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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
MR Author ID: 110640

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