On exotic group C*-algebras

TL;DR

This paper investigates exotic group C*-algebras, showing they generally lack many desirable local properties and are not amenable, by analyzing their structural and factorization characteristics.

Contribution

It demonstrates that a broad class of exotic C*-algebras fail key local properties and introduces a factorization result for certain algebraic subgroups.

Findings

Exotic C*-algebras lack local reflexivity, exactness, and local lifting property.

They do not admit amenable traces, quasidiagonality, or the weak expectation property.

A factorization property is established for algebraic subgroups of locally compact amenable groups.

Abstract

Let $\Gamma$ be a discrete group. A $C^*$-algebra $A$ is an exotic $C^*$-algebra (associated to $\Gamma$) if there exist proper surjective $C^*$-quotients $C^*(\Gamma)\to A\to C^*_r(\Gamma)$. In this paper, we show that a large class of exotic $C^*$-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 $C^*$-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.