Our initial data is a transfer operator L for a continuous, countable-to-one map \(\varphi :\Delta \rightarrow X\) defined on an open subset of a locally compact Hausdorff space X. Then L may be identified with a ‘potential’, i.e. a map \(\varrho :\Delta \rightarrow X\) that need not be continuous unless \(\varphi \) is a local homeomorphism. We define the crossed product \(C_0(X)\rtimes L\) as a universal \(C^*\) -algebra with explicit generators and relations, and give an explicit faithful representation of \(C_0(X)\rtimes L\) 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 \(C_0(X)\rtimes L\) , prove uniqueness theorems for \(C_0(X)\rtimes L\) and characterize simplicity of \(C_0(X)\rtimes L\) . We give efficient criteria for \(C_0(X)\rtimes L\) to be purely infinite simple and in particular a Kirchberg algebra.
