C*-crossed products and shift spaces

We use Exel's C*-crossed products associated to non-invertible dynamical systems to associate a C*-algebra to arbitrary shift space. We show that this C*-algebra is canonically isomorphic to the C*-algebra associated to a shift space given by Carlsen [Cuntz-Pimsner C*-algebras associated with subshifts, Internal. J. Math. (2004) 28, to appear, available at arXiv: math. OA/0505503], has the C*-algebra defined by Carlsen and Matsumoto [Some remarks on the C*-algebras associated with subshifts, Math. Scand. 95 (1) (2004) 145-160] 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 and its relationship to other C*-algebras associated to shift spaces. We show that it can be viewed as a generalization of the universal Cuntz-Krieger algebra, discuss uniqueness and present a faithful representation, show that it is nuclear and satisfies the Universal Coefficient Theorem, provide conditions for it being simple and purely infinite, show that the constructed C*-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 present a description of the structure of gauge invariant ideals. (C) 2007 Elsevier GmbH. All rights reserved.