A construction of finite index C*-algebra inclusions from free actions of compact quantum groups

TL;DR

This paper constructs finite index inclusions of C*-algebras from free actions of compact quantum groups and proves the equivalence of two key notions of freeness, advancing the understanding of quantum symmetry actions.

Contribution

It introduces a method to produce finite index inclusions from free quantum group actions and establishes the equivalence of two freeness conditions.

Findings

Inclusion is finite index when the quantum group acts freely.

Ellwood and saturatedness conditions are equivalent for freeness.

Provides a framework for analyzing quantum symmetries in C*-algebras.

Abstract

Given an action of a compact quantum group on a unital C*-algebra, one can amplify the action with an adjoint representation of the quantum group on a finite dimensional matrix algebra, and consider the resulting inclusion of fixed point algebras. We show that this inclusion is a finite index inclusion of C*-algebras when the quantum group acts freely. We show that two natural definitions for a quantum group to act freely, namely the Ellwood condition and the saturatedness condition, are equivalent.