The Rokhlin property for inclusions of C*-algebras

TL;DR

This paper introduces the Rokhlin property for conditional expectations in inclusions of σ-unital C*-algebras and demonstrates its inheritance of various structural properties under certain conditions.

Contribution

It defines the Rokhlin property for conditional expectations and proves its inheritance of key properties in simple C*-algebras with finite index inclusions.

Findings

Rokhlin property for conditional expectations is introduced.

Inheritance of properties like simplicity, nuclearity, and absorption under Rokhlin property.

Fixed point and crossed product algebras inherit these properties when group actions have Rokhlin property.

Abstract

Let $P \subset A$ be an inclusion of $\sigma$-unital C*-algebras with a finite index in the sense of Izumi. Then we introduce the Rokhlin property for a conditional expectation $E$ from $A$ onto $P$ and show that if $A$ is simple and satisfies any of the property $(1) \sim (12)$ listed in the below, and $E$ has the Rokhlin property, then so does $P$. (1) Simplicity;(2) Nuclearity;(3) C*-algebras that absorb a given strongly self-absorbing C*-algebra $\mathcal{D}$; (4)C*-algebras of stable rank one; (5) C*-algebras of real rank zero;(6) C*-algebras of nuclear dimension at most $n$, where $n \in Z^+$; (7)C*-algebras of decomposition rank at most $n$, where $n \in Z^+$; (8) Separable simple C*-algebras that are stably isomorphic to AF algebras; (9) Separable simple C*-algebras that are stably isomorphic to AI algebras; (10) Separable simple C*-algebras that are stably isomorphic to AT algebras; (11) Separable simple C*-algebras that are stably isomorphic to sequential direct limits of one dimensional NCCW complexes; (12) Separable C*-algebras with strict comparison of positive elements. In particular, when $\alpha : G \rightarrow \rm{Aut}(A)$ is an action of a finite group $G$ on $A$ with the Rokhlin property in the sense of Nawata, the properties $(1) \sim (12)$ are inherited to the fixed point algebra $A^\alpha$ and the crossed product algebra $A \rtimes_\alpha G$ from $A$.