Symmetries of a generic coaction

Abstract.

If B is ${\bf C}^*$-algebra of dimension $4\leq n<\infty$ then the finite dimensional irreducible representations of the compact quantum automorphism group of B, say $G_{aut}(\widehat{B})$, have the same fusion rules as the ones of SO(3). As consequences, we get (1) a structure result for $G_{aut}(\widehat{B})$ in the case where B is a matrix algebra (2) if $n\geq 5$ then the dual $\widehat{G}_{aut}(\widehat{B})$ is not amenable (3) if $n\geq 4$ then the fixed point subfactor $P^{G_{aut}(\widehat{B})}\subset (B\otimes P)^{G_{aut}(\widehat{B})}$ has index n and principal graph $A_\infty$.