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Transactions of the American Mathematical Society

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



Relative weak compactness of orbits in Banach spaces associated with locally compact groups

Authors: Colin C. Graham and Anthony T. M. Lau
Journal: Trans. Amer. Math. Soc. 359 (2007), 1129-1160
MSC (2000): Primary 43A15, 43A10; Secondary 46L10
Published electronically: September 11, 2006
MathSciNet review: 2262845
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Abstract: We study analogues of weak almost periodicity in Banach spaces on locally compact groups.

i) If $ \mu$ is a continous measure on the locally compact abelian group $ G$ and $ f\in L^\infty(\mu)$, then $ \{\gamma f:\gamma\in\widehat G\}$ is not relatively weakly compact.

ii) If $ G$ is a discrete abelian group and $ f\in \ell^\infty(G)\backslash C_o(G)$, then $ \{\gamma f:\gamma\in E\}$ is not relatively weakly compact if $ E\subset \widehat G$ has non-empty interior. That result will follow from an existence theorem for $ I_o$-sets, as follows.

iii) Every infinite subset of a discrete abelian group $ \Gamma$ contains an infinite $ I_o$-set such that for every neighbourhood $ U$ of the identity of $ \widehat\Gamma$ the interpolation (except at a finite subset depending on $ U$) can be done using at most 4 point masses.

iv) A new proof that $ B(G)\subset WAP(G)$ for abelian groups is given that identifies the weak limits of translates of Fourier-Stieltjes transforms.

v) Analogous results for $ C_o(G)$, $ A_p(G)$, and $ M_p(G)$ are given.

vi) Semigroup compactifications of groups are studied, both abelian and non-abelian: the weak* closure of $ \widehat G$ in $ L^\infty(\mu)$, for abelian $ G$; and when $ \rho$ is a continuous homomorphism of the locally compact group $ \Gamma$ into the unitary elements of a von Neumann algebra $ \mathcal{M}$, the weak* closure of $ \rho(\Gamma)$ is studied.

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Additional Information

Colin C. Graham
Affiliation: Department of Mathematics, University of British Columbia, RR #1 – D-156, Bowen Island, British Columbia, Canada V0N 1G0

Anthony T. M. Lau
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Keywords: Almost periodic functions, $I_o$-sets, locally compact abelian groups, $p$-multipliers, weak closure of the characters, weak and weak* closures of translates, weakly almost periodic functions
Received by editor(s): January 23, 2003
Received by editor(s) in revised form: December 8, 2004
Published electronically: September 11, 2006
Additional Notes: Both authors were partially supported by NSERC grants
Article copyright: © Copyright 2006 American Mathematical Society