Conjugate convolution operators and inner amenability

Abstract

Let $G$ be a group and $L^\infty(G)$ be the $C^*$-algebra of bounded complex-valued functions on $G$. $G$ is called inner amenable if there exists a positive norm 1 functional $m$ on $L^\infty(G)$ such that $m(\rho(y)f) = m(f)$ for each $y \in G, f \in L^\infty(G)$ (where $\rho(y)f(x) = f(yxy^{-1})$); the functional $m$ is called an inner invariant mean. In this paper, among the other things, we prove a variety of characterizations of inner amenable groups. We also give sufficient conditions on an inner invariant mean to be a topologically inner invariant mean.