Completely Sidon sets in C∗-algebras
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© 2018, Springer-Verlag GmbH Austria, part of Springer Nature. 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 ℓ1 (i.e. ℓ1 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.
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