Outer actions of finite groups on prime C*-algebras

TL;DR

This paper investigates the properties of finite group actions on prime C*-algebras, establishing equivalences between different notions of outerness and expanding the understanding of their structural implications.

Contribution

The paper extends previous results by adding new properties to the list of equivalent conditions for finite abelian group actions on prime C*-algebras.

Findings

Properly outer and strictly outer actions are equivalent for finite groups on prime C*-algebras.

For finite abelian groups, these properties are also equivalent to additional structural conditions.

The work broadens the classification of group actions on prime C*-algebras by identifying new equivalent properties.

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 and strictly outer if the commutant of the algebra in the algebra of local mutipliers of the cross product consists of scalars [11]. In [11, Theorem 11] I proved that for finite groups and prime C*-algebras (not necessarily separable), the two notions are equivalent. I also proved that for finite abelian groups this is equivalent to other relevant properties of the action [11 Theorem 14]. In this paper I add other properties to the list in [11, Theorem 14].