Representations of certain normed algebras

Abstract

We show that for a normal locally-$\mathcal{P}$ space $X$ (where $\mathcal{P}$ is a topological property subject to some mild requirements) the subset $C_{\mathcal{P}}(X)$ of $C_b(X)$ consisting of those elements whose support has a neighborhood with $\mathcal{P}$, 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--Čech compactification $\beta X$ of $X$ and contains $X$ as a dense subspace. Under certain conditions, $C_{\mathcal{P}}(X)$ coincides with the set of those elements of $C_b(X)$ whose support has $\mathcal{P}$, it moreover becomes a Banach algebra, and simultaneously, $Y$ satisfies $C_c(Y)=C_0(Y)$. This includes the cases when $\mathcal{P}$ is the Lindelöf property and $X$ is either a locally compact paracompact space or a locally-$\mathcal{P}$ metrizable space. In either of the latter cases, if $X$ is non-$\mathcal{P}$, $Y$ is non-normal, and $C_{\mathcal{P}}(X)$ 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 $C_{\mathcal{P}}(X)$ or $Y$. Specifically, when $\mathcal{P}$ is the Lindelöf property and $X$ is a locally-$\mathcal{P}$ metrizable space, we show that \[\dim C_{\mathscr P}(X)=\ell(X)^{\aleph_0},\] where $\ell (X)$ is the Lindelöf number of $X$, and when $\mathcal{P}$ is countable compactness and $X$ is a normal space, we show that \[Y=\mathrm{int}_{βX}\upsilon X\] where $\upsilon X$ is the Hewitt realcompactification of $X$.

19 pages

References in corpus (2)

• Connectedness modulo a topological property
• Continuous mappings with null support

Cited by in corpus (2)

• Continuous mappings with null support
• Representations of certain Banach algebras