On bornologicalness in locally convex algebras
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ORIGINAL ARTICLE
On bornologicalness in locally convex algebras Marina Haralampidou1 • Mohamed Oudadess2 • Lourdes Palacios3 Carlos Signoret3
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Received: 28 November 2019 / Accepted: 30 March 2020 Ó Sociedad Matemática Mexicana 2020
Abstract We give a synthetic presentation of several bornologicalness notions in locally convex algebras. Some of these notions are new. As in locally convex spaces, characterizations are given via (algebra) seminorms. A classification and examples are provided. Keywords Bornology Bornological space i-Bornological algebra
Mathematics Subject Classification Primary 46H05 46H20 Secondary 46J10
1 Introduction This is a presentation of six classes of locally convex algebras defined in terms of bornology. We introduce these notions and we give an overview and a classification of them. They are identified and characterized via the continuity of appropriate To the memory of our good friend and coauthor Mohamed Oudadess. & Lourdes Palacios [email protected] Marina Haralampidou [email protected] Mohamed Oudadess [email protected] Carlos Signoret [email protected] 1
Department of Mathematics, University of Athens Panepistimioupolis, 15784 Athens, Greece
2
E´cole Normale Supe´rieure, Takadoum, BP 5118, 1000 Rabat, Morocco
3
Universidad Auto´noma Metropolitana Iztapalapa, 09340 Mexico City, Mexico
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algebra seminorms. Notions are given, all together, in Definition 1. Two of these are known, the usual bornologicalness for a locally convex space and S. Warner’s i-bornologicity (Proposition 1). Two are new ones and they are strongly related to bornologicalness, and the other two express conditions on the locally convex topology. Characterizations are given in Propositions 3–6. Plenty of examples and counter-examples are displayed in Examples 1–13 in Sect. 5. Section 6 is devoted to provide the properties of locally A-convex algebras in terms of bornologicalness. In Sect. 7, some conditions are given for a pseudo-Banach algebra to be of a certain type concerning bornology (Proposition 11). A diagram is given to illustrate relations between the classes of bornological algebras involved. This manuscript is the analog of [6] dealing with barrelledness.
2 Preliminaries Let ðE; sÞ be a locally convex space. The bounded structure (bornology) of ðE; sÞ, denoted by Bs, is the collection of all subsets B of E which are bounded in the sense of Kolmogorov–von Neumann, namely B is absorbed by every neighborhood of zero. Concerning bornological notions, we refer to [7]. A disc in a locally convex space E is a balanced and convex subset of E. A disc is said to be bornivorous if it absorbs every bounded subset of E. We remind that a locally convex space E is called bornological, if every bornivorous disc is a neighborhood of zero (see [18, p. 61]). A set S in an algebra is multiplicative if S S S. The term idempotent set is also used instead of multiplicative set. A topological algebra is an associative algebra E endowed with a topological vector space topology
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