Russo-Dye and standard operator space arguments (see [16, p. 155-156] for more background). ∎

Remark 1.4*.*

For simplicity we state our results for sequences indexed by $\mathbb{N}$, but actually they hold with obvious adaptation of the proofs for families indexed by an arbitrary set, finite or not, with bounds independent of the number of elements.

Examples :

(i) The fundamental example of a completely Sidon set (with $C=1$) is of course any free subset in a group. More precisely, if we add the unit to a free set, the resulting augmented set is still completely Sidon with $C=1$. Moreover, any left or right translate of a completely Sidon set is completely Sidon (with the same $C$). Therefore any left translate of a free set augmented by the unit is completely Sidon with $C=1$. The converse also holds and is easy to prove, see [19].

(ii) It is proved in [16, Th. 8.2 p.150] that for any $G$ the diagonal mapping $t\mapsto\lambda_{G}(t)\otimes\lambda_{G}(t)$ defines an isometric embedding of $C^{*}(G)$ into $C_{\lambda}^{*}(G)\otimes_{\max}C_{\lambda}^{*}(G)$. It follows that a subset $\Lambda\subset G$ is completely Sidon iff the set $\{\lambda_{G}(t)\mid t\in\Lambda\}$ is completely $\otimes_{\max}^{2}$-Sidon in $C_{\lambda}^{*}(G)$. Let $M_{G}$ be the von Neumann algebra of $G$ (i.e. the one generated by $\lambda_{G}$). Similar arguments show that the same diagonal embedding embeds $C^{*}(G)$ also into $M_{G}\otimes_{\max}M_{G}$. In particular the set of free generators is a completely $\otimes_{\max}^{2}$-Sidon set in $C_{\lambda}^{*}(\mathbb{F}_{\infty})$ (and also in $M_{\mathbb{F}_{\infty}}$).

We will be interested in another property, namely the following one:

Let $X_{1},X_{2}$ be preduals of $C^{*}$-algebras (so-called non-commutative $L_{1}$-spaces).

We say that a bounded linear map $v:X_{2}\to X_{1}$ is completely positive (in short c.p.) if $v^{*}:X_{1}^{*}\to X_{2}^{*}$ is c.p..

Let $A,B$ be $C^{*}$-algebras. Let $CP(A,B)$ be the set of c.p. maps from $A$ to $B$. We say that a bounded linear map $u:A\to B$ is decomposable if there are $u_{j}\in CP(A,B)\quad(j=1,2,3,4)$ such that

[TABLE]

We use the dec-norm as defined by Haagerup [11]. We denote

[TABLE]

where the infimum runs over all maps $S_{1},S_{2}\in CP(A,B)$ such that the map

[TABLE]

is in $CP(A,M_{2}(B))$.

A mapping $v:X_{2}\to X_{1}$ is said to be decomposable if its adjoint $v^{*}:X_{1}^{*}\to X_{2}^{*}$ is decomposable in the preceding sense (linear combination of c.p. maps), and we set by convention

[TABLE]

We use the term $c$-decomposable for maps that are decomposable with dec-norm $\leq c$.

The crucial property of a decomposable map $v:A\to B$ between $C^{*}$-algebras is that for any other $C^{*}$-algebra $C$ the mapping $id_{C}\otimes v$ extends to a bounded (actually decomposable) map from $C\otimes_{\max}A$ to $C\otimes_{\max}B$. Moreover we have

[TABLE]

Consequently, for any pair $v_{j}:A_{j}\to B_{j}$ ($j=1,2$) of decomposable maps between $C^{*}$-algebras, we have

[TABLE]

[TABLE]

Definition 1.5**.**

Let $I,J$ be any sets. (i) Let $(x_{n}^{1})_{n\in I}$ (resp. $(x_{n}^{2})_{n\in I}$ ) be a family in $X_{1}$ (resp. $X_{2}$). Let us say that $(x_{n}^{1})_{n\in I}$ is $c$-dominated (or “decomposably $c$-dominated”) by $(x_{n}^{2})_{n\in I}$ if there is a decomposable mapping $v:X_{2}\to X_{1}$ with $\|v\|_{dec}\leq c$ such that $v(x_{n}^{2})=x_{n}^{1}$ for any $n\in I$.

We simply say “dominated” for $c$-dominated for some $c$.

(ii) We say that $(x_{n}^{1})_{n\in I}$ and $(x_{n}^{2})_{n\in J}$ are “ decomposably equivalent ” if there is a bijection $f:I\to J$ such that each of the families $(x_{n}^{1})_{n\in I}$ and $(x_{f(n)}^{2})_{n\in I}$ is dominated by the other.

Let $Y$ be the predual of a $C^{*}$-algebra. The positive cone in $M_{k}(Y)$ is the polar of the positive cone $M_{k}(Y^{*})_{+}$ in the $C^{*}$-algebra $M_{k}(Y^{*})$. More precisely $y\in M_{k}(Y)_{+}$ iff

[TABLE]

Clearly $v:X_{2}\to X_{1}$ is c.p. iff for any $k$ the mapping $id_{M_{k}}\otimes v:M_{k}(X_{2})\to M_{k}(X_{1})$ is positivity preserving.

More generally, since we have positive cones on both $M_{k}(X^{*})$ and $M_{k}(Y)$, we can extend the definition of complete positivity to maps from a $C^{*}$-algebra to $Y$ or from $Y$ to a $C^{*}$-algebra. In particular, a map $T:X^{*}\to Y$ is called c.p. if $id_{M_{k}}\otimes T:M_{k}(X^{*})\to M_{k}(Y)$ is positivity preserving for any $k$.

Remark 1.6*.*

[Opposite von Neumann algebra] The opposite von Neumann ${M^{op}}$ is the same linear space as $M$ but with the reverse product. Let $\Phi:{M^{op}}\to M$ be the identity map, viewed as acting from ${M^{op}}$ to $M$, so that $\Phi^{*}:M^{*}\to{M^{op}}^{*}$ also acts as the identity.

When $M$ is a von Neumann algebra equipped with a normal faithful tracial state $\tau$, there is a minor problem that needs clarification. We have a natural inclusion $J:M\to M^{*}$ denoted by $y\mapsto Jy$ and defined by $J{y}(x)=\tau(yx)$. In general this is not c.p, but it is c.p. when viewed as a mapping either from $M^{op}\to M^{*}$ or from $M\to{M^{op}}^{*}$. Indeed, for all $x,y\in M_{k}(M)_{+}$ we have $[{\rm tr}\otimes\tau](xy)=\sum_{ij}\tau(x_{ij}y_{ji})\geq 0$ but in general it is not true that for $\sum_{ij}\tau(x_{ij}y_{ij})=\sum_{ij}J{y_{ij}}(x_{ij})$.

Then the content of the preceding observation is that $\Phi^{*}J:M\to{M^{op}}^{*}$ is c.p. (but $J$ in general fails this).

Remark 1.7*.*

[About preduals of finite vN algebras] Let $(M^{1},\tau^{1})$ be here any noncommutative probability space, i.e. a von Neumann algebra equipped with a normal faithful tracial state. The predual ${M^{1}}_{*}$ is the subset of ${M^{1}}^{*}$ formed of the weak* continuous functionals on $M^{1}$. It can be isometrically identified with the space $L_{1}(\tau_{1})$ defined as the completion of $M_{1}$ for the norm $\|x\|_{1}=\tau_{1}(|x|)$. Thus we have a natural inclusion with dense range $M^{1}\subset L_{1}(\tau_{1})$. We need to observe the following fact. Let $({M}^{2},\tau^{2})$ be another noncommutative probability space. Let $V:L_{1}(\tau_{1})\to L_{1}(\tau_{2})$ be a linear map that is a $*$-homomorphism from $M^{1}$ to $M^{2}$ when restricted to $M^{1}$. Then $V$ is completely positive and hence 1-decomposable.

2 Analysis of the free group case

We denote by $M$ the von Neumann algebra of the free group $\mathbb{F}_{\infty}$ equipped with its usual trace $\tau$.

We denote by $(\varphi_{n})_{n\geq 1}$ the elements of $M=\lambda_{\mathbb{F}_{\infty}}(\mathbb{F}_{\infty})^{\prime\prime}$ corresponding to the free generators $(g_{n})$ in $\mathbb{F}_{\infty}$, i.e. $\varphi_{n}=\lambda_{\mathbb{F}_{\infty}}(g_{n})$.

For convenience we set

[TABLE]

Although this is a bit pedantic, it is wise to distinguish the elements of $M$ from the linear functionals on $M$ that they determine. Thus we let $(y_{n})_{n\in\mathbb{Z}_{*}}$ be the sequence in $M_{*}\subset M^{*}$ that is biorthogonal to the sequence $({\varphi}_{n})_{n\in\mathbb{Z}_{*}}$, and defined for all $n\in\mathbb{Z}_{*}=\mathbb{Z}\setminus\{0\}$ by

[TABLE]

We also define $y_{n}^{*}\in M_{*}\subset M^{*}$ as follows

[TABLE]

Again let $J:M\to M^{*}$ be the inclusion mapping defined by $Ja(b)=\tau(ab)$. With this notation

[TABLE]

For future reference, we record here a simple observation:

Lemma 2.1**.**

Recall $\mathbb{Z}_{*}=\mathbb{Z}\setminus\{0\}$. The families $(y_{n})_{n\geq 1}$ and $(y_{n})_{n\in\mathbb{Z}_{*}}$ are decomposably equivalent in the sense of Definition 1.5.

Proof.

Let $(z_{n})_{n\in\mathbb{Z}_{*}}$ be a sequence such that each $(z_{n})_{n>0}$ and $(z_{n})_{n<0}$ are mutually free, each one being a free Haar unitary sequence. Then $(z_{n})_{n\in\mathbb{Z}_{*}}$ and $(\varphi_{n})_{n\geq 1}$ are trivially decomposably equivalent. Let $L$ be a copy of the von Neumann algebra of $\mathbb{Z}$. Let $N=M\ast L$. Let $U$ denote the generator of $L$ viewed as a subalgebra of $N$. We also view $M\subset N$. Then the family $(Uy_{n})_{n\in\mathbb{Z}_{*}}$ viewed as sitting in $N_{*}$ is a family of free Haar unitaries. Therefore $(Uy_{n})_{n\in\mathbb{Z}_{*}}$ and $(y_{n})_{n\geq 1}$ are decomposably equivalent. But $(Uy_{n})_{n\in\mathbb{Z}_{*}}$ and $(y_{n})_{n\in\mathbb{Z}_{*}}$ are also decomposably equivalent in $N_{*}$, because the multiplication by $U$ or $U^{-1}$ is decomposable (roughly because, since $x\mapsto axa^{*}$ is c.p., $x\mapsto axb^{*}$ is decomposable by the polarization formula). Lastly using conditional expectations it is easy to see that the families $(y_{n})_{n\in\mathbb{Z}_{*}}\subset N_{*}$ and $(y_{n})_{n\in\mathbb{Z}_{*}}\subset M_{*}$ (identical families viewed as sitting in $N_{*}$ or $M_{*}$) are decomposably equivalent. ∎

Let $\mathcal{A}$ be the algebra generated by $(\varphi_{n})_{n\in\mathbb{Z}_{*}}$. Note for further reference that the orthogonal projection $P_{1}$ onto the closed span in $L_{2}(\tau)$ of $(\varphi_{n})_{n\in\mathbb{Z}_{*}}$ is defined by

[TABLE]

We use ingredients analogous to those of [17] but in [17] the free group is replaced by the free Abelian group, and an ordinary gaussian sequence is used (we could probably use analogously a free semicircular sequence here).

Let $U_{n}\in C^{*}(\mathbb{F}_{\infty})$ be the unitaries coming from the free generators. We set again by convention $U_{-n}=U_{n}^{-1}$ ($n\geq 1$).

Let $\mathcal{E}={\rm span}[U_{n}\mid n\in\mathbb{Z}_{*}]\subset C^{*}(\mathbb{F}_{\infty})$. Consider the natural linear map $\pi:\mathcal{E}\to M$ such that

[TABLE]

Its key property is that for some Hilbert space $H$ there is a factorization of the form

[TABLE]

such that

[TABLE]

where $\|\pi_{1}\|_{cb}\leq 1$, and $\pi_{2}$ is a decomposable map with $\|\pi_{2}\|_{dec}=1$. To check this note that $M$ embeds in a trace preserving way into an ultraproduct $\mathcal{M}$ of matrix algebras, and there is a c.p. conditional expectation from $\mathcal{M}$ onto $M$. Therefore there is a completely positive surjection $\pi_{2}$ from $B=\prod\nolimits_{k}M_{k}$ to $M$ and a $*$-homomorphism $\pi_{1}:C^{*}(\mathbb{F}_{\infty})\to B$ such that ${\pi_{2}\pi_{1}}_{|{\mathcal{E}}}=\pi$. To complete the argument we need to replace $B$ by $B(H)$. Since $B$ embeds in $B(H)$ for some $H$ and there is a conditional expectation from $B(H)$ to $B$, this is immediate. We refer the reader to [16, §9.10] for more details.

The following statement on the free group factor $M$ is the key for our results.

Theorem 2.2**.**

*The sequence $(\varphi_{n})_{n\in{\mathbb{Z}_{*}}}$ in $M$ satisfies the following property:

any bounded sequence $(z_{n})_{n\in{\mathbb{Z}_{*}}}$ in $M$ that is biorthogonal to $(\varphi_{n})_{n\in{\mathbb{Z}_{*}}}$ in $L_{2}(\tau)$ meaning that*

[TABLE]

is completely $\otimes^{2}_{\max}$-Sidon. More generally, if $(z^{1}_{n})_{n\in{\mathbb{Z}_{*}}}$ and $(z^{2}_{n})_{n\in{\mathbb{Z}_{*}}}$ are bounded in $M$ and each biorthogonal to $(y_{n})_{n\in{\mathbb{Z}_{*}}}$, then $(z^{1}_{n}\otimes z^{2}_{n})_{n\in{\mathbb{Z}_{*}}}$ is completely Sidon in $M\otimes_{\max}M$.

Let $(z^{1}_{n})$ and $(z^{2}_{n})$ be as in Theorem 2.2. Assume $\|z^{j}_{n}\|\leq C^{\prime}_{j}$ for all $n\in{\mathbb{Z}_{*}}$ ($j=1,2$).

Fix integers $k,k^{\prime}\geq 1$. Let $(a_{n})$ be a family in $M_{k}$ with only finitely many $n$’s for which $a_{n}\not=0$. Let $(u_{n})_{n\in{\mathbb{Z}_{*}}}$ be unitaries in $M_{k}$ such that $u_{-n}=u_{n}^{-1}$ for all $n$. Our goal is to show that there is a constant $\alpha$ depending only on $C^{\prime}_{1},C^{\prime}_{2}$ such that

[TABLE]

This will prove the key Theorem 2.2 with $M\otimes_{\max}M^{op}$ instead of $M\otimes_{\max}M$. Then a simple elementary argument will allow us to replace $M^{op}$ by $M$.

Remark 2.3*.*

Let $T:M\to M^{*}$ be a c.p. map such that $T(1)(1)=1$. We associate to it a state $f$ on $M\otimes_{\max}M$ by setting

[TABLE]

A matrix $x\in M_{k}(M^{*})$ is defined as $\geq 0$ if $\sum_{ij}x_{ij}(y_{ij})\geq 0$ for all $y\in M_{k}(M)_{+}$.

More generally, any decomposable operator $T$ on $M$ (in particular any finite rank one) determines an element $\Phi_{T}$ of $(M\otimes_{\max}M^{op})^{*}$, defined by for $x,y\in M$ by

[TABLE]

Indeed, the bilinear form $(x,y)\mapsto\tau(xy)$ is of unit norm in $(M\otimes_{\max}M^{op})^{*}$ and

[TABLE]

Furthermore, for any pair of $C^{*}$-algebras $A,B$, we have a 1-1-correspondence between the set of decomposable maps $T:A\to B^{*}$ and $(A\otimes_{\max}B)^{*}$.

Remark 2.4*.*

We will need the free analogue of Riesz products.

Recall we set $M=M_{\mathbb{F}_{\infty}}$. Let $0\leq\varepsilon\leq 1$. Let $P_{\ell}$ the orthogonal projection on $L_{2}(\tau)$ onto the span of the words of length $\ell$ in $\mathbb{F}_{\infty}$. Let ${\theta}_{\varepsilon}=\sum_{\ell\geq 0}\varepsilon^{\ell}P_{\ell}$. By Haagerup’s well known result [10], ${\theta}_{\varepsilon}$ is a c.p. map on $M$. Composing it with the inclusion $M\subset M_{*}$, we find a unital c.p. map from $M$ to ${M^{op}}^{*}$, and hence ${\theta}_{\varepsilon}$ determines a state $f_{\varepsilon}$ on $M\otimes_{\max}M^{op}$.

We view $\theta_{\varepsilon}$ as acting from $M$ to $L_{2}(\tau)$. We can also consider it as a map taking $\mathcal{A}$ to itself.

We will crucially use the decomposition $({\theta}_{\varepsilon}-{\theta}_{0})/\varepsilon=P_{1}+\sum_{{\ell}\geq 2}\varepsilon^{{\ell}-1}P_{\ell}.$ We set

[TABLE]

so that

[TABLE]

We have

[TABLE]

and

[TABLE]

Lemma 2.5**.**

With the preceding notation, we have

[TABLE]

Proof.

This follows from (1.5). ∎

Proof of Theorem 2.2.

Fix $\varepsilon<1$ (to be determined later). We have decompositions

[TABLE]

[TABLE]

where $r^{1}_{n}=-R_{\varepsilon}(z^{1}_{n})$ and $r^{2}_{n}$ are orthogonal to $(\varphi_{n})_{n\in{\mathbb{Z}_{*}}}$ and moreover

[TABLE]

[TABLE]

We have

[TABLE]

The idea will be to reduce this product to the simplest term $\varphi_{n}\otimes\varphi_{n}^{*}$.

Let $V:M\to M_{k}(M)$ be the isometric $*$-homomorphism taking $\varphi_{n}$ to $u_{n}\otimes\varphi_{n}$. Note that $V$ is decomposable with $\|V\|_{dec}=1$. We observe

[TABLE]

Let $\gamma:M\otimes M^{op}\to\mathbb{C}$ be the bilinear form defined by $\gamma(a\otimes a^{\prime})=\tau(aa^{\prime})$. It is a classical fact that $\gamma$ is a state on $M\otimes_{\max}M^{op}$. We claim

[TABLE]

Indeed, let $\mathbb{F}=\mathbb{F}_{\infty}$ for simplicity. We may develop in $L_{2}(\tau)$

[TABLE]

Let $\sigma$ be the unitary representation on $\mathbb{F}$ taking $g_{n}$ to $u_{n}\in M_{k^{\prime}}$. For simplicity we denote $u_{t}=\sigma(t)$ for any $t\in\mathbb{F}$. With this notation $V(\lambda_{\mathbb{F}}(t))=u_{t}\otimes\lambda_{\mathbb{F}}(t).$ Then

[TABLE]

[TABLE]

and hence (triangle inequality and Cauchy-Schwarz)

[TABLE]

This proves our claim. Let

[TABLE]

Recalling the orthogonality relations $\varphi_{n}\perp r^{1}_{n}$ and $\varphi_{n}\perp r^{2}_{n}$ we see that

[TABLE]

[TABLE]

We now go back to (2.6): we have

[TABLE]

Therefore (the norm $\|\ \|$ is the norm in ${M_{k^{\prime}}\otimes_{\min}M_{k}}$)

[TABLE]

and hence by (2.6)

[TABLE]

By the triangle inequality

[TABLE]

Recalling (2.7) and (2.8) we find

[TABLE]

Taking the sup over all $u_{n}$’s and using (1.1) (recall $B_{M_{k^{\prime}}}$ is the convex hull of $U(k^{\prime})$) we find

[TABLE]

This completes the proof for $M\otimes_{\max}M^{op}$, since if we choose, say, $\varepsilon=\varepsilon_{0}$ with $\varepsilon_{0}=(4C^{\prime}_{1}C^{\prime}_{2})^{-1}$ we obtain the announced result with $\alpha=8/\varepsilon_{0}=32C^{\prime}_{1}C^{\prime}_{2}$.

It remains to justify the replacement of $M^{op}$ by $M$. For this it suffices to exhibit a (normal) $\mathbb{C}$-linear $*$-isomorphism $\chi:M^{op}\to M$ such that $\chi(\varphi^{*}_{n})=\varphi_{n}$ for all $n\in{\mathbb{Z}_{*}}$. Indeed, let us view $M\subset B(H)$ with $H=\ell_{2}(\mathbb{F}_{\infty})$. Then since ${}^{t}\varphi_{n}=\varphi_{n}^{*}$ for all $n\in{\mathbb{Z}_{*}}$ (these are matrices with real entries), the matrix transposition $x\mapsto{}^{t}x$ is the required $*$-isomorphism $\chi:M^{op}\to M$. ∎

Corollary 2.6**.**

Let $(z^{1}_{n})_{n\geq 1}$ and $(z^{2}_{n})_{n\geq 1}$ be bounded in $M$ and each biorthogonal to $(y_{n})_{n\geq 1}$, then $(z^{1}_{n}\otimes z^{2}_{n})_{n\geq 1}$ is completely Sidon in $M\otimes_{\max}M$.

Proof.

By Lemma 2.1 we know that $(y_{n})_{n\geq 1}$ is dominated by $(y_{n})_{n\in{\mathbb{Z}_{*}}}$. Let $v:M_{*}\to M_{*}$ decomposable taking $(y_{n})_{n\in{\mathbb{Z}_{*}}}$ to $(y_{n})_{n\geq 1}$ (modulo a suitable bijection $f:{\mathbb{Z}_{*}}\to\mathbb{N}_{*}$). Then $(v^{*}(z^{j}_{f(n)}))_{n\in{\mathbb{Z}_{*}}}$ ($j=1,2$) is biorthogonal to $(y_{f(n)})_{n\in{\mathbb{Z}_{*}}}$. By Theorem 2.2, $(v^{*}(z^{1}_{f(n)})\otimes v^{*}(z^{2}_{f(n)}))_{n\in{\mathbb{Z}_{*}}}$ is completely Sidon in $M\otimes_{\max}M$. By (1.5) $(z^{1}_{f(n)}\otimes z^{2}_{f(n)})_{n\in{\mathbb{Z}_{*}}}$ is completely Sidon in $M\otimes_{\max}M$. Equivalently since this is obviously invariant under permutation, we conclude $(z^{1}_{n}\otimes z^{2}_{n})_{n\in\mathbb{N}_{*}}$ is completely Sidon in $M\otimes_{\max}M$.

∎

3 Main results. Free unitary domination

We start with a simple but crucial observation that links completely Sidon sets with the free analogues of Rademacher functions or independent gaussian random variables.

Proposition 3.1**.**

Let $\Lambda=\{\psi_{n}\mid n\geq 1\}$ be a completely Sidon set in $A$ with constant $C$. Then there is a biorthogonal system $(x_{n})_{n\geq 1}$ in $A^{*}$ that is $C$-dominated by $(y_{n})_{n\geq 1}$.

Proof.

Let $E\subset A$ be the linear span of $\{\psi_{n}\}$. Let $\alpha:E\to\mathcal{E}$ be the linear map such that $\alpha(\psi_{n})=U_{n}$. By our assumption $\|\alpha\|_{cb}\leq C$. We have $\|\pi_{1}\alpha:E\to B(H)\|_{cb}\leq C$. By the injectivity of $B(H)$, $\pi_{1}\alpha$ admits an extension $\beta:A\to B(H)$ with $\|\beta\|_{cb}\leq C$. Note (see [11]) that $\|\beta\|_{dec}=\|\beta\|_{cb}$. Let $V=\pi_{2}\beta:A\to M$. Then $V$ is a $C$-decomposable map. Its adjoint $V_{*}:M_{*}\to A^{*}$ is also $C$-decomposable. Let ${y_{n}}\in M_{*}$ be the functionals biorthogonal to the sequence $({\varphi}_{n})$ defined above in $M$. We have ${y_{n}}({\varphi}_{m})=\delta_{nm}$. Therefore since $V(\psi_{m})=\pi_{2}\beta(\psi_{m})=\pi_{2}\pi_{1}\alpha(\psi_{m})={\varphi}_{m}$

[TABLE]

Thus setting $x_{n}=V_{*}({y_{n}})$ we find ${x_{n}}(\psi_{m})=\delta_{nm}.$ This shows that $(x_{n})$, which is by definition $C$-dominated by $(y_{n})$, is biorthogonal to $(\psi_{n})$. ∎

Theorem 3.2**.**

*Let $A_{1}$, $A_{2}$ be $C^{*}$-algebras. Let $(\psi^{1}_{n})_{n\geq 1}$, $(\psi^{2}_{n})_{n\geq 1}$ be bounded sequences in $A_{1}$, $A_{2}$ bounded by $C_{1}^{\prime}$ and $C_{2}^{\prime}$ respectively. Let $(x^{1}_{n})_{n\geq 1}$ be a sequence in $A_{1}^{*}$ biorthogonal to $(\psi^{1}_{n})_{n\geq 1}$, and let $(x^{2}_{n})_{n\geq 1}$ be a sequence in $A_{2}^{*}$ biorthogonal to $(\psi^{2}_{n})_{n\geq 1}$. If both are dominated by $(y_{n})_{n\geq 1}$, then $(\psi^{1}_{n}\otimes\psi^{2}_{n})_{n\geq 1}$ is completely Sidon in $A_{1}\otimes_{\max}A_{2}$.

More precisely, if $(x^{j}_{n})_{n\geq 1}$ is $c_{j}$-dominated by $(y_{n})_{n\geq 1}$, $(\psi^{1}_{n}\otimes\psi^{2}_{n})_{n\geq 1}$ is completely Sidon in $A_{1}\otimes_{\max}A_{2}$ with a constant $C$ depending only on $C_{1}^{\prime},C_{2}^{\prime},c_{1},c_{2}$.*

Proof.

The key ingredient is Corollary 2.6. Assume $(x_{n}^{j})$ dominated by $(y_{n})$. Let $v_{j}:M_{*}\to A^{*}_{j}$ be decomposable such that $v_{j}(y_{n})=x_{n}^{j}$ ($j=1,2$), with $(x_{n}^{j})$ biorthogonal to $(\psi_{n}^{j})$ and $\|v_{j}\|_{dec}\leq c_{j}$. Moreover let $w_{j}:A_{j}\to M$ be the restriction of $v_{j}^{*}:A_{j}^{**}\to M$ to $A_{j}$. Note that $(v_{j}^{*}(\psi^{j}_{n}))$, or equivalently $(w_{j}(\psi^{j}_{n}))$, is obviously biorthogonal to $(y_{n})$ for each $j=1,2$. Let $z_{n}^{j}=w_{j}(\psi^{j}_{n})$. By Corollary 2.6 the sequence $(z_{n}^{1}\otimes z_{n}^{2})$ is completely Sidon in $M\otimes_{\max}M$. But since $w_{j}\in D(A_{j},M)$ we see by (1.5) that this implies that $(\psi^{1}_{n}\otimes\psi^{2}_{n})$ is completely Sidon in $A_{1}\otimes_{\max}A_{2}$. The assertion on the constants is easy to check by going over the argument. ∎

Remark 3.3* (On “pseudo-free” sequences).*

Let us say that a sequence $({\kappa}_{n})$ in the predual $N_{*}$ of a von Neumann algebra $N$ is pseudo-free if $(y_{n})$ and $({\kappa}_{n})$ are decomposably equivalent. Clearly, we may replace $(y_{n})$ by any other pseudo-free sequence in what precedes. Note that any sequence of free Haar unitaries, (or free Rademacher) or of free semicircular variables is pseudo-free. More generally, any free sequence $({\kappa}_{n})$ with mean 0 in a non-commutative tracial probability space such that $\inf\|{\kappa}_{n}\|_{1}>0$ and $\sup\|{\kappa}_{n}\|_{\infty}<\infty$ is pseudo-free.

Indeed, this can be deduced from the fact that trace preserving unital c.p. maps extend to trace preserving c.p. maps on reduced free products. The latter fact reduces the problem to the commutative case (one first checks the result for a single variable with unital c.p. maps instead of decomposable ones).

4 The union problem

It is high time to formalize a bit more the central notion of this paper.

Definition 4.1**.**

Let $(x_{n})_{n\geq 1}$ be a sequence in the predual $X$ of a von Neumann algebra. Let $(y_{n})$ be as before in $M_{*}$. We will say that $(x_{n})_{n\geq 1}$ is free-gaussian dominated in $X$ (or dominated by free-gaussians in $X$) if it is (decomposably) dominated by the sequence $(y_{n})$ in $M_{*}$, or equivalently (see Remark 3.3) if it is (decomposably) dominated by a free-gaussian sequence (or any pseudo-free sequence) in $M_{*}$. Here “(decomposably) dominated” is meant in the sense of Definition 1.5.

For convenience we define the associated constant using the (unitary) sequence $(y_{n})$: we say that $(x_{n})_{n\geq 1}$ is free-gaussian $c$-dominated in $X$ if it is $c$-dominated by $\{y_{n}\mid n\geq 1\}\subset M_{*}$, so that we have $T:M_{*}\to X$ with $\|T\|_{dec}\leq c$ such that $T(y_{n})=x_{n}$.

Remark 4.2*.*

By classical results (see [20, p. 126]) for any von Neumann algebra ${\mathcal{M}}$, there is a c.p. projection (with dec-norm equal to 1) from ${\mathcal{M}}^{*}=({\mathcal{M}}_{*})^{**}$ to ${\mathcal{M}}_{*}$. Therefore the notions of domination in ${\mathcal{M}}_{*}$ or in ${\mathcal{M}}^{*}$ are equivalent for sequences sitting in ${\mathcal{M}}_{*}$.

Of course we frame the preceding definition to emphasize the analogy with the sequences dominated by i.i.d gaussians in [17]. Note that in the latter, with independence in place of freeness, dominated by gaussians does not imply dominated by i.i.d. Haar unitaries, (indeed gaussians themselves fail this) but it holds in the free case because free-gaussians are bounded. Note in passing that bounded linear maps between $L_{1}$-spaces of commutative (and hence injective) von Neumann algebras are automatically decomposable.

Lemma 4.3**.**

Let ${\mathcal{M}},{\mathcal{N}}$ be von Neumann algebras. Assume that ${\mathcal{N}}$ is equipped with a normal faithful tracial state $\tau^{\prime}$. For $a\in{\mathcal{N}}$ we denote by $\widehat{a}\in{\mathcal{N}}_{*}$ the associated linear form on ${\mathcal{N}}$ defined by $\widehat{a}(x)=\tau^{\prime}(a^{*}x)$. Let $(v_{n})_{n\geq 1}$ be unitaries in ${\mathcal{N}}$, so that $\widehat{v}_{n}\in N_{*}$. Let $(x_{n})_{n\geq 1}\in{\mathcal{M}}_{*}$ be free-gaussian $c$-dominated. Then the sequence $(x_{n}\otimes\widehat{v}_{n})\in({\mathcal{M}}\overline{\otimes}{\mathcal{N}})_{*}$ is also $c$-dominated by $(y_{n})$.

Proof.

Let $T$ be as in Definition 4.1 (here $X={\mathcal{M}}_{*}$). Since $(y_{n}\otimes\widehat{v}_{n})$ and $(y_{n})$ have the same $*$-distribution, the linear mapping $W$ taking $y_{n}$ to $y_{n}\otimes\widehat{v}_{n}$ extends to a c.p. (isometric, unital and trace preserving) map $W$ from ${M}_{*}$ to $({M}\overline{\otimes}{\mathcal{N}})_{*}$ (see Remark 1.7). Then the composition $T_{1}=(T\otimes id_{{\mathcal{N}}_{*}})W$ takes $y_{n}$ to $x_{n}\otimes\widehat{v}_{n}$. Since $W^{*}$ is c.p. and $(T\otimes id_{{\mathcal{N}}_{*}})^{*}=T^{*}\otimes id_{{\mathcal{N}}}$, with dec-norm $\leq c$, $T_{1}$ is $c$-decomposable. ∎

Remark 4.4*.*

By the Russo-Dye theorem the unit ball of ${\mathcal{N}}$ is the closed convex hull of its unitaries. Actually, for any fixed $0<\delta<1$ there is an integer $K_{\delta}$ such that any $v\in{\mathcal{N}}$ with norm $<\delta$ can be written as an average of $K_{\delta}$ unitaries, this is due to Kadison and Pedersen, see [12] for a proof with $\delta=1-2/n$ and $K_{\delta}=n$. Using this, we can extend Lemma 4.3 to sequences $(v_{n})$ in the unit ball of ${\mathcal{N}}$. Indeed, the set of sequences $(v_{n})$ in ${\mathcal{N}}$ such that the sequence $(x_{n}\otimes\widehat{v}_{n})$ in $({\mathcal{M}}\overline{\otimes}{\mathcal{N}})_{*}$ is $c$-dominated by $(y_{n})$ is obviously a convex set. By Lemma 4.3 it contains the set of sequences of unitaries in ${\mathcal{N}}$. By the Kadison-Pedersen result it contains any family $(v_{n})$ with $\sup\|v_{n}\|<1$. Therefore if $\|v_{n}\|\leq 1$ for all $n$ the sequence $(x_{n}\otimes\widehat{v}_{n})\in({\mathcal{M}}\overline{\otimes}{\mathcal{N}})_{*}$ is $c(1+\varepsilon)$-dominated by $(y_{n})$ for any $\varepsilon>0$.

Lemma 4.5**.**

Let $\Lambda=\{\psi_{n}\}$ be a completely Sidon set in ${\mathcal{M}}$ with constant $C$. There is a biorthogonal system $(x_{n})$ in ${\mathcal{M}}^{*}$ such that, for any $(\mathcal{N},\tau^{\prime})$ as before and any $(v_{n})$ with $\sup\nolimits_{n}\|v_{n}\|<c^{\prime}$, the sequence $(x_{n}\otimes\widehat{v}_{n})\in({\mathcal{M}}^{**}\overline{\otimes}{\mathcal{N}})_{*}$ is free-gaussian $Ccc^{\prime}$-dominated.

Proof.

This follows from Proposition 3.1 and Lemma 4.3 with the variant described in Remark 4.4, applied to $(v_{n}/c^{\prime})$. ∎

Remark 4.6*.*

At this point it is useful to observe the following: consider two sequences in $X$ (a von Neumann algebra predual) each of which is free-gaussian $C$-dominated, we claim that their union is free-gaussian $2C$-dominated. Indeed, if we have $x_{n}^{j}=T_{j}(y_{n})$, $x_{n}^{j}\in X$, with $T_{j}:M_{*}\to X$, $\|T_{j}\|_{dec}\leq C$ ($j=1,2$). Let $M\ast M$ be the free product, and let $E_{j}$ ($j=1,2$) be the conditional expectation onto each copy of $M_{*}$ in $(M\ast M)_{*}$. We can form the operator $T:(M\ast M)_{*}\to X$ defined by $T(a)=T_{1}E_{1}(x)+T_{2}E_{2}(x)$. Clearly $T$ is decomposable with $\|T\|_{dec}\leq 2C$. Let $(y^{1}_{n})$ and $(y^{2}_{n})$ denote the sequences corresponding to $(y_{n})$ in each copy of $M_{*}$ in $(M\ast M)_{*}$. We have $T(y^{j}_{n})=x^{j}_{n}$ for all $n$ and all $j=1,2$. But since the sequence $\{y^{1}_{n}\}\cup\{y^{2}_{n}\}$ is clearly equivalent to our original sequence $\{y_{n}\}$, this proves the claim.

We now come to a non-commutative generalization of our result from [17].

Theorem 4.7**.**

Let ${\mathcal{M}},{\mathcal{N}}$ be von Neumann algebras, with $\tau^{\prime}$ as before. Suppose $\Lambda_{1}=\{\psi^{1}_{n}\}$ and $\Lambda_{2}=\{\psi^{2}_{n}\}$ are two completely Sidon sets in a $C^{*}$-subalgebra $A\subset\mathcal{M}$. Assume there is a representation $\pi:A\to\mathcal{N}$ such that for some $\delta>0$ we have

[TABLE]

We assume that $\pi(\Lambda_{1})$ and $\pi(\Lambda_{2})$ are mutually orthogonal in $L_{2}(\tau^{\prime})$. Then the union $\Lambda_{1}\cup\Lambda_{2}$ is completely $\otimes_{\max}^{4}$-Sidon.

Proof.

We first observe that since $\pi$ extends to a (normal) representation from $A^{**}$ to $\mathcal{N}$, we may assume without loss of generality that $\mathcal{M}=A^{**}$ and that $\pi$ is extended to $\mathcal{M}$. Note that by our assumption $\Lambda_{1}\cup\Lambda_{2}$ is bounded in $\mathcal{M}$. By a simple homogeneity argument, we may assume without loss of generality that

[TABLE]

By Lemma 4.5 there are $x_{n}^{1}\in\mathcal{M}^{*}$ biorthogonal to $(\psi^{1}_{n})$ such that $(x_{n}^{1}\otimes\widehat{\pi(\psi^{1}_{n})})$ is free-gaussian dominated in $({\mathcal{M}}^{**}\overline{\otimes}{\mathcal{N}})_{*}$. Note that the latter is also biorthogonal to $(\psi^{1}_{n}\otimes\widehat{\pi(\psi^{1}_{n})})$. Similarly there are $x_{n}^{2}\in\mathcal{M}^{*}$ such that the same holds for $(x_{n}^{2}\otimes\widehat{\pi(\psi^{2}_{n})})$. By Remark 4.6, the union $\{x_{n}^{1}\otimes\widehat{\pi(\psi^{1}_{n})}\}\cup\{x_{n}^{2}\otimes\widehat{\pi(\psi^{2}_{n})}\}\subset({\mathcal{M}}^{**}\overline{\otimes}{\mathcal{N}})_{*}$ is free-gaussian dominated. But now the latter system is biorthogonal to $\{\psi_{n}^{1}\otimes{\pi(\psi^{1}_{n})}\}\cup\{\psi_{n}^{2}\otimes{\pi(\psi^{2}_{n})}\}\subset{\mathcal{M}}^{**}\overline{\otimes}{\mathcal{N}}$. Indeed, this holds because, by our orthogonality assumption, $\widehat{\pi\psi^{1}_{n}}(\pi\psi_{n}^{2})=\widehat{\pi\psi^{2}_{n}}(\pi\psi_{n}^{1})=0$ for all $m,n$. By Theorem 3.2 we conclude that the latter system, which can be described as $([id:{\mathcal{M}}\to{\mathcal{M}}^{**}]\otimes\pi)(\{\psi_{n}^{1}\otimes\psi_{n}^{1}\}\cup\{\psi_{n}^{2}\otimes\psi_{n}^{2}\})$, is completely $\otimes_{\max}^{2}$-Sidon. Using (1.5) to remove

[TABLE]

we see that this implies that $\Lambda_{1}\cup\Lambda_{2}$ itself is completely $\otimes_{\max}^{4}$-Sidon. ∎

Remark 4.8*.*

As the reader may have noticed the preceding proof actually shows that $\{\psi\otimes\psi\otimes\psi\otimes\psi\mid\psi\in\Lambda_{1}\cup\Lambda_{2}\}$ is completely Sidon in $(A\otimes_{\min}A)\otimes_{\max}(A\otimes_{\min}A)$.

Let $G$ be any discrete group. We say that subset $\Lambda\subset G$ is completely Sidon if the set $\{U_{G}(t)\mid t\in\Lambda\}$ is completely Sidon in $C^{*}(G)$. In this setting we recover our recent generalization [19] of Drury’s classical commutative result.

Corollary 4.9**.**

Let $G$ be any discrete group. The union of two completely Sidon subsets of $G$ is completely Sidon.

Proof.

We claim that any completely $\otimes^{4}_{\max}$-Sidon set in $G$ is completely Sidon. With this claim, the Corollary follows from Theorem 4.7 applied with $A_{1}=A_{2}=C^{*}(G)$. To check this claim, we use (ii) in Proposition 1.3. Let $U_{G}$ be the universal representation on $G$. Assuming $\Lambda=\{t_{n}\}$. Let $\psi_{n}=U_{G}(t_{n})$. For any unitary representation $\pi$ on $G$ with values in a unital $C^{*}$ algebra $A_{\pi}$, with the same notation as in (ii), we have obviously (since $\pi$ extends completely contractively to $C^{*}(G)$)

[TABLE]

Applying this with $\pi=U_{G}\otimes U_{G}\otimes U_{G}\otimes U_{G}$, and $A_{\pi}=C^{*}(G)\otimes_{\max}C^{*}(G)\otimes_{\max}C^{*}(G)\otimes_{\max}C^{*}(G)$, the claim becomes immediate. ∎

We refer to [19] for several complementary results, in particular for “completely Sidon” versions of the interpolation and Fatou-Zygmund properties of Sidon sets, and for a discussion of the closed span of a completely Sidon set in the reduced $C^{*}$-algebra of $G$.

Remark 4.10*.*

By analogy with the commutative case, we propose the following definition: Let $(y_{n})$ be a free-gaussian (i.e. free semicircular) sequence in $M_{*}$. We say that $(x_{n})$ in $A^{*}$ is free-subgaussian if there is $C$ such that for any $k$ the union of the sequences $\{x_{n}^{1}\},\cdots,\{x_{n}^{k}\}$ in $(A^{\ast k})^{*}$ is $C$-dominated by $(y_{n})$. Here $A{\ast}\cdots\ast A$ is the (full) free product of $k$ copies of $A$, and $x_{n}^{1},\cdots,x_{n}^{k}$ are the copies of $x_{n}$ in each of the free factors of $A\ast\cdots\ast A$. Note that with the same notation the sequence ${y_{n}^{1},\cdots,y_{n}^{k}}$ in $(M{\ast}\cdots\ast M)^{*}$ has the same distribution as the original sequence $(y_{n})$.

In the commutative case, when $(x_{n})$ lies in $L_{1}$ over some probability space and freeness is replaced by independence, this is the same as subgaussian in the usual sense, see [17, Prop. 2.10] for details. See [18] for a survey on subgaussian systems.

Acknowledgement. Thanks are due to Marek Bożejko, Simeng Wang and Mateusz Wasilewski for useful communications.