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

## TL;DR

This paper extends Bekka-type amenability concepts to locally compact quantum groups, establishing new characterizations of co-amenability and amenability via corepresentations and fundamental multiplicative unitaries.

## Contribution

It generalizes Ng's results to broader quantum group settings, linking Bekka amenability to corepresentation properties and duality.

## Key findings

- Co-amenability characterized by Bekka amenability of the contra-corepresentation.
- Amenability characterized by weak Bekka amenability of the dual quantum group's fundamental unitary.
- Extension of Bekka-type amenability concepts to general locally compact quantum groups.

## Abstract

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