Weak amenability of locally compact quantum groups and approximation properties of extended quantum SU(1,1)

TL;DR

This paper investigates weak amenability and approximation properties of certain non-discrete locally compact quantum groups, specifically extended quantum SU(1,1), establishing their weak amenability, coamenability, and Haagerup property.

Contribution

It provides new criteria for weak amenability in quantum groups and applies these to show extended quantum SU(1,1) is weakly amenable, coamenable, and has the Haagerup property.

Findings

Extended quantum SU(1,1) is weakly amenable.

Extended quantum SU(1,1) is coamenable.

Extended quantum SU(1,1) has the Haagerup property.

Abstract

We study weak amenability for locally compact quantum groups in the sense of Kustermans and Vaes. In particular, we focus on non-discrete examples. We prove that a coamenable quantum group is weakly amenable if there exists a net of positive, scaling invariant elements in the Fourier algebra A(G) whose representing multipliers form an approximate identity in C_0(G) that is bounded in the M_0A(G) norm; the bound being an upper estimate for the associated Cowling-Haagerup constant. As an application, we find the appropriate approximation properties of the extended quantum SU(1,1) group and its dual. That is, we prove that it is weakly amenable and coamenable. Furthermore, it has the Haagerup property in the quantum group sense, introduced by Daws, Fima, Skalski and White.