Crossed Product Equivalence of Quantum Automorphism Groups

TL;DR

This paper demonstrates that certain algebraic constructions like matrix amplification and crossed products preserve key properties of quantum automorphism groups, enabling property transfer between related algebras.

Contribution

It establishes isomorphisms between algebras of quantum automorphism groups under specific operations, facilitating the transfer of properties like Connes embeddability.

Findings

Matrix amplification preserves algebraic properties.

Crossed products by finite Abelian groups yield isomorphic algebras.

Properties such as inner unitarity and strong 1-boundedness are transferable.

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.