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Reflexive representability of topological groups
by
Michael Megrelishvili
Bar-Ilan University, Israel
Theorem 1. Let G=H+[0, 1] be the topological group of all orientation-preserving homeomorphisms of the closed interval endowed with the usual compact open topology. Then every weakly almost periodic function on G is constant.
This result provides an example of a topological group G such that the algebra WAP(G) does not separate the points from the closed subsets, answering a question of Ruppert [Ru].
Corollary 1. There exists a Hausdorff topological group G such that every semitopological semigroup compactification of G is trivial.
Corollary 2. There exists a Hausdorff topological group G such that every weakly continuous representation of G in a reflexive Banach space (by linear isometries)is trivial.
Threorem 2. There exists a reflexive representable topological group G (namely, the additive group of the classical Banach space L4) which is not unitary representable. That is, there exists a reflexive Banach space E such that G is a topological subgroup of Is(E)s, the group of all linear isometries of E endowed with the strong operator topology, and there is no such Hilbert space E.
This answers a question of A. Shtern [Sh].
References
[Ru] W. Ruppert, Compact semitopological semigroups: An intrinsic theory, Lecture Notes in Math., vol. 1079, 1984, Springer-Verlag.
[Sh] A. Shtern, Compact semitopological semigroups and reflexive representability of topological groups, Russian J. of Math. Physics 2, 1994, 131-132.
Date received: June 22, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeh-35.