On exotic group C*-algebras

Abstract

Let Γ be a discrete group. A C*-algebra A is an exotic C*-algebra (associated to Γ) if there exist proper surjective C*-quotients C⁎(Γ)→A→Cr⁎(Γ) which compose to the canonical quotient C⁎(Γ)→Cr⁎(Γ). In this paper, we show that a large class of exotic C*-algebras has poor local properties. More precisely, we demonstrate the failure of local reflexivity, 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*-algebras defined by Brown and Guentner (see [8]). 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.