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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48 .

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Lebesgue type decomposition of subspaces of Fourier-Stieltjes algebras
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by E. Kaniuth, A. T. Lau and G. Schlichting PDF
Trans. Amer. Math. Soc. 355 (2003), 1467-1490 Request permission

Abstract:

Let $G$ be a locally compact group and let $A(G)$ and $B(G)$ be the Fourier algebra and the Fourier-Stieltjes algebra of $G$, respectively. For any unitary representation $\pi$ of $G$, let $B_\pi (G)$ denote the $w^\ast$-closed linear subspace of $B(G)$ generated by all coefficient functions of $\pi$, and $B_\pi ^0(G)$ the closure of $B_\pi (G) \cap A_c(G)$, where $A_c(G)$ consists of all functions in $A(G)$ with compact support. In this paper we present descriptions of $B_\pi ^0(G)$ and its orthogonal complement $B_\pi ^s(G)$ in $B_\pi (G)$, generalizing a recent result of T. Miao. We show that for some classes of locally compact groups $G$, there is a dichotomy in the sense that for arbitrary $\pi$, either $B_\pi ^0(G) = \{0\}$ or $B_\pi ^0(G) = A(G)$. We also characterize functions in ${\mathcal B}_\pi ^0(G) = A_c(G) + B_\pi ^0(G)$ and study the question of whether ${\mathcal B}_\pi ^0(G) = A(G)$ implies that $\pi$ weakly contains the regular representation.
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Additional Information
  • E. Kaniuth
  • Affiliation: Fachbereich Mathematik/Informatik, Universität Paderborn, D-33095 Paderborn, Germany
  • Email: kaniuth@math.uni-paderborn.de
  • A. T. Lau
  • Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada T6G 2G1
  • MR Author ID: 110640
  • Email: tlau@math.ualberta.ca
  • G. Schlichting
  • Affiliation: Zentrum Mathematik, Technische Universität München, D-80290 München, Germany
  • Email: schlicht@mathematik.tu-muenchen.de
  • Received by editor(s): July 9, 2002
  • Published electronically: November 22, 2002
  • Additional Notes: The second author was supported by an NSERC grant.
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 1467-1490
  • MSC (2000): Primary 43A15; Secondary 22D10
  • DOI: https://doi.org/10.1090/S0002-9947-02-03203-8
  • MathSciNet review: 1946400