Locally Compact Pro-C*-Algebras

Amini, Massoud

Canadian journal of mathematics, Tome 56 (2004) no. 1, pp. 3-22

Résumé

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.

Détail

DOI : 10.4153/CJM-2004-001-6

Mots-clés : 46L05, 46M40, pro-C* -algebras, projective limit, multipliers of Pedersen's ideal

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