The ideal structure of algebraic partial crossed products

TL;DR

This paper investigates the ideal structure of algebraic partial crossed products arising from partial group actions on totally disconnected spaces, proving an algebraic version of the Effros-Hahn conjecture.

Contribution

It develops a theory of induced ideals and shows that all ideals are intersections of those induced from isotropy groups, confirming the algebraic Effros-Hahn conjecture.

Findings

Every ideal is an intersection of ideals induced from isotropy groups.

Established an algebraic version of the Effros-Hahn conjecture.

Provided a detailed structure of ideals in algebraic partial crossed products.

Abstract

Given a partial action of a discrete group $G$ on a Hausdorff, locally compact, totally disconnected topological space $X$, we consider the correponding partial action of $G$ on the algebra $L_c(X)$ consisting of all locally constant, compactly supported functions on $X$, taking values in a given field $K$. We then study the ideal structure of the algebraic partial crossed product $L_c(X)\rtimes G$. After developping a theory of induced ideals, we show that every ideal in $L_c(X)\rtimes G$ may be obtained as the intersection of ideals induced from isotropy groups, thus proving an algebraic version of the Effros-Hahn conjecture.