Operator theoretic characterizations of $[IN]$-groups and inner amenability

Abstract:

Let $G$ be a locally compact group and $p \in [1,\infty ]$. Let ${\pi _p}$ be the isometric representation of $G$ on ${L_p}(G)$ given by ${\pi _p}(x)f(t) = f({x^{ - 1}}tx)\Delta {(x)^{1/p}}$. Let ${\mathcal {A}’_p}$ be the commutant of ${\mathcal {A}_p}$ in $B({L_p}(G))$. In this paper we determine those $G$ for which: (*) ${\mathcal {A}’_p}$ contains a nonzero compact operator. We prove, among other things, that if $p \in [1,\infty )$, then (*) holds if and only if $G \in [IN]$, and that if $p = \infty$, then (*) holds if and only if $G$ is inner amenable.