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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Relative weak compactness of orbits in Banach spaces associated with locally compact groups
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by Colin C. Graham and Anthony T. M. Lau PDF
Trans. Amer. Math. Soc. 359 (2007), 1129-1160 Request permission

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
  • Email: ccgraham@alum.mit.edu
  • Anthony T. M. Lau
  • Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
  • MR Author ID: 110640
  • Email: tlau@math.ualberta.ca
  • 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
  • © Copyright 2006 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 1129-1160
  • MSC (2000): Primary 43A15, 43A10; Secondary 46L10
  • DOI: https://doi.org/10.1090/S0002-9947-06-04039-6
  • MathSciNet review: 2262845