Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The Radon-Nikodym property for some Banach algebras related to the Fourier algebra


Author: Edmond E. Granirer
Journal: Proc. Amer. Math. Soc. 139 (2011), 4377-4384
MSC (2010): Primary 43A15, 46J10, 43A25, 46B22; Secondary 46J20, 43A30, 43A80, 22E30
DOI: https://doi.org/10.1090/S0002-9939-2011-10853-0
Published electronically: April 22, 2011
MathSciNet review: 2823083
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Radon-Nikodym property for the Banach algebras $ A_p^r(G)=A_p\cap L^r(G)$, where $ A_2(G)$ is the Fourier algebra, is investigated. A complete solution is given for amenable groups $ G$ if $ 1<p<\infty $ and for arbitrary $ G$ if $ p=2$ and $ A_2(G)$ has a multiplier bounded approximate identity. The results are new even for $ G=\mathbb{R}^n$.


References [Enhancements On Off] (What's this?)

  • [Br] W. Braun: Einige Bemerkungen $ ZuS_0(G)$ und $ A_p(G)\cap L^1(G)$. Preprint (1983).
  • [BrFei] W. Braun and Hans G. Feichtinger: Banach spaces of distributions having two module structures. J. Funct. Analysis 51 (1983), 174-212. MR 701055 (84h:46062)
  • [CaHa] J. De Carrière and U. Haagerup: Multipliers of the Fourier algebra of some simple Lie groups and their discrete subgroups. Amer. J. Math. 107 (1985), 455-500. MR 784292 (86m:43002)
  • [Co1] Michael Cowling: The Kunze-Stein phenomenon. Ann. of Math. 107 (1978), 209-234. MR 0507240 (58:22398)
  • [Co2] Michael Cowling: An application of Littlewood-Paley theory in harmonic analysis. Math. Ann. 241 (1979), 83-96. MR 531153 (81f:43003)
  • [DiU] J. Diestel and J.J. Uhl: Vector Measures. Math. Surveys, No. 15, Amer. Math. Soc., 1977. MR 0453964 (56:12216)
  • [Dix] J. Dixmier: Les $ C^\ast$ algebres et leur representations. Gauthier-Villars, 1969. MR 0246136 (39:7442)
  • [Do] B. Dorofaeff: The Fourier algebra of $ \mathrm{SL}(2,\mathbb{R}) \propto R^n$, $ n\ge 2$ has no multiplier bounded approximate unit. Math. Ann. 297 (1993), 707-724. MR 1245415 (94k:43005)
  • [DS] N. Dunford and J. Schwartz: Linear Operators. Part 1. General Theory. Interscience Publishers, New York, 1958. MR 0117523 (22:8302)
  • [Ey1] P. Eymard: L'algebre de Fourier d'un groupe localement compacte. Bull. Soc. Math. France 92 (1964), 181-236. MR 0228628 (37:4208)
  • [Ey2] P. Eymard: Algebre $ A_p$ et convoluteurs de $ L_p$. Lecture Notes in Math., No. 180, Springer, 1971, 364-381.
  • [Fe] Gero Fendler: An $ L^p$-version of a theorem of D.A. Raikov. Ann. Inst. Fourier, Grenoble 35 (1985), 125-135. MR 781782 (86h:43003)
  • [FT] A. Figa-Talamanca: Translation invariant operators on $ L^p$. Duke Math. J. 32 (1965), 495-501. MR 0181869 (31:6095)
  • [FTP] A. Figa-Talamanca and M. Picardello: Multiplicateurs de $ A(G)$ qui ne sont pas dans $ B(G)$. C.R. Acad. Sci. Paris 277 (1973), 117-119. MR 0333597 (48:11922)
  • [Ga] Steven A. Gaal: Linear Analysis and Representation Theory. Springer Verlag, New York, Heidelberg, Berlin, 1973. MR 0447465 (56:5777)
  • [GMc] C.C. Graham and O.C. McGehee: Essays in Commutative Harmonic Analysis. Springer Verlag, 1979. MR 550606 (81d:43001)
  • [Gr1] Edmond E. Granirer: The Figa-Talamanca-Herz-Lebesgue Banach algebras $ A_p^r(G)=A_p(G)\cap L^r(G)$. Math. Proc. Camb. Phil. Soc. 140 (2006), 401-416. MR 2225639 (2007f:46049)
  • [Gr2] Edmond E. Granirer: An application of the Radon Nikodym property in harmonic analysis. Bull. U.M.I. (5) 18-B (1981), 663-671. MR 629430 (83b:43004)
  • [Gr3] Edmond E. Granirer: Strong and extremely strong Ditkin sets for the Banach algebras $ A_p^r(G)=A_p(G)\cap L^r(G)$. Canad. J. Math. 63 (2011).
  • [GrL] 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 0722579 (85f:43009)
  • [HRI] E. Hewitt and K.A. Ross: Abstract Harmonic Analysis, Vols. 1 and 2. Springer Verlag, 1963, 1970. MR 0156915 (28:158), MR 0262773 (41:7378)
  • [HZ] Edwin Hewitt and Herbert Zuckerman: Singular measures with absolutely continuous convolution squares. Proc. Camb. Phil. Soc. 62 (1966), 399-420. MR 0193435 (33:1655)
  • [Hz1] C. Herz: Harmonic synthesis for subgroups. Ann. Inst. Fourier, Grenoble 23 (1973), 91-123. MR 0355482 (50:7956)
  • [Hz2] C. Herz: The theory of $ p$ spaces with an application to convolution operators. Trans. Amer. Math. Soc. 154 (1971), 69-82. MR 0272952 (42:7833)
  • [Hz3] C. Herz: On the asymmetry of norms of convolution operators. I.J. Functional Analysis 23 (1976), 11-22. MR 0420138 (54:8153)
  • [Kn] Anthony W. Knapp: Representation Theory for Semisimple Groups. Princeton University Press, Princeton, New Jersey, 1986. MR 855239 (87j:22022)
  • [KuS] R.A. Kunze and E.M. Stein: Uniformly bounded representations and harmonic analysis on the $ 2\times 2$ unimodular group. Amer. J. Math. 82 (1960), 47-66. MR 0163988 (29:1287)
  • [LCh] Hang-Chin Lai and Ing-Sheun Chen: Harmonic analysis on the Fourier algebras $ A_{1,p}(G)$. J. Austral. Math. Soc. (Series A) 30 (1981), 438-452. MR 621559 (82k:43002)
  • [Li] R.L. Lipsman: Harmonic analysis on $ \mathrm{SL}(n,\mathbb{C})$. J. Funct. Anal. 3 (1969), 126-155. MR 0237716 (38:5997)
  • [LiR] Teng-sun Liu and Arnoud van Rooij: Sums and intersections of normed linear spaces. Math. Nachrichten. 42 (1969), 29-42. MR 0273370 (42:8249)
  • [LLW] Ron Larsen, Teng-sun Liu and Ju-kwei Wang: On functions with Fourier transforms in $ Lp$. Mich. Math. J. 11 (1964), 369-378. MR 0170173 (30:412)
  • [Lu] Francoise Lust-Piquard: Means on $ CV_p(G)$-subspaces of $ CV_p (G)$ with RNP and Schur property. Ann. Inst. Fourier, Grenoble 39 (1989), 969-1006. MR 1036340 (91d:43002)
  • [RS] H. Reiter and J.T. Stegeman: Classical harmonic analysis and locally compact groups. London Math. Soc. Monographs, New Series, 22, Oxford Sci. Publ., 2000. MR 1802924 (2002d:43005)
  • [Ri] N.W. Rickert: Convolutions of $ L_p$ functions. Proc. Amer. Math. Soc. 18 (1967), 762-763. MR 0216301 (35:7136)
  • [Ru1] W. Rudin: Fourier analysis on groups. Interscience Publ., 1962. MR 0152834 (27:2808)
  • [Ru2] W. Rudin: Functional analysis. McGraw-Hill, 1973. MR 0365062 (51:1315)
  • [Tay] Keith F. Taylor: Geometry of the Fourier algebras and locally compact groups with atomic unitary representations. Math. Ann. 262 (1983), 183-190. MR 690194 (84h:43020)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 43A15, 46J10, 43A25, 46B22, 46J20, 43A30, 43A80, 22E30

Retrieve articles in all journals with MSC (2010): 43A15, 46J10, 43A25, 46B22, 46J20, 43A30, 43A80, 22E30


Additional Information

Edmond E. Granirer
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z4
Email: granirer@math.ubc.ca

DOI: https://doi.org/10.1090/S0002-9939-2011-10853-0
Received by editor(s): November 15, 2009
Received by editor(s) in revised form: October 14, 2010, and October 20, 2010
Published electronically: April 22, 2011
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society