Pontryagin duality in the theory of topological vector spaces and in topological algebra
This long paper presents in a unified way the recent research of the author in the theory of stereotype topological vector spaces and algebras. Let \(X\) be a Hausdorff locally convex space over \(C\), and let \(X^{*}\) denote the dual of \(X\) endowed with the topology of uniform convergence on the precompact subsets of \(X\). The author defines \(X\) to be stereotype if the canonical map \(X\to X^{**}\) is a topological isomorphism. Accordingly, a space is stereotype if and only if it is polar reflexive in the sense of \textit{G.\ Köthe} [``Topological vector spaces I'' (Springer Verlag) (1969; Zbl 0179.17001)]. The class of stereotype spaces is rather large; it contains, in particular, all quasicomplete barrelled spaces. Stereotype spaces form a category (denoted by \(\mathcal{S}\)) with a number of useful properties; in particular, it is preabelian and it is monoidal with respect to a suitably defined tensor product functor. Stereotype spaces satisfy a reasonable completeness-type condition (each precompact set is relatively compact). The paper provides a systematic study of stereotype locally convex spaces, algebras, and modules. Sections 1-9 are devoted to stereotype spaces. The author introduces the operations of saturation, pseudosaturation, and pseudocompletion of a locally convex space, and studies their basic properties. He gives examples of stereotype spaces and studies relations between various properties of a stereotype space and its dual. Next he considers spaces of linear and bilinear mappings and introduces stereotype versions of projective and injective tensor products. The author discusses integration theory for functions with values in stereotype spaces, and applies this theory to tensor products of stereotype spaces consisting of functions, measures, and distributions. Section 9 is devoted to the stereotype approximation property. Homological characterizations of the stereotype approximation property are given. The results presented in these first nine sections are related to classical results: Grothendieck's precompactness lemma (see, e.g. [\textit{G.\ Köthe}, ``Topological vector spaces II'' (Grundlehren 237, Springer Verlag) (1979; Zbl 0417.46001)]), precompactness of sets of operators, which started with the important long article of \textit{L. Schwartz} [Ann. Inst. Fourier 7, 1-141 (1957; Zbl 0089.09601), ibid. 8, 1-209 (1958; Zbl 0089.09801)] and was continued by Ruess, Defant and Floret, and Buchwalter duality of tensor products is also relevant in connection with the author's discussions; it can be found in the books by Köthe or by Jarchow. The second part of the paper (Sections 10-16) is devoted to stereotype algebras and modules. Group algebras (i.e., algebras of measures and distributions on groups) and their representations are treated. Stereotype Hopf algebras and their duality are also discussed. Due to the self-duality of \(\mathcal{S}\), the two categories of stereotype Hopf algebras turn out to be anti-equivalent to one another via the duality functor. The categories of left and right stereotype modules over a stereotype algebra \(A\) are introduced and studied in Sections 11 and 12; they are shown to be preabelian, complete, and cocomplete. In Section 13, the author obtains some results on algebras of endomorphisms of stereotype modules. The next section is devoted to stereotype algebras with reflection. Examples of algebras with reflection include algebras of operators \(L(X)\) and algebras of measures and distributions on compact groups. For each algebra \(A\) with reflection, the author defines finite-rank morphisms between stereotype \(A\)-modules and introduces an \(A\)-module version of the approximation property. In the last two sections, the author develops absolute and relative homological algebra in the categories of stereotype modules. Using some specific properties of stereotype spaces, he obtains results that have no analogues in the classical homological theory for topological algebras developed by \textit{A. Ya. Helemskii} [``The homology of Banach and topological algebras'' (Kluwer, Dordrecht) (1989; Zbl 0695.46033)] and \textit{J. L. Taylor} [Adv. Math. 9, 137-182 (1972; Zbl 0271.46040)]. Reviewer's remark: The author writes in his introduction: ``The theory of topological vector spaces, being a foundation of modern functional analysis, is now considered as a completely mature, or, to be more specific, dead mathematical discipline. [...] The number of papers on the theory of topological vector spaces is declining, while the attitude of other mathematicians now consists of perplexity or lenient irony.'' A look at the list of references of Akbarov's paper reveals that he does not mention any publication about locally convex spaces after 1980, he misses some classical references, and in particular all the work since the mid-eighties about Fréchet spaces and their analytic applications developed by Bierstedt, Domanski, Meise, Taskinen, Vogt and others. In any case, the article under review constitutes an interesting addition to the theory of locally convex spaces, algebras and homological methods.
scientific article; zbMATH DE number 1981995