Crossed products by finite group actions with the Rokhlin property

TL;DR

This paper demonstrates that various classes of separable unital C*-algebras remain stable under crossed products with finite group actions possessing the Rokhlin property, expanding understanding of their structural robustness.

Contribution

It establishes the closure of multiple important C*-algebra classes under such crossed products, including AI, AT, AH algebras, and those with specific rank and quotient properties.

Findings

Closure of AI, AT, and related classes under crossed products.

Stability of simple unital AH algebras with slow dimension growth.

Preservation of real rank zero and stable rank one in crossed products.

Abstract

We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: (1) AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. (2) Simple unital AH algebras with slow dimension growth and real rank zero. (3) C*-algebras with real rank zero or stable rank one. (4) C*-algebras whose quotients all satisfy the Universal Coefficient Theorem. Along the way, we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective.