On localizing topological algebras

Details On localizing topological algebras On localizing topological algebras

2002-11-11

arxiv.org/abs/gr-qc/0211032v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

23 pages

Scientific paper

Through the subsequent discussion we consider a certain particular sort of (topological) algebras, which may substitute the `` structure sheaf algebras'' in many--in point of fact, in all--the situations of a geometrical character that occur, thus far, in several mathematical disciplines, as for instance, differential and/or algebraic geometry, complex (geometric) analysis etc. It is proved that at the basis of this type of algebras lies the sheaf-theoretic notion of (functional) localization, which, in the particular case of a given topological algebra, refers to the respective ``Gel'fand transform algebra'' over the spectrum of the initial algebra. As a result, one further considers the so-called ``geometric topological algebras'', having special cohomological properties, in terms of their ``Gel'fand sheaves'', being also of a particular significance for (abstract) differential-geometric applications; yet, the same class of algebras is still ``closed'', with respect to appropriate inductive limits, a fact which thus considerably broadens the sort of the topological algebras involved, hence, as we shall see, their potential applications as well.

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