The Banach algebra of continuous bounded functions with separable support (Q2919696)

For a completely regular space \(X\), let \(C_s(X)\) be the set of all real-valued or complex-valued bounded and continuous functions on \(X\) having separable support. In this nice paper, it is proved as a main result (Theorem 2.10) that, if \(X\) is a locally separable metrizable space, then \(C_s(X)\) (endowed with the supremum norm) is a Banach algebra isometrically isomorphic to the Banach algebra \(C_0(Y)\) of functions vanishing at infinity, for some unique (up to homeomorphism) locally compact space \(Y\). Moreover, this space \(Y\) can be constructed as a subspace of the Stone-Čech compactification of \(X,\) and it is shown to be countably compact and \(\infty\)-compact. Recall that a topological space \(Y\) is said to be \(\infty\)-compact whenever \(C_0(Y)=C_{00}(Y)\), that is, when every function vanishing at infinity has compact support. We refer to \textit{A. R. Aliabad} et al. [Commentat. Math. Univ. Carol. 45, No. 3, 519--533 (2004; Zbl 1097.54021)], where this notion of compactness was introduced.NEWLINENEWLINEThe paper under review also contains another description of the above mentioned space \(Y\) as the spectrum of the \(C^*\)-algebra \(C_s(X)\), in the complex case (Theorem 4.1).NEWLINENEWLINEReviewer's remark. Note that the hypothesis of metrizability of the space \(X\) is only needed when the representation theorem of Alexandroff is considered. Hence, we can assert that all the results contained in this paper remain true for those spaces (not necessarily metrizable) for which such a representation exists; for instance, for \(X\) being any uncountable disjoint union of Sorgenfrey lines.