Restriction and extension of partial actions

Abstract

Given a partial action α=(Ag,αg)g∈G of a connected groupoid G on a ring A and an object x of G, the isotropy group G(x) acts partially on the ideal Ax of A by the restriction of α. In this paper we investigate the following reverse question: under which conditions a partial group action of G(x) on an ideal of A can be extended to a partial groupoid action of G on A? The globalization problem and some applications to the Morita and Galois theories are also considered.