The three space problem for locally pseudoconvex algebras
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The three space problem for locally pseudoconvex algebras Mati Abel1 · Reyna María Pérez-Tiscareño1 Received: 9 April 2019 / Accepted: 31 October 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract The three space problem for locally pseudoconvex algebras is considered in this paper. It is shown that a locally pseudoconvex algebra E, with jointly continuous multiplication, is locally m-pseudoconvex, if E contains a two-sided ideal I such that I (in the subset topology) and E/I (in the quotient topology) are locally m-pseudoconvex algebras. Keywords Topological algebras · Locally pseudoconvex algebras · Locally m-pseudoconvex algebras · Three space problem Mathematics Subject Classification Primary 46H05; Secondary 46H20
1 Introduction Let E be a topological algebra . The property P is called a three space property for topological algebras if whenever I is a two-sided ideal in E and I (with the subset topology) and E/I (with the quotient topology) have the property P , then E has the property P too. The problem of determine if a property is a three space property is known as three space problem. It was shown in [6, p. 80, Proposition 6.14] that the property ”to be a Q-algebra” (that is, the set of quasi-invertible elements in the topological algebra is open) is a three space property for topological algebras and in [4] that the property, ”to be locally m-convex” is a three space property for locally convex algebras with jointly continuous multiplication. Moreover, the three space problem for Banach algebras has been studied in [10]. Some results about the three space problem for topological linear spaces (in this case, instead of ideals, are considered linear subspaces) have been considered in [9] and, in particular cases, in [7] for locally bounded F-spaces and in [3] and [5] for Banach spaces. Moreover, the three space problem for topological groups is considered in [2] and for commutative algebras in [1].
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Reyna María Pérez-Tiscareño [email protected] Mati Abel [email protected]
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Institute of Mathematics and Statistics, University of Tartu, 2 J. Liivi Str., room 615, 50409 Tartu, Estonia
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M. Abel, R. M. Pérez-Tiscareño
The three space problem for locally pseudoconvex algebras is considered in the present paper. It is showed that any locally pseudoconvex algebra E with jointly continuous multiplication is locally m-pseudoconvex algebra if it contains a two-sided ideal such that I is locally m-pseudoconvex in the subset topology and E/I is locally m-pseudoconvex in the quotient topology.
2 Preliminaries An algebra, E, over K, the field of the real numbers R or complex numbers C, is called topological algebra, if E is equipped with a topology such that E is a topological linear space with a separately continuous multiplication (it is for each a ∈ E, the maps la , ra : E → E, la (x) = ax, ra (x) = xa are continuous). If the map (a, b) → ab from E × E into E is continuous, then it is said that the multiplication is jointly continuous in E. If the underlying topological linear space of
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