Cancellation for inclusions of C*-algebras of finite depth

TL;DR

This paper investigates conditions under which inclusions of C*-algebras with finite depth and index-finite type ensure cancellation properties, especially in crossed products by finite groups or integers.

Contribution

It establishes a new criterion linking stable rank, Property (SP), and finite depth to cancellation in C*-algebra inclusions and crossed products.

Findings

Topological stable rank bound for inclusions with conditional expectations

Cancellation results for crossed products by finite groups with specific properties

Cancellation in crossed products by integers under outer action and K_0-triviality

Abstract

Let B be a unital C*-algebra, let A be a unital subalgebra, and let E be a conditional expectation from B to A with index-finite type and a quasi-basis of n elements. Then the topological stable rank satisfies \tsr (B) \leq \tsr (A) + n - 1. As an application, we show that if a unital inclusion A \subset B of C*-algebras has index-finite type and finite depth, and A is simple with stable rank one and Property (SP), then B has cancellation. In particular, if A is a simple unital C*-algebra with stable rank one and Property (SP), and a finite group G acts on A, then the crossed product has cancellation. Separately, if the group is the integers, we obtain cancellation under the additional hypotheses that the group action is outer and is trivial on K_0 (A).