Reduced operator algebras of trace-preserving quantum automorphism groups

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 $\mathbb{G}$ of $B$. We prove that the discrete dual quantum group $\hat{\mathbb{G}}$ has the property of rapid decay, the reduced von Neumann algebra $L^{\infty }(\mathbb{G})$ has the Haagerup property and is solid, and that $L^{\infty }(\mathbb{G})$ 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 $C_r(\mathbb{G})$, and the existence of a multiplier-bounded approximate identity for the convolution algebra $L^1(\mathbb{G})$.