Coactions of compact groups on \(M_n\)

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 \(M_n\).