Strongly and weakly almost periodic linear maps on semigroup algebras (Q2480760)

Let \(A\) be a Banach algebra with a bounded approximate identity bounded by 1 and let \(A^*\) be the dual of \(A\). For each \(u\in A\), define the seminorm \(\rho_u(f):=| \langle f,u\rangle| \), \(f\in A^*\). The topology defined on \(A^*\) by the seminorms \(\rho_u\), \(u\in A\), is denoted by \(\tau_c\). The \( (A^*, \tau_c)^*\)-topology of \(A^*\) is denoted by \(\tau_w\). Let \(S\) be a locally compact, Hausdorff topological semigroup with identity \(e\). Let \(M(S)\) be the space of all complex Borel measures on \(S\) and let \(M_a(S)\) be the set of all measures \(\mu\in M(S)\) for which both the mappings \(x\mapsto \delta_x*\mu\) and \(x\mapsto \mu*\delta_x\), \(x\in S\), are weakly continuous. A topological semigroup \(S\) is called a foundation semigroup if \(S\) coincides with the closure of \(\cup\{\text{supp} (\mu): \mu\in M_a(S)\}\). Note that \(M_a(S)\) is a closed two-sided ideal of \(M(S)\). Let \(S\) be a foundation locally compact topological semigroup. In the paper under review the author studies \(\tau_c\)- and \(\tau_w\)-almost periodic functionals in \(M_a(S)^*\) and compares them with each other and with the norm almost periodic and weakly almost periodic functionals. Finally if \(P(S)\) (the set of all probability measures in \(M_a(S)\)) has the semiright invariant isometry property, it is shown that the set of \(\tau_w\)-almost periodic functionals has a topological left invariant mean.