The extension of the operation in the Stone-Čech compactification of semigroups
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 \(s\in S\), then \(S\) has the right \(\beta\)-extension property if \(S\) is either discrete or countably compact. Furthermore, for the class of semigroups which are neither discrete nor countably compact, it has been shown that the quotient \({C_b(S)\over{\mathcal L}{\mathcal M}{\mathcal C}(S)}\) contains a linear isometric copy of \(l_\infty\).
scientific article; zbMATH DE number 5614950