Transformation semigroup compactifications and norm continuity of weakly almost periodic functions (Q1971655)

For a topological space \(T\), let \(\mathcal C(T)\) denote the \(C^{*}\)-algebra of all bounded continuous complex-valued functions on \(T\). In the sequel, \(S\) is a semigroup and a topological space and \(X\) is a topological space such that \(S\) acts on \(X\). For \(s\in S\), let \(L_s\) and \(R_s\) denote operators on \(\mathcal C(S)\) corresponding to left and right inner translations of \(s\). For \(s\in S\) and \(x\in X\), let \(\overline {L}_s\) and \(\overline {R}_x\) denote operators on \(\mathcal C(X)\) corresponding to the action of \(S\) on \(X\). We say that \((S,X)\) is semitopological if all inner translations and all actions of \(S\) on \(X\) are continuous. Let \(\mathcal W\mathcal A\mathcal P(X)\) denote the set of all \(f\in \mathcal C(X)\) such that \(\{\overline {L}_sf;s\in S\}\) is weak relatively compact in \(\mathcal C(X)\), let \(\mathcal L\mathcal C(X)\) denote the set of all \(f\in \mathcal C (X)\) such that the map \(s\mapsto\overline {L}_sf:S\to\mathcal C(X)\) is norm continuous, and let \(\mathcal R \mathcal C(X)\) denote the set of all \(f\in \mathcal C(X)\) such that the map \(x\mapsto\overline {R}_xf:X\to \mathcal C(S)\) is norm continuous. It is known that \(\mathcal L\mathcal C(X)\subseteq \mathcal W\mathcal A\mathcal P(X)\) (or \(\mathcal R\mathcal C (X)\subseteq \mathcal W\mathcal A\mathcal P(X)\)) whenever \(S\) (or \(X\)) is compact. For semitopological \((S,X)\) the following is proved: if \(S\) is topologically right simple and \(sX\) is a dense set in \(X\) for all \(s\in S\) then \(\mathcal W\mathcal A\mathcal P(X)\subseteq \mathcal L\mathcal C (X)\); if \(S\) is topologically left simple and \(Sx\) is a dense set in \(X\) for all \(x\in X\) then \(\mathcal W\mathcal A\mathcal P(X)\subseteq \mathcal R\mathcal C (X)\). Hence if \((S,X)\) is semitopological and \(S\) is a group then \(\mathcal W\mathcal A\mathcal P(X)\subseteq \mathcal L\mathcal C(X)\), and if, moreover, \( Sx=X\) for some \(x\in X\) then \(\mathcal W\mathcal A\mathcal P(X)\subseteq \mathcal L\mathcal C(X)\cap \mathcal R\mathcal C (X)\).