Orthogonal free quantum group factors are strongly 1-bounded

We prove that the orthogonal free quantum group factors $\mathcal{L}(\mathbb{F}O_N)$ are strongly $1$-bounded in the sense of Jung. In particular, they are not isomorphic to free group factors. This result is obtained by establishing a spectral regularity result for the edge reversing operator on the quantum Cayley tree associated to $\mathbb{F}O_N$, and combining this result with a recent free entropy dimension rank theorem of Jung and Shlyakhtenko.Comment: v3: accepted versio

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).Comment: Section 6 removed and replaced by a more general solidity resul