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 $M'\cap M^{\omega}$ 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 $C_{\rm{red}}^*(\mathbb{F}_2)$ is non-commutative.