On exotic group C*-algebras

Let $\mathrm{\Gamma }$ be a discrete group. A $C^{\ast }$-algebra $A$ is an exotic $C^{\ast }$-algebra (associated to $\mathrm{\Gamma }$) if there exist proper surjective $C^{\ast }$-quotients $C^{\ast }(\mathrm{\Gamma })\to A\to C_r^{\ast }(\mathrm{\Gamma })$. 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.