Reduced operator algebras of trace-preserving quantum automorphism groups

Abstract

Let B be a finite dimensional C ∗ -algebra equipped with its canonical trace induced by the regular representation of B on it- self. 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 b G has the property of rapid decay, the reduced von Neumann algebra L ∞ (G) has the Haagerup property and is solid, and that L ∞ (G) is (in most cases) a prime type II1-factor. As applica- tions of these and other results, we deduce the metric approximation property, exactness, simplicity and uniqueness of trace for the reduced C ∗ -algebra Cr(G), and the existence of a multiplier-bounded approx- imate identity for the convolution algebra L 1 (G).