## The von Neumann kernel and minimally almost periodic groups

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- by Sheldon Rothman PDF
- Trans. Amer. Math. Soc.
**259**(1980), 401-421 Request permission

## 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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## Additional Information

- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**259**(1980), 401-421 - MSC: Primary 22E15; Secondary 22D05, 43A60
- DOI: https://doi.org/10.1090/S0002-9947-1980-0567087-9
- MathSciNet review: 567087