C*C*-crossed products and shift spaces

We use Exel's C*C*-crossed products associated to non-invertible dynamical systems to associate a C*C*-algebra to arbitrary shift space. We show that this C*C*-algebra is canonically isomorphic to the C*C*-algebra associated to a shift space given by Carlsen [Cuntz–Pimsner C*C*-algebras associated with subshifts, Internat. J. Math. (2004) 28, to appear, available at arXiv:math.OA/0505503], has the C*C*-algebra defined by Carlsen and Matsumoto [Some remarks on the C*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*C*-algebra associated to a shift space.We also consider its representations and its relationship to other C*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*C*-algebras and thus their KK-theory, K0K0 and K1K1, are conjugacy invariants of one-sided shift spaces, present formulas for those invariants, and present a description of the structure of gauge invariant ideals.