2003/06/28 by N. Christopher Phillips, Phillips, N. Christopher · 3 citations
Mathematics · #46L35 #46L40 (Secondary) #46L55 (Primary) 22D25 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:22D25 #msc:46L35 #msc:46L40 #msc:46L55
paper · pdf · doi:10.48550/arxiv.math/0306410
90 pages, AMSLaTeX
arxiv created 2003/06/28 · openalex publication_date 2003/06/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define the tracial Rokhlin property for actions of finite cyclic groups on stably finite simple unital C*-algebras. We prove that when the algebra is in addition simple and has tracial rank zero, then the crossed product again has tracial rank zero. Under a kind of weak approximate innerness assumption and one other technical condition, we prove that if the action has the the tracial Rokhlin property and the crossed product has tracial rank zero, then the original algebra has tracial rank zero. We give examples showing how the tracial Rokhlin property differs from the Rokhlin property of Izumi. We use these results, together with work of Elliott-Evans and Kishimoto, to give an inductive proof that every simple higher dimensional noncommutative torus is an AT algebra. We further prove that the crossed product of every simple higher dimensional noncommutative torus by the flip is an AF algebra, and that the crossed products of irrational rotation algebras by the standard actions of the cyclic groups of orders 3, 4, and 6 are simple AH algebras with real rank zero.