Linearization of proper actions of locally compact groups on Tychonoff spaces (Q409666)

This paper continues the work in [\textit{N. Antonyan, S. Antonyan} and \textit{L. Rodríguez-Medina}, Topology Appl. 156, No. 11, 1946--1956 (2009; Zbl 1171.54015)]. Its purpose is to solve the linearization problem for proper actions of locally compact groups on Tychonoff spaces. An action of a locally compact group \(G\) on a Tychonoff space \(X\) is called proper if each \(x\in X\) has a neighborhood \(V_x\) such that any \(y\in X\) has a neighborhood \(V_y\) with the property that \(\{g\in G\,|\,gV_x\cap V_y\neq\emptyset\}\) has compact closure in \(G\). In this case we call \(X\) a proper \(G\)-space.NEWLINENEWLINEThe main result is:NEWLINENEWLINE\textbf{Theorem 1.1.} Let \(G\) be a locally compact group. Then for each proper \(G\)-space \(X\), there exist a linear \(G\)-space \(L\) and a \(G\)-embedding \(X\hookrightarrow L\setminus\{0\}\) such that \(L\setminus\{0\}\) is a proper \(G\)-space. Moreover, \(L\) is the product \(\prod_{j\in\mathcal{J}}L_j\) of normed linear \(G\)-spaces \(L_j\) such that the complement \(L_j\setminus\{0\}\) is a proper \(G\)-space.NEWLINENEWLINEBy a ``normed linear \(G\)-space'' is meant a \(G\)-space which is a normed linear space on which \(G\) acts continuously by means of linear isometries.