Title: Amenability of the sequence of unitary groups associated with a $C*$-algebra

Authors: Ping Wong NG

Issue: Volume 55 (2006), Issue 4, 1389-1400

Abstract: We study norm topology amenability of the unitary group of a unital $C^*$-algebra, and its relations with nuclearity and stable finiteness. These considerations lead to an operator-space version of group amenability. The main result is the following: \textit{Let $\mathcal{A}$ be a unital separable simple $C^*$-algebra. 1. If $\mathcal{A}$ is nuclear and quasidiagonal, then the unitary group sequence $\{ U( \mathbb{M}_{n} (\mathcal{A})) \}_{n=1}^{\infty}$ is amenable. 2. If $\{ U(\mathbb{M}_{n} (\mathcal{A})) \}_{n=1}^{\infty}$ is amenable, then $\mathcal{A}$ is nuclear and stably finite.}