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

TL;DR

This paper introduces the tracial Rokhlin property for finite cyclic group actions on certain C*-algebras, showing how it preserves tracial rank zero in crossed products and applying it to classify noncommutative tori.

Contribution

It defines the tracial Rokhlin property for cyclic group actions, compares it with Izumi's Rokhlin property, and applies these concepts to classify higher dimensional noncommutative tori.

Findings

Crossed products preserve tracial rank zero under certain conditions.

Examples show differences between tracial Rokhlin and Rokhlin properties.

Every simple higher dimensional noncommutative torus is an AT algebra.

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.