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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The von Neumann kernel and minimally almost periodic groups

Author: Sheldon Rothman
Journal: Trans. Amer. Math. Soc. 259 (1980), 401-421
MSC: Primary 22E15; Secondary 22D05, 43A60
MathSciNet review: 567087
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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.

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Keywords: Von Neumann kernel, minimally almost periodic, maximally almost periodic, automorphism of bounded displacement
Article copyright: © Copyright 1980 American Mathematical Society