Coactions of compact groups on Mₙ

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 Mₙ 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 Mₙ reveals that SO(3) has them, while SU(2) does not. We give explicit examples of the two smallest finite nonabelian groups having effective ergodic coactions on Mₙ.