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On algebra spaces. (English) Zbl 1122.46034
Mallios, Anastasios (ed.) et al., Topological algebras and their applications. Proceedings of the 5th international conference, Athens, Greece, June 27–July 1, 2005. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-3868-6/pbk). Contemporary Mathematics 427, 263-283 (2007).
Summary: The present account constitutes a further elaboration of the idea that “geometry/space” is always the result of functions that characterize it, which is also in accordance with the point of view that “physical geometry” is the outcome of the physical laws. The same lies also at the root of a “relativistic quantization”, where functions are transformed into sections of appropriate (topological) algebra sheaves, corresponding to a “relativistic localization”, while the indispensable, in that context, “dynamics” is further accomplished, by employing an abstract form, à la Leibniz, viz. no “space”, at all(!), of the classical differential geometric machinery, that is, in terms of the so-called “abstract (alias, ‘modern’) differential geometry” (ADG). This enables one to exercise herewith, even a quite general context, as, for instance, topos theory, or what we may call topological algebra schemes, an “Einstein (topological) algebra space”, being, in effect, a particular instance, thereat, in terms thus of which one can formulate, for example, Einstein’s quantized equation (in vacuo).
46J25Representations of commutative topological algebras
46M20Methods of algebraic topology in functional analysis
18F20Categorical methods in sheaf theory
14F10Special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14A20Generalizations (algebraic spaces, stacks)
81Q70Differential-geometric methods in quantum mechanics
81T05Axiomatic quantum field theory; operator algebras
83C05Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C45Quantization of the gravitational field