Approximation properties and entropy estimates for crossed products by actions of amenable discrete quantum groups

TL;DR

This paper demonstrates that key approximation properties and noncommutative entropy are preserved when forming crossed products of C*-algebras by actions of amenable discrete quantum groups, extending to twisted cases.

Contribution

It provides explicit approximation nets for crossed products and shows preservation of nuclearity, exactness, and entropy under quantum group actions, including twisted cases.

Findings

Approximate nets constructed for crossed products.

Approximation properties are preserved under quantum group actions.

Noncommutative entropy remains unchanged in crossed products.

Abstract

We construct explicit approximating nets for crossed products of C*-algebras by actions of discrete quantum groups. This implies that C*-algebraic approximation properties such as nuclearity, exactness or completely bounded approximation property are preserved by taking crossed products by actions of amenable discrete quantum groups. We also show that the noncommutative topological entropy of a transformation commuting with the quantum group action does not change when we pass to the canonical extension to the crossed product. Both these results are extended to twisted crossed products via a stabilisation trick.