Representations of certain normed algebras

Abstract

We show that for a normal locally-${\mathscr P}$ space $X$ (where ${\mathscr P}$ is a topological property subject to some mild requirements) the subset $C_{\mathscr P}(X)$ of $C_b(X)$ consisting of those elements whose support has a neighborhood with ${\mathscr P}$, is a subalgebra of $C_b(X)$ isometrically isomorphic to $C_c(Y)$ for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of the Stone--\v{C}ech compactification $\beta X$ of $X$ and contains $X$ as a dense subspace. Under certain conditions, $C_{\mathscr P}(X)$ coincides with the set of those elements of $C_b(X)$ whose support has ${\mathscr P}$, it moreover becomes a Banach algebra, and simultaneously, $Y$ satisfies $C_c(Y)=C_0(Y)$. This includes the cases when ${\mathscr P}$ is the Lindel\"{o}f property and $X$ is either a locally compact paracompact space or a locally-${\mathscr P}$ metrizable space. In either of the latter cases, if $X$ is non-${\mathscr P}$, $Y$ is non-normal, and $C_{\mathscr P}(X)$ fits properly between $C_0(X)$ and $C_b(X)$; even more, we can fit a chain of ideals of certain length between $C_0(X)$ and $C_b(X)$. The known construction of $Y$ enables us to derive a few further properties of either $C_{\mathscr P}(X)$ or $Y$. Specifically, when ${\mathscr P}$ is the Lindel\"{o}f property and $X$ is a locally-${\mathscr P}$ metrizable space, we show that \[\dim C_{\mathscr P}(X)=\ell(X)^{\aleph_0},\] where $\ell(X)$ is the Lindel\"{o}f number of $X$, and when ${\mathscr P}$ is countable compactness and $X$ is a normal space, we show that \[Y=\mathrm{int}_{\beta X}\upsilon X\] where $\upsilon X$ is the Hewitt realcompactification of $X$.