Generalization of Segal algebras for arbitrary topological algebras
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Generalization of Segal algebras for arbitrary topological algebras Mart Abel1,2
© Akadémiai Kiadó, Budapest, Hungary 2017
Abstract In the present paper we introduce the definition of a topological Segal algebra, which generalizes most of the earlier known definitions for Segal algebras. We also generalize some results about Segal algebras and algebras of continuous functions vanishing at infinity for the case of topological Segal algebras. Keywords Segal algebra · Dense ideal · Algebra of continuous functions vanishing at infinity
1 Introduction The notion of a Segal algebra is already more than 50 years old (see [8]). Although the notion of a Segal algebra was first introduced in the context of subalgebras of L 1 (G), where G was a locally compact group, it was soon generalized to the case of subalgebras of arbitrary Banach algebras. The algebra was called Segal algebra by Reiter in honour of Irwin Ezra Segal, who had written down in [9] a set of axioms (called Segal’s axioms) which was used later in the definition. Segal himself tried to describe a general algebraic structure underlying Wiener’s
The research was supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research. The author would like to thank the referee for many valuable remarks and suggestions. Many thanks also to the participants of the seminar on topological algebras held at the University of Tartu during the academic year 2016/2017 for posing the right questions which helped to improve the contents of this paper.
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Mart Abel [email protected]; [email protected]
1
Institute of Digital Technologies, Tallinn University, 29 Narva Str., Room A-416, 10120 Tallinn, Estonia
2
Institute of Mathematics, University of Tartu, 2 Liivi Str., Room 615, 50409 Tartu, Estonia
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M. Abel
algebra, but he did not exploit this algebraic structure in systematic fashion. About the history of Segal algebras, one could also consult [3]. The usual context of Segal algebras during the last 20 years is the setting of Banach algebras. Recently, there have been some generalizations to the case of locally multiplicatively convex (shortly, lmc) Frechet ´ algebras by Abtahi et al. (see [1] and [2]), who named their generalization Segal Fré chet algebra, as well as to the case of complete lmc algebras by Yousofzadeh (see [10] and [11]). Actually one could take a much wider approach by considering any topological algebra (over R or C) instead of limiting the study to Banach or lmc (Frechet ´ or just complete) algebras. The present paper gives a way to generalize the notion of Segal algebra to the case of arbitrary (real or complex) topological algebras. In order to show that this context is meaningful, we will prove some results generalizing results from [4] and [1] about the ideals in C ∗ -Segal algebras or Segal Frechet ´ algebras and from [6] about the algebra of (Banach) Segal algebra valued continuous functions on a Hausdorff locally compact space. Interested readers can find many results about (Banach) Segal algebras, C ∗
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