Tracial Rokhlin property and non-commutative dimensions

TL;DR

This paper explores the tracial Rokhlin property in C*-algebra actions, extending its classification and inheritance properties to non-simple algebras and introducing a weaker version for broader applicability.

Contribution

It provides a complete classification of the tracial Rokhlin property for product-type actions and introduces the weak tracial Rokhlin property for non-simple algebras, analyzing inheritance of structural properties.

Findings

Classification of tracial Rokhlin property for product-type actions

Introduction of weak tracial Rokhlin property for non-simple algebras

Inheritance of structural properties under certain conditions

Abstract

Tracial Rokhlin property was introduced by Phillips to prove various structure theorems for crossed product. But it is defined for actions on simple C*-algebras only. This paper consists of two major parts: In section 2 and 3, we study the permanence properties and give a complete classification of tracial Rokhlin property for product-type actions; In section 4 and 5, we introduce the weak tracial Rokhlin property for actions on non-simple C*-algebras. We prove that when the action has the weak tracial Rokhlin property and the crossed product is simple, the properties on $A$ of having tracial rank $\leq k$, or real rank 0, or stable rank 1, can be inherited by the crossed product.