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.
The analysis of Lie groups depends to a large extent on their maximal tori. For a compact connected topological group G, the subgroups analogous to the maximal tori are the maximal connected Abelian subgroups. As in Hofmann and Morris [7] we call them maximal protori. We sharpen some results of [7] by showing that each maximal protorus is in a natural way the projective limit of maximal tori Tα in the corresponding Gα, where G= projGα. This sharpened characterization together with some methods of Moskowitz [4], [10] will be used to show that a number of well‐known theorems concerning Lie groups extend in a natural way to all compact connected groups.
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