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: X 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: X ϱ: Δ → X that need not be continuous unless φ is a local homeomorphism. We define the crossed product C₀ (X) L C 0 (X) ⋊ L as a universal C^* C ∗ -algebra with explicit generators and relations, and give an explicit faithful representation of C₀ (X) L C 0 (X) ⋊ L under which it is generated by weighted composition operators. We explain its relationship with Exel–Royer’s crossed products, quiver C^* C ∗ -algebras of Muhly and Tomforde, C^* C ∗ -algebras associated to complex or self-similar dynamics by Kajiwara and Watatani, and groupoid C^* C ∗ -algebras associated to Deaconu–Renault groupoids. We describe spectra of core subalgebras of C₀ (X) L C 0 (X) ⋊ L, prove uniqueness theorems for C₀ (X) L C 0 (X) ⋊ L and characterize simplicity of C₀ (X) L C 0 (X) ⋊ L. We give efficient criteria for C₀ (X) L C 0 (X) ⋊ L to be purely infinite simple and in particular a Kirchberg algebra.