C*-crossed products and shift spaces

TL;DR

This paper constructs a universal C*-algebra for shift spaces using Exel's method, exploring its properties, representations, and invariants, and establishing its role as a generalization of known algebras in symbolic dynamics.

Contribution

It introduces a new C*-algebra associated with shift spaces, generalizing the universal Cuntz-Krieger algebra, and analyzes its properties, invariants, and relation to existing algebras.

Findings

The algebra is a quotient of a previously defined C*-algebra.

It is a universal C*-algebra for shift spaces.

K-theory is a conjugacy invariant of shift spaces.

Abstract

In this article, we use Exel's construction to associate a C*-algebra to every shift space. We show that it has the C*-algebra defined in [Carlsen and Matsumoto: Some remarks on the C*-algebras associated with subshifts] as a quotient, and possesses properties indicating that it can be thought of as the universal C*-algebra associated to a shift space. We also consider its representations, relationship to other C*-algebras associated to shift spaces, show that it can be viewed as a generalization of the universal Cuntz-Krieger algebra, discuss uniqueness and a faithful representation, provide conditions for it being nuclear, for satisfying the UCT, for being simple, and for being purely infinite, show that the constructed algebras and thus their K-theory, $K_0$ and $K_1$, are conjugacy invariants of one-sided shift spaces, present formulas for those invariants, and also present a description of the structure of gauge invariant ideals.