  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 $L^1(\G)$.
