  For a topological group G the intersection KO(G) of all kernels of ordinary
representations is studied. We show that KO(G) is contained in the center of G
if G is a connected pro-Lie group. The class KO(C) is determined explicitly if
C is the class ConnLie of connected Lie groups or the class almConnLie 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 KO(G) for some connected pro-Lie group G.
However, the dimension of KO(G) is bounded by the cardinality of the continuum
if G is locally compact and connected. Examples are given to show that KO(C)
becomes complicated if C contains groups with infinitely many connected
components.
