Some Permanence properties for crossed products by compact group actions with the tracial Rokhlin property

TL;DR

This paper investigates how certain structural properties of a simple infinite-dimensional C*-algebra are preserved when taking fixed point algebras and crossed products under actions of compact groups with the tracial Rokhlin property.

Contribution

It establishes that properties like stable rank one, real rank zero, and various comparison and divisibility conditions are retained in the fixed point algebra and crossed product under these group actions.

Findings

Fixed point algebra and crossed product preserve stable rank one.

Real rank zero is maintained in the crossed product.

Comparison and divisibility properties are inherited by the crossed product.

Abstract

In this paper, we give some properties of the fixed point algebra and the crossed product of a unital separable simple infinite dimensional C*-algebra by an action of a second-countable compact group with the tracial Rokhlin property with comparison that could be deduced from the properties of its original algebra: (1) stable rank one; (2) real rank zero; (3) $\beta$-comparison; (4) Winter's $n$-comparison; (5) $m$-almost divisible; (6) weakly ($m$,$n$)-divisible.