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

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 ℓ 1 (i.e. ℓ 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 ∞ 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. MSC Classif. 43A46, 46L06Recently, following the impulse of Bourgain and Lewko [1], we studied in [17] the uniformly bounded orthonormal systems that span in L ∞ a subspace isomorphic to ℓ 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 {ψ n } in L 2 of a probability space (Ω, P) is the union {ψ 1 n } ∪ {ψ 2 n } of two Sidon sequences, then the sequence {ψ n ⊗ ψ n ⊗ ψ n ⊗ ψ n } or simply {ψ ⊗ 4 n } is Sidon in L ∞ (Ω 4 , 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 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 ∞ . 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 Z ∞ formed of all the finitely supported functions f : N → Z. This is sometimes referred to as the free Abelian group with countably infinitely many generators. The dual of the discrete group Z ∞ is the compact group T N , and the von Neumann algebra of Z ∞ can be identified with L ∞ (T N ). The analogue of this for our work is the free group F ∞ with countably infinitely many generators, and its von Neumann algebra M F∞ . In the commutative case the generators of Z ∞ correspond in L ∞ (T N ) to independent random variables uniformly distributed over T. In classical Sidon set theory, the associated Riesz pr...