Partial crossed products as equivalence relation algebras

TL;DR

This paper characterizes ideals in partial skew group rings by representing them as algebras of functions over equivalence relations, generalizing known C*-algebra results to a purely algebraic context.

Contribution

It provides a new algebraic realization of partial skew group rings as function algebras over equivalence relations, extending the groupoid C*-algebra characterization.

Findings

Realization of partial skew group rings as function algebras over equivalence relations

Characterization of ideals in partial skew group rings

Extension of C*-algebra results to algebraic setting

Abstract

For a free partial action of a group in a set we realize the associated partial skew group ring as an algebra of functions with finite support over an equivalence relation and we use this result to characterize the ideals in the partial skew group ring. This generalizes, to the purely algebraic setting, the known characterization of partial C*-crossed products as groupoid C*-algebras. For completeness we include a new proof of the C* result for free partial actions.