Kernels of linear representations of Lie groups, locally compact groups, and pro-Lie Groups

Abstract

Abstract. For a topological group the intersection of all kernels of ordinary representations is studied. We show that is contained in the center of if is a connected pro-Lie group. The class is determined explicitly if is the class of connected Lie groups or the class of almost-connected Lie groups: in both cases, it consists of all compactly-generated abelian Lie groups. Every compact abelian group and every connected abelian pro-Lie group occurs as for some connected pro-Lie group . However, the dimension of is bounded by the cardinality of the continuum if is locally compact and connected. Examples are given to show that becomes complicated if contains groups with infinitely many connected components.