The von Neumann kernel and minimally almost periodic groups

Abstract

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 a is an automorphism of G with disp(a, H) bounded, then a 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.

Compactifications, Hartman functions and (weak) almost periodicity

“…Let G1 be a topological group such that bG1 = {e} is trivial (such groups exist in abundance, cf. [15,35,37], they are called minimally almost periodic) and G2 = T, the torus. Denote by µ the Haar measure on T. Consider G := G1 × G2.…”

Representations of minimally almost periodic groups

“…Let G be a topological group; the intersection «(G) of the kernels of the finite-dimensional continuous unitary representations of G is the von Neumann kernel of G; this closed normal subgroup of G can be completely characterized when G is locally compact and connected (see [13], [14]). G is said to be minimally almost periodic (m.a.p.)…”

The von Neumann kernel of a locally compact group

“…For a locally compact group G, the von Neumann kernel, n(G), is the intersection of the kernels of the finite dimensional continuous complex unitary representations of G. Rothman [5] has calculated n(G) when G is a connected Lie group. Every such group has a Levi decomposition G -RL, where R is the radical and L = KS is the decomposition of the Levi factor into compact and noncompact parts.…”

“…Then the von Neumann kernel of G' is Vj-S', and the von Neumann kernel of G itself is the preimage of Vj-S' in G. That is, if w: G -* G' is the projection, «(G) = •n~\V^ • w(5)). Unless otherwise mentioned, results in the Lie case are to be found in [5].…”

“…Consequently, we may assume that G has abelian radical. Using Rothman's characterization for the von Neumann kernel of a connected Lie group ( [5], Theorem 1.2), we have n(G a ) = V^fS a . We know that S = limS a .…”