×

Order structure of the Figà-Talamanca-Herz algebra. (English) Zbl 1219.43002

Summary: We study the interplay between the order structure and the \(p\)-operator space structure of the Figà-Talamanca-Herz algebra \(A_p(G)\) of a locally compact group \(G\). We show that for amenable groups, an order and algebra isomorphism of Figà-Talamanca-Herz algebras yields an isomorphism or anti-isomorphism of the underlying groups. We also give a bound for the norm of a \(p\)-completely positive linear map from a Figà-Talamanca-Herz algebra to its dual space.

MSC:

43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
43A07 Means on groups, semigroups, etc.; amenable groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] P. Eymard, “L’algèbre de Fourier d’un groupe localement compact,” Bulletin de la Société Mathématique de France, vol. 92, pp. 181-236, 1964. · Zbl 0169.46403
[2] A. Figà-Talamanca, “Translation invariant operators in Lp,” Duke Mathematical Journal, vol. 32, pp. 495-501, 1965. · Zbl 0142.10403 · doi:10.1215/S0012-7094-65-03250-3
[3] C. Herz, “The theory of p-spaces with an application to convolution operators,” Transactions of the American Mathematical Society, vol. 154, pp. 69-82, 1971. · Zbl 0216.15606 · doi:10.2307/1995427
[4] C. Herz, “Harmonic synthesis for subgroups,” Annales de l’Institut Fourier, vol. 23, no. 3, pp. 91-123, 1973. · Zbl 0257.43007 · doi:10.5802/aif.473
[5] M. Cowling, “An application of Littlewood-Paley theory in harmonic analysis,” Mathematische Annalen, vol. 241, pp. 83-96, 1972. · Zbl 0399.43004 · doi:10.1007/BF01406711
[6] J. P. Pier, Amenable Locally Compact Groups, John Wiley & Sons, New York, NY, USA, 1984. · Zbl 0597.43001
[7] V. Runde, “Representations of locally compact groups on QSLp-spaces and a p-analog of the Fourier-Stieltjes algebra,” Pacific Journal of Mathematics, vol. 221, no. 2, pp. 379-397, 2005. · Zbl 1095.43001 · doi:10.2140/pjm.2005.221.379
[8] M. Shams Yousefi, M. Amini, and F. Sady, “Complete order amenability of the Fourier algebra,” Indian Journal of Pure and Applied Mathematics, vol. 41, no. 3, pp. 485-504, 2010. · Zbl 1217.43001 · doi:10.1007/s13226-010-0028-7
[9] W. Arendt and J. D. Cannière, “Order isomorphisms of Fourier algebras,” Journal of Functional Analysis, vol. 50, no. 1, pp. 1-7, 1983. · Zbl 0506.43001 · doi:10.1016/0022-1236(83)90057-5
[10] E. G. Effros and Z. J. Ruan, Operator Spaces, vol. 23, Oxford University Press, Oxford, UK, 2000.
[11] G. Pisier, Introduction to Operator Space Theory, vol. 294 of London Mathematical Society Lecture Note Series 294, Cambridge University Press, Cambridge, 2003. · Zbl 1093.46001
[12] G. Wittstock, “What are operator spaces?” http://www.math.uni-sb.de/ag/wittstock/OperatorSpace/. · Zbl 0228.46003
[13] M. Daws, “p-operator spaces and Figà-Talamanca-Herz algebras,” Journal of Operator Theory, vol. 63, no. 1, pp. 47-83, 2010. · Zbl 1199.46125
[14] V. Runde, “Operator Figà-Talamanca-Herz algebras,” Studia Mathematica, vol. 155, no. 2, pp. 153-170, 2003. · Zbl 1032.47048 · doi:10.4064/sm155-2-5
[15] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin, 1974. · Zbl 0296.47023
[16] D. P. Blecher, “Positivity in operator algebras and operator spaces,” in Positivity, Trends in Mathematics, pp. 27-71, Birkhauser, Basle, Switzerland, 2007. · Zbl 1146.46031
[17] C. Le Merdy, “Factorization of p-completely bounded multilinear maps,” Pacific Journal of Mathematics, vol. 172, no. 1, pp. 187-213, 1996. · Zbl 0853.46054
[18] J. Bergh and J. Löfström, Interpolation Spaces. An introduction, Die Grundlehren der Mathematischen Wis-senschaften 223, Springer, Berlin, 1976. · Zbl 0344.46071
[19] W. J. Schreiner, “Matrix regular operator spaces,” Journal of Functional Analysis, vol. 152, no. 1, pp. 136-175, 1997. · Zbl 0898.46017 · doi:10.1006/jfan.1997.3160
[20] M. S. Monfared, “Extensions and isomorphisms for the generalized Fourier algebras of a locally compact group,” Journal of Functional Analysis, vol. 198, no. 2, pp. 413-444, 2003. · Zbl 1019.43001 · doi:10.1016/S0022-1236(02)00040-X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.