M. Filali, J. Galindo, Algebraic structure of semigroup compactifications: Pym's and Veech's Theorems and strongly prime points, Journal of Mathematical Analysis and Applications, Volume 456, Issue 1, 2017, Pages 117-150, ISSN 0022-247X, https://doi.org/10.1016/j.jmaa.2017.06.038

### Algebraic structure of semigroup compactifications : Pym’s and Veech’s Theorems and strongly prime points

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Author: Filali, M.1; Galindo, J.2
Organizations: 1Department of Mathematical Sciences, University of Oulu, Oulu, Finland
2Instituto Universitario de Matemáticas y Aplicaciones (IMAC), Universidad Jaume I, E-12071, Castellón, Spain
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.5 MB)
Language: English
Published: Elsevier, 2017
Publish Date: 2019-07-04
Description:

# Abstract

The spectrum of an admissible subalgebra $$\mathscr{A}(G)$$ of $$LUC(G)(G)$$, the algebra of right uniformly continuous functions on a locally compact group $$G$$, constitutes a semigroup compactification $$G^{\mathscr{A}}$$ of $$G$$. In this paper we analyze the algebraic behaviour of those points of $$G^{\mathscr{A}}$$ that lie in the closure of $$\mathscr{A}(G)$$-sets, sets whose characteristic function can be approximated by functions in $$\mathscr{A}(G)$$. This analysis provides a common ground for far reaching generalizations of Veech’s property (the action of $$G$$ on $$G^{LUC(G)}$$ is free) and Pym’s Local Structure Theorem. This approach is linked to the concept of translation-compact set, recently developed by the authors, and leads to characterizations of strongly prime points in $$G^{\mathscr{A}}$$, points that do not belong to the closure of $$G^⁎G^⁎$$, where $$G^⁎ = G^{\mathscr{A}}\setminus G$$. All these results will be applied to show that, in many of the most important algebras, left invariant means of $$\mathscr{A}(G)$$ (when such means are present) are supported in the closure of $$G^⁎G^⁎$$.

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Series: Journal of mathematical analysis and applications
ISSN: 0022-247X
ISSN-E: 1096-0813
ISSN-L: 0022-247X
Volume: 456
Issue: 1
Pages: 117 - 150
DOI: 10.1016/j.jmaa.2017.06.038