Reduced operator algebras of trace-preserving quantum automorphism groups

TL;DR

This paper investigates the properties of trace-preserving quantum automorphism groups of finite-dimensional C*-algebras, establishing key properties like rapid decay, the Haagerup property, and solidity of their associated von Neumann algebras.

Contribution

It provides new results on the operator algebraic properties of quantum automorphism groups, including rapid decay, Haagerup property, and conditions for primeness and type classification.

Findings

The dual quantum group has rapid decay.

The von Neumann algebra has the Haagerup property and is solid.

The von Neumann algebra is often a prime type II₁ factor.

Abstract

Let $B$ be a finite dimensional C$^\ast$-algebra equipped with its canonical trace induced by the regular representation of $B$ on itself. In this paper, we study various properties of the trace-preserving quantum automorphism group $\G$ of $B$. We prove that the discrete dual quantum group $\hG$ has the property of rapid decay, the reduced von Neumann algebra $L^\infty(\G)$ has the Haagerup property and is solid, and that $L^\infty(\G)$ is (in most cases) a prime type II$_1$-factor. As applications of these and other results, we deduce the metric approximation property, exactness, simplicity and uniqueness of trace for the reduced $C^\ast$-algebra $C_r(\G)$, and the existence of a multiplier-bounded approximate identity for the convolution algebra $L^1(\G)$.