Maximal closed ideals of the Colombeau Algebra of Generalized functions
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Maximal closed ideals of the Colombeau Algebra of Generalized functions A. Khelif1 · D. Scarpalezos1 Received: 25 March 2019 / Accepted: 15 October 2020 © The Author(s) 2020
Abstract In this paper we investigate the structure of the set of maximal ideals of G(). The method of investigation passes through the use of the m−reduction and the ideas are analoguous to those in Gillman and Jerison (Rings of Continuous Functions, N.J. Van Nostrand, Princeton, 1960) for the investigation of maximal ideals of continuous functions on a Hausdorff space K . Keywords Ideal · Net · Ultrafilter · Functions · Distributions · Algebra Mathematics Subject Classification Primary 54C40 · 46F30; Secondary 46E10 · 46E25
Introduction The algebra of Colombeau generalized functions has been used for non linear P.D.E. (see for example [4,6–8,22,24]) as well as for linear P.D.E. with irregular coefficients [21]. Much work has been done on this field from the point view of Analysis. However the Algebraic properties of this Algebra are still a largely open field of investigation. In the case of continuous functions (with complex or real values over a compact set K it is known that the set of points are in one-one correspondence with the set of maximal ideals. While in the more general case of the algebra of continuous functions on a Hausdorff regular space A, the set of maximal ideals is in a one to one correspondence ˇ with the Cech-compactification (β(A)) (see [16]). It is known that generalized functions G() have a double structure: (a) They constitute a sheaf of algebras (see [4,6]) and
Communicated by Adrian Constantin.
B 1
A. Khelif [email protected] Institut de mathematiques de Jussieu Paris Rive Gauche, Paris, France
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A. Khelif, D. Scarpalezos
(b) They are continuous mappings of c (the set of compactly supported generalized points) into the ring K of generalized constants. (This result follows immediately from definitions.) This leads to the notion of ‘localisation’ of ideals, the notion of ‘support’ of an ideal and to the notion of generalized trace of ideals (see [2]). Trying to obtain a better knowledge of prime and maximal ideals of G() we make an extensive use of the results concerning the algebraic and topological properties of the ring of generalized constants [3]. In order to use ideas analogous to those in [16], we first use the m−reduction procedure [27] which gives a canonical surjective mapping of G() onto Gm (), but Gm () is a topological algebra on the field of m−reduced generalized constants Km = K/m where m is maximal ideal of K. We now consider a very large class of maximal ideals called regular maximal ideals. The set of regular maximal ideals is proved to contain all closed maximal ideals (closed for the natural Hausdorff topology of G()). Our main result is the following: the set of regular maximal ideals is in one to one correspondence with (m, d) where m is a maximal ideal of K and d ∈ γ (m,c ) where γ (m,c ) is a special compactification of m,c called the g−compactification of
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