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Transactions of the American Mathematical Society

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



Lebesgue type decomposition of subspaces of Fourier-Stieltjes algebras

Authors: E. Kaniuth, A. T. Lau and G. Schlichting
Journal: Trans. Amer. Math. Soc. 355 (2003), 1467-1490
MSC (2000): Primary 43A15; Secondary 22D10
Published electronically: November 22, 2002
MathSciNet review: 1946400
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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

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

G. Schlichting
Affiliation: Zentrum Mathematik, Technische Universität München, D-80290 München, Germany

Keywords: Locally compact group, Fourier-Stieltjes algebra, Fourier algebra, unitary representation, coefficient function space, Lebesgue decomposition
Received by editor(s): July 9, 2002
Published electronically: November 22, 2002
Additional Notes: The second author was supported by an NSERC grant.
Article copyright: © Copyright 2002 American Mathematical Society