Crossed Product Equivalence of Quantum Automorphism Groups

Abstract

We compare the algebras of the quantum automorphism group of finite-dimensional C${}^{\ast }$-algebra $B$, which includes the quantum permutation group $S_N^{+}$, where $N=\dim B$. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group $\Gamma$ lead to isomorphic $\ast$-algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong $1$-boundedness between the various algebras associated with these quantum groups.

27 pages. Revised version to appear in IMRN