The structure of locally bounded finite-dimensional representations of connected locally compact groups

Abstract

An analogue of a Lie theorem is obtained for (not necessarily continuous) finite-dimensional representations of soluble finite-dimensional locally compact groups with connected quotient group by the centre. As a corollary, the following automatic continuity proposition is obtained for locally bounded finite-dimensional representations of connected locally compact groups: if is a connected locally compact group, is a compact normal subgroup of such that the quotient group is a Lie group, is the connected identity component in , is the family of elements of commuting with every element of , and is a (not necessarily continuous) locally bounded finite-dimensional representation of , then is continuous on the commutator subgroup of (in the intrinsic topology of the smallest analytic subgroup of containing this commutator subgroup). Bibliography: 23 titles.