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REPRESENTATION THEOREMS FOR NORMED ALGEBRAS

Part of: Fairly general properties Topological algebras, normed rings and algebras, Banach algebras Commutative Banach algebras and commutative topological algebras Rings and algebras arising under various constructions Linear function spaces and their duals

Published online by Cambridge University Press:  17 June 2013

M. R. KOUSHESH
M. R. KOUSHESH*
Affiliation:
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156–83111, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), PO Box: 19395–5746, Tehran, Iran
*
[email protected]
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Abstract

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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–Čech 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 Lindelöf 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-$\mathscr{P}$, then $Y$ is nonnormal 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 Lindelöf property and $X$ is a locally-$\mathscr{P}$ metrizable space, we show that

$$\begin{eqnarray*}\dim C_{\mathscr{P}}(X)= \ell \mathop{(X)}\nolimits ^{{\aleph }_{0} } ,\end{eqnarray*}$$
where $\ell (X)$ is the Lindelöf number of $X$, and when $\mathscr{P}$ is countable compactness and $X$ is a normal space, we show that
$$\begin{eqnarray*}Y= {\mathrm{int} }_{\beta X} \upsilon X\end{eqnarray*}$$
 where $\upsilon X$ is the Hewitt realcompactification of $X$.

Keywords

Stone–Čech compactificationcommutative Gelfand–Naimark theoremreal Banach algebraGelfand theorycompact supportLindelöfHewitt realcompactificationcountably compact

MSC classification

Secondary: 46J10: Banach algebras of continuous functions, function algebras 46J25: Representations of commutative topological algebras 46E25: Rings and algebras of continuous, differentiable or analytic functions 46E15: Banach spaces of continuous, differentiable or analytic functions 54D35: Extensions of spaces (compactifications, supercompactifications, completions, etc.) 46H05: General theory of topological algebras 16S60: Rings of functions, subdirect products, sheaves of rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 95 , Issue 2 , October 2013 , pp. 201 - 222
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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