Approximation properties for locally compact quantum groups

TL;DR

This survey explores approximation properties of locally compact quantum groups, including amenability, co-amenability, Haagerup property, and weak amenability, highlighting their interrelations and implications for operator algebras.

Contribution

It provides a comprehensive overview of approximation properties in quantum groups, linking them to operator algebra concepts and monoidal equivalence.

Findings

Dual notions of amenability and co-amenability are analyzed.

Connections between approximation properties and operator algebra approximation properties are established.

Central approximation properties for discrete quantum groups are linked to monoidal equivalence.

Abstract

This is a survey of some aspects of the subject of approximation properties for locally compact quantum groups, based on lectures given at the {\it Topological Quantum Groups} Graduate School, 28 June - 11 July, 2015 in Bed\l{}ewo, Poland. We begin with a study of the dual notions of amenability and co-amenability, and then consider weakenings of these properties in the form of the Haagerup property and weak amenability. For discrete quantum groups, the interaction between these properties and various operator algebra approximation properties are investigated. We also study the connection between central approximation properties for discrete quantum groups and monoidal equivalence for their compact duals. We finish by discussing the central weak amenability and central Haagerup property for free quantum groups.