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

Abstract

AbstractOur initial data is a transfer operator L for a continuous, countable-to-one map $$\varphi :\Delta \rightarrow X$$ 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 $$\varrho :\Delta \rightarrow X$$ ϱ : Δ → X that need not be continuous unless $$\varphi $$ φ is a local homeomorphism. We define the crossed product $$C_0(X)\rtimes L$$ C 0 ( X ) ⋊ L as a universal $$C^*$$ C ∗ -algebra with explicit generators and relations, and give an explicit faithful representation of $$C_0(X)\rtimes L$$ 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 $$C_0(X)\rtimes L$$ C 0 ( X ) ⋊ L , prove uniqueness theorems for $$C_0(X)\rtimes L$$ C 0 ( X ) ⋊ L and characterize simplicity of $$C_0(X)\rtimes L$$ C 0 ( X ) ⋊ L . We give efficient criteria for $$C_0(X)\rtimes L$$ C 0 ( X ) ⋊ L to be purely infinite simple and in particular a Kirchberg algebra.