The Jiang-Su absorption for inclusions of unital C*-algebras

TL;DR

This paper introduces the tracial Rokhlin property for inclusions of unital C*-algebras and demonstrates that certain structural properties are preserved under this condition, advancing understanding of C*-algebra classifications.

Contribution

It defines the tracial Rokhlin property for algebra inclusions and proves the preservation of various structural properties, including Jiang-Su absorption, under this property.

Findings

Structural properties are preserved under the tracial Rokhlin property.

Equivalence of tracial Rokhlin property for group actions and fixed point algebras.

Supports Toms and Winter's conjecture on C*-algebra classification.

Abstract

In this paper we will introduce the tracial Rokhlin property for an inclusion of separable simple unital C*-algebras $P \subset A$ with finite index in the sense of Watatani, and prove theorems of the following type. Suppose that $A$ belongs to a class of C*-algebras characterized by some structural property, such as tracial rank zero in the sense of Lin. Then $P$ belongs to the same class. The classes we consider include:(1) Simple C*-algebras with real rank zero or stable rank one, (2) Simple C*-algebras with tracial rank zero or tracial rank less than or equal to one, (3) Simple C*-algebras with the Jiang-Su algebra $\mathcal{Z}$ absorption, (4) Simple C*-algebras for which the order on projections is determined by traces, (5) Simple C*-algebras with the strict comparison property for the Cuntz semigroup. The conditions (3) and (5) are important properties related to Toms and Winter's conjecture, that is, the properties of strict comparison, finite nuclear dimension, and Z-absorption are equivalent for separable simple infinite-dimensional nuclear unital C*-algebras. We show that an action $\alpha$ from a finite group $G$ on a simple unital C*-algebra $A$ has the tracial Rokhlin property in the sense of Phillips if and only if the canonical conditional expectation $E\colon A \rightarrow A^G$ has the tracial Rokhlin property for an inclusion $A^G \subset A$.