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)

 
 

 

An extension property for the Figà-Talamanca Herz algebra


Author: Christian Fiorillo
Journal: Proc. Amer. Math. Soc. 137 (2009), 1001-1011
MSC (2000): Primary 43A07, 43A15, 43A22; Secondary 43A32, 43A45, 46J10
DOI: https://doi.org/10.1090/S0002-9939-08-09679-2
Published electronically: October 9, 2008
MathSciNet review: 2457440
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a locally compact group and $ H$ a closed amenable subgroup of $ G$. We prove that every element in $ A_{p}(H)$ with compact support can be extended to an element of $ A_{p}(G)$ of which we control the norm and support. The result is new even for the Fourier algebra. Our approach gives us new results concerning the operator norm closure of the convolution operators of $ G$ with compact support.


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

  • [1] M. Cowling and P. Rodway, Restrictions of certain function spaces to closed subgroups of locally compact groups, Pacific J. Math. 80 (1979), 91-104. MR 534697 (80i:43008)
  • [2] J. Delaporte and A. Derighetti, Invariant projections and convolution operators, Proc. Amer. Math. Soc. 129 (2001), 1427-1435. MR 1814169 (2002m:43002)
  • [3] A. Derighetti, On Herz's projection theorem, Illinois J. Math. 48 (2004), 463-476. MR 2085420 (2006h:43002)
  • [4] C. Fiorillo, Extrapolation et synthèse, Ph.D. thesis, EPFL, Lausanne (2007).
  • [5] B. Forrest, Complemented ideals in the Fourier algebra and the Radon Nikodým property, Trans. Amer. Math. Soc. 333 (1992), 689-700. MR 1112546 (92m:43003)
  • [6] C. Herz, Harmonic synthesis for subgroups, Annales de l'Institut Fourier, tome 23, no. 3, Grenoble (1973), 91-123. MR 0355482 (50:7956)
  • [7] -, Synthèse spectrale pour les sous-groupes par rapport aux algèbres $ A_{p}$, C. R. Acad. Sci. Paris Sér. A-B 271, 1970, A316-A318. MR 0273429 (42:8307b)
  • [8] E. Kaniuth and A. Lau, Spectral synthesis for $ A(G)$ and subspaces of $ VN(G)$, Proc. Amer. Math. Soc. 129 (2001), 3253-3263. MR 1845000 (2002h:43003)
  • [9] -, Extension and separation properties of positive definite functions on locally compact groups, Trans. Amer. Math. Soc. 359 (2007), 447-463. MR 2247899 (2007k:43008)
  • [10] J.R. McMullen, Extensions of positive-definite functions, Memoirs of the American Mathematical Society 117 (1972), 1-71. MR 0445229 (56:3573)
  • [11] H. Reiter and J.D. Stegeman, Classical harmonic analysis and locally compact groups, The Clarendon Press, Oxford University Press, New York (2000). MR 1802924 (2002d:43005)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 43A07, 43A15, 43A22, 43A32, 43A45, 46J10

Retrieve articles in all journals with MSC (2000): 43A07, 43A15, 43A22, 43A32, 43A45, 46J10


Additional Information

Christian Fiorillo
Affiliation: Department of Mathematics, École Polytechnique Federale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland
Address at time of publication: Via Pratocarasso 31, 6500 Bellinzona (TI), Switzerland

DOI: https://doi.org/10.1090/S0002-9939-08-09679-2
Keywords: Amenable groups, Fig\`a-Talamanca Herz algebra, Fourier algebra, convolution operators.
Received by editor(s): March 5, 2008
Published electronically: October 9, 2008
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society