Reduced operator algebras of trace-preserving quantum automorphism groups

Details Reduced operator algebras of trace-preserving quantum automorphism groups Reduced operator algebras of trace-preserving quantum automorphism groups

2012-02-22

arxiv.org/abs/1202.5020v1

Mathematics

Operator Algebras

Scientific paper

Let $B$ be a finite dimensional C$^*$-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 that $L^\infty(\G)$ is (in most cases) a full 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 $C^*$-algebra $C_r(\G)$, and the existence of a multiplier-bounded approximate identity for the convolution algebra $L^1(\G)$. We also show that when $B$ is a full matrix algebra, $L^\infty(\G)$ is an index 2 subfactor of the reduced von Neumann algebra of a free orthogonal quantum group and thus solid and prime.

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