The von Neumann kernel and minimally almost periodic groups

Abstract

We calculate the von Neumann kernel n ( G ) n(G) of an arbitrary connected Lie group. As a consequence we see that the closed characteristic subgroup n ( G ) n(G) is also connected. It is shown that any Levi factor of a connected Lie group is closed. Then, various characterizations of minimal almost periodicity for a connected Lie group are given. Among them is the following. A connected Lie group G with radical R is minimally almost periodic (m.a.p.) if and only if G / R G/R is semisimple without compact factors and G = [ G , G ] − G\, = \,{[G,\,G]^ - } . In the special case where R is also simply connected it is proven that G = [ G , G ] G\, = \,[G,\,G] . This has the corollary that if the radical of a connected m.a.p. Lie group is simply connected then it is nilpotent. Next we prove that a connected m.a.p. Lie group has no nontrivial automorphisms of bounded displacement. As a consequence, if G is a m.a.p. connected Lie group, H is a closed subgroup of G such that G / H G/H has finite volume, and α \alpha is an automorphism of G with disp ( α , H ) {\text {disp}}(\alpha ,\,H) bounded, then α \alpha is trivial. Using projective limits of Lie groups we extend most of our results on the characterization of m.a.p. connected Lie groups to arbitrary locally compact connected topological groups, and finally get a new and relatively simple proof of the Freudenthal-Weil theorem.