Algebraic structure of semigroup compactifications : Pym’s and Veech’s Theorems and strongly prime points
Filali, M.; Galindo, J. (2017-07-04)
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
© 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/.
https://creativecommons.org/licenses/by-nc-nd/4.0/
https://urn.fi/URN:NBN:fi-fe2019040811485
Tiivistelmä
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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