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We introduce the notion of continuous orbit equivalence for partial dynamicalsystems, and give an equivalent characterization in terms ofCartan-isomorphisms for partial C*-crossed products. Both graph C*-algebras andsemigroup C*-algebras can be described as C*-algebras attached to partialdynamical systems. As applications, for graphs, we generalize and explain aresult of Matsumoto and Matui relating orbit equivalence andCartan-isomorphism, and for semigroups, we strengthen several structuralresults for semigroup C*-algebras concerning amenability, nuclearity as well assimplicity of boundary quotients. We also discuss pure infiniteness for partialtransformation groupoids arising from graphs and semigroups.

Partial transformation groupoids attached to graphs and semigroups