Relating properties of crossed products to those of fixed point algebras

TL;DR

This paper establishes that for various properties of C*-algebras, the fixed point algebra and the crossed product share these properties under actions of second countable compact abelian groups, linking their structural characteristics.

Contribution

It proves an equivalence of properties between fixed point algebras and crossed products for a broad class of C*-algebra properties under specific group actions.

Findings

Fixed point algebra and crossed product share properties like real rank zero and pure infiniteness.

Equivalence holds for properties such as stable rank one and residual hereditary infiniteness.

Results apply to actions of second countable compact abelian groups.

Abstract

For a number of properties of C*-algebras, including real rank zero, stable rank one, pure infiniteness, residual hereditary infiniteness, the combination of pure infiniteness and the ideal property, the property of being an AT algebra with real rank zero, and stability under tensoring with a strongly selfabsorbing C*-algebra, we prove the following. Consider an arbitrary action of a second countable compact abelian group on a separable C*-algebra. Then the fixed point algebra under the action has the given property if and only if the crossed product has the same property.