Purely infinite simple C*-algebras associated to integer dilation matrices

TL;DR

This paper investigates the structure of certain C*-algebras derived from integer matrices with eigenvalues greater than one, demonstrating their simplicity, pure infiniteness, and computing their K-theory groups.

Contribution

It introduces a new class of purely infinite simple C*-algebras associated with integer dilation matrices and analyzes their properties and K-theory.

Findings

The associated crossed-product C*-algebra is simple and purely infinite.

K-theory groups of these algebras are explicitly computed.

The construction links matrix dilation properties to operator algebra structure.

Abstract

Given an n x n integer matrix A whose eigenvalues are strictly greater than 1 in absolute value, let \sigma_A be the transformation of the n-torus T^n=R^n/Z^n defined by \sigma_A(e^{2\pi ix})=e^{2\pi iAx} for x\in R^n. We study the associated crossed-product C*-algebra, which is defined using a certain transfer operator for \sigma_A, proving it to be simple and purely infinite and computing its K-theory groups.