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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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  • 1. G. Arsac, Sur l'espace de Banach engendré par les coefficients d'une représentation unitaire, Publ. Dép. Math. (Lyon) 13 (1976), 1-101. MR 56:3188
  • 2. L. Baggett and K. F. Taylor, Groups with completely reducible regular representation, Proc. Amer. Math. Soc. 72 (1978), 593-600. MR 80b:22009
  • 3. L. Baggett and K. F. Taylor, A sufficient condition for the complete reducibility of the regular representation, J. Funct. Anal. 34 (1979), 250-265. MR 81f:22005
  • 4. A. Belanger and B. Forrest, Geometric properties of coefficient function spaces determined by unitary representations of a locally compact group, J. Math. Anal. Appl. 193 (1995), 390-405. MR 96f:22005
  • 5. M. 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), 513-522. MR 93k:43006
  • 6. D. Bernier and K. F. Taylor, Wavelets from square-integrable representation, SIAM J. Math. Anal. 27 (1996), 594-608. MR 97h:22004
  • 7. L. C. Bunce, The Dunford-Pettis property in the predual of a von Neumann algebra, Proc. Amer. Math. Soc. 116 (1992), 99-100. MR 92k:46100
  • 8. R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach, 1970. MR 41:8562
  • 9. C. H. Chu, A note on scattered $C^\ast$-algebras and the Radon-Nikodym property, J. London Math. Soc. 24 (1981), 533-536. MR 82k:46086
  • 10. L. Corwin and F. P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part 1: Basic theory and examples, Cambridge, 1990. MR 92b:22007
  • 11. J. Dixmier, $C^*$-algebras, North-Holland, 1977. MR 56:16388
  • 12. J. Dixmier, Von Neumann algebras, North-Holland, 1981. MR 83a:46004
  • 13. R. Doss, On the transform of a singular or an absolutely continuous measure, Proc. Amer. Math. Soc. 19 (1968), 361-363. MR 36:5619
  • 14. M. Duflo and C. C. Moore, On the regular representation of a nonunimodular locally compact group, J. Funct. Anal. 21 (1976), 209-243. MR 52:14145
  • 15. P. Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236. MR 37:4208
  • 16. J. M. G. Fell, Weak containment and induced representations of groups. II, Trans. Amer. Math. Soc. 110 (1964), 424-447. MR 28:3114
  • 17. V. Flory, On the Fourier algebra of a locally compact amenable group, Proc. Amer. Math. Soc. 29 (1971), 603-606. MR 44:371
  • 18. V. Flory, Eine Lebesgue-Zerlegung und funktorielle Eigenschaften der Fourier-Stieltjes Algebra, Inaugural Dissertation, Universität Heidelberg, 1972.
  • 19. E. E. Granirer and M. Leinert, On some topologies which coincide on the unit sphere of the Fourier-Stieltjes algebra $B(G)$ and the measure algebra $M(G)$, Rocky Mountain J. Math. 11 (1981), 459-472. MR 85f:43009
  • 20. F. P. Greenleaf, Amenable actions of locally compact groups, J. Funct. Anal. 4 (1969), 295-315. MR 40:268
  • 21. S. Grosser and M. Moskowitz, Compactness conditions in topological groups, J. reine angew. Math. 246 (1971), 1-40. MR 44:1766
  • 22. C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier 23 (1973), 91-123. MR 50:7956
  • 23. E. Hewitt and K. A. Ross, Abstract harmonic analysis. II, Springer-Verlag, 1970. MR 41:7378
  • 24. R. Howe, The Fourier transform for nilpotent locally compact groups, Pacific J. Math. 73 (1977), 307-327. MR 58:11215
  • 25. E. Kaniuth, On primary ideals in group algebras, Monatsh. Math. 93 (1982), 293-302. MR 84i:43003
  • 26. A. T. Lau, Closed convex invariant subsets of $L_p(G)$, Trans. Amer. Math. Soc. 232 (1977), 131-142. MR 57:17122
  • 27. A. T. Lau and V. Losert, The $C^\ast$-algebra generated by operators with compact support on a locally compact group, J. Funct. Anal. 112 (1993), 1-30. MR 94d:22005
  • 28. A. T. Lau and A. 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), 321-359. MR 93g:22007
  • 29. P. F. Mah and T. Miao, Extreme points of the unit ball of the Fourier-Stieltjes algebra, Proc. Amer. Math. Soc. 128 (2000), 1097-1103. MR 2000i:43003
  • 30. T. Miao, Decomposition of $B(G)$, Trans. Amer. Math. Soc. 351 (1999), 4675-4692. MR 2000a:43006
  • 31. C. C. Moore, Groups with finite dimensional irreducible representations, Trans. Amer. Math. Soc. 166 (1972), 401-410. MR 46:1960
  • 32. R. 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 45:2096
  • 33. A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, RI, 1988. MR 90e:43001
  • 34. J. P. Pier, Amenable locally compact groups, John Wiley and Sons, 1984. MR 86a:43001
  • 35. M. A. Rieffel, Unitary representations of group extensions. An algebraic approach to the theory of Mackey and Blattner, Adv. Math. Suppl. Stud. 4 (1979), 43-82. MR 81h:22004
  • 36. M. Takesaki, Theory of operator algebras.I, Springer-Verlag, 1979. MR 81e:46038
  • 37. K. F. Taylor, Geometry of Fourier algebras and locally compact groups with atomic decomposition, Math. Ann. 262 (1983), 183-190. MR 84h:43020
  • 38. C. R. Warner, Harmonic analysis on semi-simple Lie groups. II, Springer-Verlag, 1972. MR 58:16980

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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