Dimension of spaces of polynomials on abelian topological semigroups

2012/06/28 by Bolis Basit, Basit, Bolis, A. J. Pryde +1
Mathematics · #05A19 (Secondary) #11C08 #22B05 #39A99 (Primary) #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Dynamics and Fractals #math.FA #msc:05A19 #msc:11C08 #msc:22B05 #msc:39A99

paper · pdf · doi:10.48550/arxiv.1207.0410

9 pages

arxiv created 2012/06/28 · openalex publication_date 2012/06/28 · arxiv updated 2012/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study (continuous) polynomials p: J→ X, where J is an abelian topological semigroup and X is a topological vector space. If J is a subsemigroup with non-empty interior of a locally compact abelian group G and G=J-J, then every polynomial p on J extends uniquely to a polynomial on G. It is of particular interest to know when the spaces Pⁿ (J,X) of polynomials of order at most n are finite dimensional. For example we show that for some semigroups the subspace Pⁿ_R (J,C) of Riss polynomials (those generated by a finite number of homomorphisms α: J→ R) is properly contained in Pⁿ (G,C). However, if P¹ (J,C) is finite dimensional then Pⁿ_R (J,C)= Pⁿ (J,C). Finally we exhibit a large family of groups for which Pⁿ (G,C) is finite dimensional.

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