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On localizing topological algebras. (English) Zbl 1094.46045
Arizmendi, Hugo (ed.) et al., Topological algebras and their applications. Proceedings of the 4th international conference, Oaxaca, Mexico, July 1--5, 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3556-4/pbk). Contemp. Math. 341, 79-95 (2004).
The author continues his previous study on geometric (alias, structure) topological algebras by means of the sheaf-theoretic notion of “functional localization”, in particular applied to the Gel’fand transform algebra $\hat{A}$ over the (non void) spectrum ${\mathcal M}(A)$ of a given topological algebra $A$ (i.e., another aspect of sheafification). Thus, $A$ is said to be localizable, if the so resulting presheaf of it is already a sheaf. The aforesaid algebras, being represented as the global section spaces of the above sheaves on their spectra, are now subject to specific cohomological conditions and seem to be of special bearing on various situations of a geometrical character. It is shown that in the case that they have compact spectra, they are closed under the formation of inductive limits. As an outcome, a formal notion of a “geometric algebra space” $(X,\cal A)$, consisting of a topological space $X$ and a ${\mathbb C}$-algebra sheaf $\cal A$ on $X$ satisfying the corresponding conditions, is proposed. Related results can also be found in [{\it A. Mallios} and {\it A. Oukhouya}, Sci. Math. Jpn. 61, No. 3, 391--396 (2005; Zbl 1080.46033)] and in the second author’s thesis. For the entire collection see [Zbl 1031.46002].

46M20Methods of algebraic topology in functional analysis
46H05General theory of topological algebras
18F20Categorical methods in sheaf theory
55N30Sheaf cohomology (algebraic topology)