#### DOCUMENTA MATHEMATICA, Vol. 18 (2013), 1349-1402

M. Brannan

Reduced Operator Algebras of Trace-Preserving Quantum Automorphism Groups

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 $L^1(\G)$.

2010 Mathematics Subject Classification: Primary 46L65, 20G42; Secondary 46L54.

Keywords and Phrases: Quantum automorphism groups, approximation properties, property of rapid decay, II_1-factor, solid von Neumann algebra, Temperley-Lieb algebra.

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