A new class of semigroups of bounded linear operators on \(C_b(E)\), the Banach space of all real, uniformly continuous and bounded functions with the supremum norm on a separable metric space \(E\), is studied. These semigroups are called \(\pi\)-semigroups and they are characterized by the following assumptions:
(i) for any \(f\in C_b(E)\) and \(x\in e\), the map \([0,\infty[\to \mathbb{R}\), \(t\mapsto P_t f(x)\), is continuous;
(ii) for any bounded sequence \((f_n)\subset C_b(E)\) such that \(f_n\) converges pointwise to \(f\in C_b(E)\) (we briefly write \(f_n@>\pi>> f\)), we have \(P_t f_n@>\pi>> P_tf\), \(t\geq 0\);
(iii) there exist \(M\geq 1\) and \(\omega\geq 0\) such that \(\|P_tf\|\leq Me^{\omega t}\|f\|\), \(f\in C_b(E)\), \(t\geq 0\).
\(\pi\)-semigroups are not strongly continuous in general.
Results of this paper allow to characterize the generators of Markov transition semigroups in infinite dimensions such as the heat and the Ornstein-Uhlenbeck semigroups.