2004/02/06 by Huaxin Lin, Hiroyuki Osaka, Lin, Huaxin +1
Mathematics · #46L05 #46L35 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA) #math.OA #msc:46L05 #msc:46L35
paper · pdf · doi:10.48550/arxiv.math/0402094
21 pages
openalex publication_date 2004/02/06 · arxiv created 2004/03/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a unital separable simple \CA with \tr(A)≤ 1 and α be an automorphism. We show that if α satisfies the tracially cyclic Rokhlin property then \tr(A\rtimesα\Z)≤ 1. We also show that whenever A has a unique tracial state and αm is uniformly outer for each m (\not= 0) and αr is approximately inner for some r>0, α satisfies the tracial cyclic Rokhlin property. By applying the classification theory of nuclear \CA s, we use the above result to prove a conjecture of Kishimoto: if A is a unital simple A\mathbb T-algebra of real rank zero and α∈ \Aut(A) which is approximately inner and if α satisfies some Rokhlin property, then the crossed product A\rtimesα\Z is again an A\mathbb T -algebra of real rank zero. As a by-product, we find that one can construct a large class of simple \CA s with tracial rank one (and zero) from crossed products.