On exotic group C*-algebras

Abstract

Let $\Gamma$ be a discrete group. A $C^{\ast }$-algebra $A$ is an exotic $C^{\ast }$-algebra (associated to $\Gamma$) if there exist proper surjective $C^{\ast }$-quotients $C^{\ast }(\Gamma )\to A\to C_r^{\ast }(\Gamma )$. In this paper, we show that a large class of exotic $C^{\ast }$-algebras have poor local properties. More precisely, we demonstrate the failure of local reflexitity, exactness, and local lifting property. Additionally, $A$ does not admit an amenable trace and, hence, is not quasidiagonal and does not have the WEP when $A$ is from the class of exotic $C^{\ast }$-algebras defined by Brown and Guentner. In order to achieve the main results of this paper, we prove a result which implies the factorization property for the class of discrete groups which are algebraic subgroups of locally compact amenable groups.

Improvements to some proofs and several other changes, 14 pages, to appear in J. Funct. Anal

References in corpus (3)

• New C*-completions of discrete groups and related spaces
• C*-norms for tensor products of discrete group C*-algebras
• Exotic group C*-algebras

Cited by in corpus (1)

• Ordered invariant ideals of Fourier-Stieltjes algebras