Cancellation for inclusions of C*-algebras of finite depth

2007/04/27 by Ja A Jeong, Jeong, Ja A, Hiroyuki Osaka +5 · 2 citations
Mathematics · #46L37 (Secondary) #46L55 (Primary) 19A13 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.0704.3645

openalex publication_date 2007/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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) ≤ \tsr (A) + n - 1. As an application, we show that if a unital inclusion A ⊂ 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 K₀ (A).

Cited by

• Depth 2 inclusions of simple C^*-algebras and their weak C^*-Hopf algebra symmetries
• Higher reflections and entropy of canonical shifts for inclusions of C^*-algebras with finite Watatani index

Related

• Stable rank of inclusion of C*-algebras of depth 2
• Topological stable rank of inclusions of unital C*-algebras
• Cancellation does not imply stable rank one
• Stable rank for crossed products by actions of finite groups on C*-algebras
• Stable rank for inclusions of C*-algebras