Bekka-type amenabilities for unitary corepresentations of locally compact quantum groups

Abstract

In this short note, we further Ng’s work by extending Bekka amenability and weak Bekka amenability to general locally compact quantum groups, and we generalize some of Ng’s results to the general case. In particular, we show that a locally compact quantum group $\mathbb{G}$ is coamenable if and only if the contra-corepresentation of its fundamental multiplicative unitary $W_{\mathbb{G}}$ is Bekka-amenable, and that $\mathbb{G}$ is amenable if and only if its dual quantum group’s fundamental multiplicative unitary $W_{\hat{\mathbb{G}}}$ is weakly Bekka-amenable.