Non-commutativity of the central sequence algebra for separable non-type I C${}^{\ast }$-algebras

Abstract

We show that if $A$ is a separable, simple and non-type I C${}^{\ast }$ algebra, then for every properly infinite hyperfinite von Neumann algebra $M$ with separable predual, its Ocneanu ultrapower $M^{\prime }\cap M^{\omega }$ arises as a sub-quotient of the central sequence algebra $F(A)$ defined by the second named author. In particular, this answers affirmatively the question of the second named author (Abel Symposium '04): the central sequence C${}^{\ast }$-algebra of the reduced free group C${}^{\ast }$-algebra $C_{\mathrm{red}}^{\ast }({\mathbb{F}}_2)$ is non-commutative.

15 pages, same as the published version. We added Ozawa's proof (using Kishimoto-Ozawa-Sakai Theorem) of non-commutativity of F(A) in Appendix

References in corpus (2)

• Discrete amenable group actions on von Neumann algebras and invariant nuclear C*-subalgebras
• Central sequence C*-algebras and tensorial absorption of the Jiang-Su algebra

Cited by in corpus (3)

• Equivariant property (SI) revisited
• Central Sequences in Subhomogeneous Unital C*-algebras
• Maximal Tracial Algebras