Criteria for weak and strong continuity of representations

A. I. Shtern

Sbornik. Mathematics, Tome 193 (2002) no. 9, pp. 1381-1396

Résumé

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 $q$, $0\leqslant q1$, 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.