The extension of the operation in the Stone-Čech compactification of semigroups
type: Item
instance of: scholarly article
title: The extension of the operation in the Stone-Čech compactification of semigroups
zbMATH Open document ID: 1182.22002
author: M. Filali
author: Tero Vedenjuoksu
published in: Semigroup Forum
publication date: 2009-10-14T00:00:00Z
review text: The Stone-Čech compactification \(\beta S\) of an infinite discrete semigroup \(S\) is a semigroup compactification of \(S\) in the following sense: \(\beta S\) is a compact right (or left) topological semigroup which contains a dense homomorphic image of \(S\) and has the additional property that \(x\to sx:\beta S\to\beta S\) is continuous for every \(s\in S\). With \(C_b(S)\) -- the \(C^*\)-algebra of all complex-valued continuous functions on \(S\), \(C_b(S)^*\) -- the Banach dual of \(C_b(S)\), \(L_sf= f\circ l_s\), \(R_s f= f\circ r_s\) for \(f\in C_b(S)\) and \(s\in S[l_x: s\to x.s: S\to\beta S\), \(r_x: x\to x.s: S\to\beta S]\), \(f^x(s)= x(L_s f)\) and \(f_x(s)= x(R_sf)\) and finally, \({\mathcal L}{\mathcal M}{\mathcal C}(S)\) -- the right multiplicatively continuous functions defined similarly, using \(f_x\) instead -- the authors have defined, for a semitopological semigroup \(S\), the following: \(S\) is said to have the left (resp. right) \(\beta\)-extension property if \({\mathcal L}{\mathcal M}{\mathcal C}(S)= C_b(S)\) (resp. \({\mathcal R}{\mathcal M}{\mathcal C}(S)= C_b(S)\)); if a semigroup \(S\) has both the left and the right \(\beta\)-extension property, then \(S\) is said to have the \(\beta\)-extension property. The main results are the following: (i) Let \(S\) be a closed subsemigroup of a locally compact group \(G\). Then \(S\) has the left or the right \(\beta\)-extension property iff it is either discrete or compact. (ii) For a locally compact semitopological semigroup \(S\) one has: (a) If \(S\) is right cancellative, \(K^{-1}K\) is relatively compact for every compact subset \(K\) of \(S\) and \(U_s\) is open for every open subset \(U\) of \(S\) and \(s\in S\), then \(S\) has the left \(\beta\)-extension property iff \(S\) is either discrete or countably compact. (b) If \(S\) is left cancellative, \(KK^{-1}\) is relatively compact for every compact subset \(K\) of \(S\) and \(sU\) is open for every open subset \(U\) of \(S\) and \(s\in S…
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 54D35
zbMATH DE Number: 5614950
zbMATH Keywords: Stone-Čech compactification
zbMATH Keywords: multiplicatively continuous function
zbMATH Keywords: semigroup compactification
zbMATH Keywords: semitopological semigroup
zbMATH Keywords: right topological semigroup
zbMATH Keywords: extension
reviewed by: Sobhakar Ganguly
MaRDI profile type: Publication
full work available at URL: s00233-009-9171-z
OpenAlex ID: W1987740212
cites work: The Stone-Čech compactification of a topological semigroup
cites work: Q3995290
cites work: On the size of quotients of function spaces on a topological group
cites work: Nonseparability of Quotient Spaces of Function Algebras on Topological Semigroups
cites work: Q3083042
cites work: Function spaces on semitopological semigroups
DOI: 10.1007/S00233-009-9171-Z
description: scientific article; zbMATH DE number 5614950