A Galois correspondence for compact group actions on C*-algebras

TL;DR

This paper establishes a Galois correspondence linking invariant subalgebras and subgroup structures for compact group actions on C*-algebras with commuting minimal actions, extending the understanding of symmetry and invariance in operator algebras.

Contribution

It introduces a novel Galois correspondence for compact group actions on C*-algebras with commuting minimal actions, connecting invariant subalgebras to subgroup structures.

Findings

One-to-one correspondence between invariant subalgebras and closed normal subgroups.

Characterization of subalgebras containing the fixed point algebra.

Extension of Galois theory to non-commutative operator algebras.

Abstract

In this paper, we prove a Galois correspondence for compact group actions on C*-algebras in the presence of a commuting minimal action. Namely, we show that there is a one to one correspondence between the C*-subalgebras that are globally invariant under the compact action and the commuting minimal action, that in addition contain the fixed point algebra of the compact action and the closed, normal subgroups of the compact group.