Polish globalization of Polish group partial actions

Abstract

Abstract. Let X be a separable metrizable space. We establish a criteria for the existence of a metrizable globalization for a given continuous partial action of a separable metrizable group G on X. If G and X are Polish spaces, we show that the globalization is also a Polish space. We also show the existence of an universal globalization for partial actions of Polish groups.

Partial cohomology of groups

“…Proof. If n = 1, then the result follows from (29), (39), (44) and (60) and the fact that η g 1 (x) ∈ H. Let n > 1. Using the recursion whose base is (46), an intermediate step is (52) and the final step is (61), we have w(g −1 x 1 , .…”

“…Given a partial action it is natural to ask whether there exists a global action which restricts to the partial one. This question was first considered in the PhD Thesis [1] (see also [2]) and independently in [45] and [39] for partial group actions, with subsequent developments in [3,12,17,18,21,22,24,32,33,40,44]. More generally the problem was investigated for partial semigroup actions in [37,38,41,43], for partial groupoid actions in [10,11,36] and around partial Hopf (co)actions in [5,6,7,8,14,15,16].…”

Recent developments around partial actions

“…Furthermore, partial representations and cohomology based on partial actions turned out to be useful in dealing with the separation and intersection properties of ideals in global reduced C * -crossed products [206]. In addition, continuous partial actions of Polish groups on Polish spaces, and more generally, of separable metrizable groups on Hausdorff (in the majority of facts metrizable) spaces were studied in [181,182,262,263]. In particular, Effros' theorem [144] on orbits in Polish spaces under continuous actions of Polish groups is extended in [181] to the context of partial actions.…”

The open mapping principle for partial actions of Polish groups

“…They also show that there is a minimal globalization X G called the enveloping space of X. Partial actions of Polish groups on Polish spaces were studied in [11,22,23]. In particular, sufficient conditions for the space X G to be Polish were found (see Theorem 2.5 below).…”