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 \widehat{\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) .