A Dixmier type averaging property of automorphisms on a $C^*$-algebra

TL;DR

This paper characterizes when automorphisms on $C^*$-algebras have the strong averaging property, linking it to freeness, outerness, and proper outerness, and demonstrates its usefulness in simplifying and extending results on crossed product $C^*$-algebras.

Contribution

It provides a complete characterization of the strong averaging property for automorphisms on various classes of $C^*$-algebras, connecting it to existing concepts like outerness.

Findings

Automorphisms on commutative $C^*$-algebras are free iff they have the strong averaging property.

On unital separable simple $C^*$-algebras with a trace, the property holds iff the extension to the bi-dual is properly outer.

In non-tracial simple cases, the property is equivalent to the automorphism being outer.

Abstract

In his study of the relative Dixmier property for inclusions of von Neumann algebras and of $C^*$-algebras, Popa considered a certain property of automorphisms on $C^*$-algebras, that we here call the strong averaging property. In this note we characterize when an automorphism on a $C^*$-algebra has the strong averaging property. In particular, automorphisms on commutative $C^*$-algebras possess this property precisely when they are free. An automorphism on a unital separable simple $C^*$-algebra with at least one tracial state has the strong averaging property precisely when its extension to the finite part of the bi-dual of the $C^*$-algebra is properly outer, and in the simple, non-tracial case the strong averaging property is equivalent to being outer. To illustrate the usefulness of the strong averaging property we give three examples where we can provide simpler proofs of existing results on crossed product $C^*$-algebras, and we are also able to extend these results in different directions.