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
DOI:
https://doi.org/10.1090/S0002-9947-02-03203-8
Published electronically:
November 22, 2002
MathSciNet review:
1946400
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- Michel Duflo, Opérateurs différentiels bi-invariants sur un groupe de Lie, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 2, 265–288 (French, with English summary). MR 444841
- Larry Baggett and Keith Taylor, Groups with completely reducible regular representation, Proc. Amer. Math. Soc. 72 (1978), no. 3, 593–600. MR 509261, DOI https://doi.org/10.1090/S0002-9939-1978-0509261-X
- Larry Baggett and Keith Taylor, A sufficient condition for the complete reducibility of the regular representation, J. Functional Analysis 34 (1979), no. 2, 250–265. MR 552704, DOI https://doi.org/10.1016/0022-1236%2879%2990033-8
- Alain Belanger and Brian E. Forrest, Geometric properties of coefficient function spaces determined by unitary representations of a locally compact group, J. Math. Anal. Appl. 193 (1995), no. 2, 390–405. MR 1338711, DOI https://doi.org/10.1006/jmaa.1995.1242
- M. E. B. Bekka, A. T. Lau, and G. Schlichting, On invariant subalgebras of the Fourier-Stieltjes algebra of a locally compact group, Math. Ann. 294 (1992), no. 3, 513–522. MR 1188135, DOI https://doi.org/10.1007/BF01934339
- David Bernier and Keith F. Taylor, Wavelets from square-integrable representations, SIAM J. Math. Anal. 27 (1996), no. 2, 594–608. MR 1377491, DOI https://doi.org/10.1137/S0036141093256265
- L. J. Bunce, The Dunford-Pettis property in the predual of a von Neumann algebra, Proc. Amer. Math. Soc. 116 (1992), no. 1, 99–100. MR 1091177, DOI https://doi.org/10.1090/S0002-9939-1992-1091177-1
- R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach Science Publishers, New York-London-Paris, 1970. MR 0263963
- Cho-Ho Chu, A note on scattered $C^{\ast } $-algebras and the Radon-Nikodým property, J. London Math. Soc. (2) 24 (1981), no. 3, 533–536. MR 635884, DOI https://doi.org/10.1112/jlms/s2-24.3.533
- Lawrence J. Corwin and Frederick P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press, Cambridge, 1990. Basic theory and examples. MR 1070979
- Jacques Dixmier, $C^ *$-algebras, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett; North-Holland Mathematical Library, Vol. 15. MR 0458185
- Jacques Dixmier, von Neumann algebras, North-Holland Mathematical Library, vol. 27, North-Holland Publishing Co., Amsterdam-New York, 1981. With a preface by E. C. Lance; Translated from the second French edition by F. Jellett. MR 641217
- Raouf Doss, On the transform of a singular or an absolutely continuous measure, Proc. Amer. Math. Soc. 19 (1968), 361–363. MR 222569, DOI https://doi.org/10.1090/S0002-9939-1968-0222569-4
- M. Duflo and Calvin C. Moore, On the regular representation of a nonunimodular locally compact group, J. Functional Analysis 21 (1976), no. 2, 209–243. MR 0393335, DOI https://doi.org/10.1016/0022-1236%2876%2990079-3
- Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628
- J. M. G. Fell, Weak containment and induced representations of groups. II, Trans. Amer. Math. Soc. 110 (1964), 424–447. MR 159898, DOI https://doi.org/10.1090/S0002-9947-1964-0159898-X
- Volker Flory, On the Fourier-algebra of a locally compact amenable group, Proc. Amer. Math. Soc. 29 (1971), 603–606. MR 283138, DOI https://doi.org/10.1090/S0002-9939-1971-0283138-3
- V. Flory, Eine Lebesgue-Zerlegung und funktorielle Eigenschaften der Fourier-Stieltjes Algebra, Inaugural Dissertation, Universität Heidelberg, 1972.
- E. E. Granirer and M. Leinert, On some topologies which coincide on the unit sphere of the Fourier-Stieltjes algebra $B(G)$ and of the measure algebra $M(G)$, Rocky Mountain J. Math. 11 (1981), no. 3, 459–472. MR 722579, DOI https://doi.org/10.1216/RMJ-1981-11-3-459
- F. P. Greenleaf, Amenable actions of locally compact groups, J. Functional Analysis 4 (1969), 295–315. MR 0246999, DOI https://doi.org/10.1016/0022-1236%2869%2990016-0
- Siegfried Grosser and Martin Moskowitz, Compactness conditions in topological groups, J. Reine Angew. Math. 246 (1971), 1–40. MR 284541, DOI https://doi.org/10.1515/crll.1971.246.1
- Carl Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 91–123 (English, with French summary). MR 355482
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
- Roger E. Howe, The Fourier transform for nilpotent locally compact groups. I, Pacific J. Math. 73 (1977), no. 2, 307–327. MR 492059
- Eberhard Kaniuth, On primary ideals in group algebras, Monatsh. Math. 93 (1982), no. 4, 293–302. MR 666831, DOI https://doi.org/10.1007/BF01295230
- Anthony To Ming Lau, Closed convex invariant subsets of $L_{p}(G)$, Trans. Amer. Math. Soc. 232 (1977), 131–142. MR 477604, DOI https://doi.org/10.1090/S0002-9947-1977-0477604-5
- Anthony To Ming Lau and Viktor Losert, The $C^*$-algebra generated by operators with compact support on a locally compact group, J. Funct. Anal. 112 (1993), no. 1, 1–30. MR 1207935, DOI https://doi.org/10.1006/jfan.1993.1024
- Anthony To Ming Lau and Ali Ülger, Some geometric properties on the Fourier and Fourier-Stieltjes algebras of locally compact groups, Arens regularity and related problems, Trans. Amer. Math. Soc. 337 (1993), no. 1, 321–359. MR 1147402, DOI https://doi.org/10.1090/S0002-9947-1993-1147402-7
- Peter F. Mah and Tianxuan Miao, Extreme points of the unit ball of the Fourier-Stieltjes algebra, Proc. Amer. Math. Soc. 128 (2000), no. 4, 1097–1103. MR 1637396, DOI https://doi.org/10.1090/S0002-9939-99-05104-7
- Tianxuan Miao, Decomposition of $B(G)$, Trans. Amer. Math. Soc. 351 (1999), no. 11, 4675–4692. MR 1608490, DOI https://doi.org/10.1090/S0002-9947-99-02328-4
- Calvin C. Moore, Groups with finite dimensional irreducible representations, Trans. Amer. Math. Soc. 166 (1972), 401–410. MR 302817, DOI https://doi.org/10.1090/S0002-9947-1972-0302817-8
- Richard D. Mosak, The $L^{1}$- and $C^{\ast } $-algebras of $[FIA]^{-}_{B}$ groups, and their representations, Trans. Amer. Math. Soc. 163 (1972), 277–310. MR 293016, DOI https://doi.org/10.1090/S0002-9947-1972-0293016-7
- Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261
- Jean-Paul Pier, Amenable locally compact groups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 767264
- Marc A. Rieffel, Unitary representations of group extensions; an algebraic approach to the theory of Mackey and Blattner, Studies in analysis, Adv. in Math. Suppl. Stud., vol. 4, Academic Press, New York-London, 1979, pp. 43–82. MR 546802
- Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728
- Keith F. Taylor, Geometry of the Fourier algebras and locally compact groups with atomic unitary representations, Math. Ann. 262 (1983), no. 2, 183–190. MR 690194, DOI https://doi.org/10.1007/BF01455310
- Garth Warner, Harmonic analysis on semi-simple Lie groups. II, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 189. MR 0499000
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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
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