The von Neumann Kernel and Minimally Almost Periodic Groups

Abstract

We calculate the von Neumann kernel $n(G)$ of an arbitrary connected Lie group. As a consequence we see that the closed characteristic subgroup $n(G)$ 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 $G/R$ is semisimple without compact factors and $G = {[G, G]^ - }$. In the special case where R is also simply connected it is proven that $G = [G, G]$. 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 $G/H$ has finite volume, and $\alpha$ is an automorphism of G with ${\text {disp}}(\alpha , H)$ bounded, then $\alpha$ 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.