Coactions of compact groups on $M_n$

TL;DR

This paper characterizes coactions of compact groups on finite-dimensional C*-algebras, showing their relation to Fell bundles, and explores conditions for inner and ergodic coactions, including explicit examples involving finite groups.

Contribution

It establishes a correspondence between coactions and Fell bundles on finite-dimensional C*-algebras and provides criteria for inner and ergodic coactions, with explicit examples.

Findings

Every coaction on a matrix algebra is implemented by a unitary operator.

An inner coaction's fixed-point algebra contains an abelian subalgebra of dimension n.

SO(3) admits effective ergodic coactions on M_n, while SU(2) does not.

Abstract

We prove that every coaction of a compact group on a finite-dimensional $C^*$-algebra is associated with a Fell bundle. Every coaction of a compact group on a matrix algebra is implemented by a unitary operator. A coaction of a compact group on $M_n$ is inner if and only if its fixed-point algebra has an abelian $C^*$-subalgebra of dimension $n$. Investigating the existence of effective ergodic coactions on $M_n$ reveals that $\operatorname{SO}(3)$ has them, while $\operatorname{SU}(2)$ does not. We give explicit examples of the two smallest finite nonabelian groups having effective ergodic coactions on $M_n$.