C*-crossed products and 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.

29 pages

References in corpus (4)

• A construction of $C^{\ast }$-algebras from $C^{\ast }$-correspondences
• Compactly supported wavelets and representations of the Cuntz relations
• On C*-algebras Associated with Sofic Shifts
• Symbolic Dynamics, Partial Dynamical Systems, Boolean Algebras and C*-algebras Generated by Partial Isometries

Cited by in corpus (6)

• Cuntz-Pimsner C*-algebras associated with subshifts
• $C^{\ast }$-algebras associated to Boolean dynamical systems
• $C^{\ast }$-algebras of labelled graphs III - $K$-theory computations
• On the Exel crossed product of topological covering maps
• C*-algebras, groupoids and covers of shift spaces
• Orbit Representations from Linear mod 1 Transformations