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Locally Compact Pro-C*-Algebras

Published online by Cambridge University Press:  20 November 2018

Massoud Amini
Massoud Amini*
Affiliation:
Department of Mathematics, Tarbiat Modarres University, P.O. Box 14115-175, Tehran, Iran e-mail: [email protected]
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Abstract

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Let $X$ be a locally compact non-compact Hausdorff topological space. Consider the algebras $C\left( X \right),{{C}_{b}}\left( X \right),{{C}_{0}}\left( X \right),\,\text{and}\,{{C}_{00}}\left( X \right)$ of respectively arbitrary, bounded, vanishing at infinity, and compactly supported continuous functions on $X$. Of these, the second and third are ${{C}^{*}}$-algebras, the fourth is a normed algebra, whereas the first is only a topological algebra (it is indeed a pro-${{C}^{*}}$- algebra). The interesting fact about these algebras is that if one of them is given, the others can be obtained using functional analysis tools. For instance, given the ${{C}^{*}}$-algebra ${{C}_{0}}\left( X \right)$, one can get the other three algebras by ${{C}_{00}}\left( X \right)=K\left( {{C}_{0}}\left( X \right) \right),{{C}_{b}}\left( X \right)=M\left( {{C}_{0}}\left( X \right) \right),C\left( X \right)=\Gamma \left( K\left( {{C}_{0}}\left( X \right) \right) \right)$, where the right hand sides are the Pedersen ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of ${{C}_{0}}\left( X \right)$, respectively. In this article we consider the possibility of these transitions for general ${{C}^{*}}$-algebras. The difficult part is to start with a pro-${{C}^{*}}$-algebra A and to construct a ${{C}^{*}}$-algebra ${{A}_{0}}$ such that $A=\Gamma \left( K\left( {{A}_{0}} \right) \right)$. The pro-${{C}^{*}}$-algebras for which this is possible are called locally compact and we have characterized them using a concept similar to that of an approximate identity.

Keywords

46L0546M40pro-C* -algebrasprojective limitmultipliers of Pedersen's ideal
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 1 , 01 February 2004 , pp. 3 - 22
Copyright
Copyright © Canadian Mathematical Society 2004

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