C*-Algebras Associated to Transfer Operators for Countable-to-One Maps

Our initial data is a transfer operator L for a continuous, countable-to-one map phi : Delta -> X defined on an open subset of a locally compact Hausdorff space X. Then L may be identified with a 'potential', i.e. a map rho : Delta -> X that need not be continuous unless phi is a local homeomorphism. We define the crossed product C-0(X) (sic) L as a universal C*-algebra with explicit generators and relations, and give an explicit faithful representation of C-0(X) (sic) L under which it is generated by weighted composition operators. We explain its relationship with Exel-Royer's crossed products, quiver C*-algebras of Muhly and Tomforde, C*-algebras associated to complex or self-similar dynamics by Kajiwara and Watatani, and groupoid C*-algebras associated to Deaconu-Renault groupoids. We describe spectra of core subalgebras of C-0(X) (sic) L, prove uniqueness theorems for C-0(X) (sic) L and characterize simplicity of C-0(X) (sic) L. We give efficient criteria for C-0(X) (sic) L to be purely infinite simple and in particular a Kirchberg algebra.