Crossed products by finite cyclic group actions with the tracial Rokhlin property

Abstract

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.

arXiv comment90 pages, AMSLaTeX

References in corpus (1)

• The AF structure of non commutative toroidal Z/4Z orbifolds

Cited by in corpus (10)

• Minimal dynamical systems on the product of the Cantor set and the circle
• The AF structure of non commutative toroidal Z/4Z orbifolds
• Furstenberg transformations on irrational rotation algebras
• The Rokhlin Property for Automorphisms on Simple C*-Algebras
• Nuclear dimension for an inclusion of unital C*-algebras
• Crossed products by minimal homeomorphisms
• Strong Morita equivalence of higher-dimensional noncommutative tori
• Morita equivalence of smooth noncommutative tori
• The Rokhlin property and the tracial topological rank
• Tracial Rokhlin property for automorphisms on simple $A\mathbb{T}$-algebras