Properly outer and strictly outer actions of finite groups on prime C*-algebras

Abstract

An action of a compact, in particular finite group on a C*-algebra is called properly outer if no automorphism of the group that is distinct from identity is implemented by a unitary element of the algebra of local multipliers of the C*-algebra. In this paper, I define the notion of strictly outer action (similar to the definition for von Neumann factors in [S. Vaes, The unitary implementation of a locally compact group action, J. Funct. Anal. 180 (2001) 426–480]) and prove that for finite groups and prime C*-algebras, it is equivalent to the proper outerness of the action. For finite abelian groups this is equivalent to other relevant properties of the action.