Crossed Product Equivalence of Quantum Automorphism Groups of Finite Dimensional C*-Algebras

Abstract

AbstractWe 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.