The approximation property for locally compact quantum groups

TL;DR

This paper investigates the Haagerup--Kraus approximation property for locally compact quantum groups, unifying previous research, exploring its inheritance, and relating it to other approximation properties and tensor categories.

Contribution

It generalizes and unifies existing work on approximation properties for quantum groups, introduces a central variant for discrete cases, and connects to tensor category theory.

Findings

Established inheritance properties for the approximation property.

Linked the approximation property to the weak* operator approximation property.

Introduced a central approximation property for discrete quantum groups.

Abstract

We study the Haagerup--Kraus approximation property for locally compact quantum groups, generalising and unifying previous work by Kraus--Ruan and Crann. Along the way we discuss how multipliers of quantum groups interact with the $\mathrm{C}^*$-algebraic theory of locally compact quantum groups. Several inheritance properties of the approximation property are established in this setting, including passage to quantum subgroups, free products of discrete quantum groups, and duals of double crossed products. We also discuss a relation to the weak$^*$ operator approximation property. For discrete quantum groups, we introduce a central variant of the approximation property, and relate this to a version of the approximation property for rigid $\mathrm{C}^*$-tensor categories, building on work of Arano--De Laat--Wahl.