Finite symmetry group actions on substitution tiling C*-algebras

TL;DR

This paper studies the effects of finite symmetry group actions on substitution tiling C*-algebras, demonstrating properties like real rank zero, unique trace, and Rokhlin properties, and linking to AF algebras from Penrose tilings.

Contribution

It establishes new structural properties of crossed product C*-algebras under finite symmetry group actions on substitution tilings, including Rokhlin properties and connections to AF algebras.

Findings

Crossed product has real rank zero and a unique trace.

Action satisfies the weak Rokhlin property, and the tracial Rokhlin property under certain conditions.

Links between tiling C*-algebras and AF algebras from Penrose tilings.

Abstract

For a finite symmetry group $G$ of an aperiodic substitution tiling system $(\p,\omega)$, we show that the crossed product of the tiling C*-algebra $\Aw$ by $G$ has real rank zero, tracial rank one, a unique trace, and that order on its K-theory is determined by the trace. We also show that the action of $G$ on $\Aw$ satisfies the weak Rokhlin property, and that it also satisfies the tracial Rokhlin property provided that $\Aw$ has tracial rank zero. In the course of proving the latter we show that $\Aw$ is finitely generated. We also provide a link between $\Aw$ and the AF algebra Connes associated to the Penrose tilings.