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

Abstract

Our initial data is a transfer operator L for a continuous, countable-to-one map φ:Δ→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 ϱ:Δ→X that need not be continuous unless φ is a local homeomorphism. We define the crossed product C0(X)⋊L as a universal C∗-algebra with explicit generators and relations, and give an explicit faithful representation of C0(X)⋊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 C0(X)⋊L, prove uniqueness theorems for C0(X)⋊L and characterize simplicity of C0(X)⋊L. We give efficient criteria for C0(X)⋊L to be purely infinite simple and in particular a Kirchberg algebra.