Criteria for weak and strong continuity of representations of topological groups in Banach spaces

Several necessary and sufficient conditions for weak and strong continuity of representations of topological groups in Banach spaces are obtained. In particular, it is shown that a representation $S$ of a locally compact group $G$ in a Banach space is continuous in the strong (or, equivalently, in the weak) operator topology if and only if for some real number $q$, $0\leqslant q<1$, and each unit vector $\xi$ in the representation space of $S$ there exists a neighbourhood $U=U(\xi)\subset G$ of the identity element $e\in G$ such that $\|S(g)\xi-\xi\|\leqslant q$ for all $g\in U$. Versions of this criterion for other classes of groups (including not necessarily locally compact groups) and refinements for finite-dimensional representations are obtained; examples are discussed. Applications to the theory of quasirepresentations of topological groups are presented.