New Deformations of Convolution Algebras and Fourier Algebras on Locally Compact Groups

Lee, Hun Hee ; Youn, Sang-gyun

Canadian journal of mathematics, Tome 69 (2017) no. 2, pp. 434-452

Résumé

In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some information about the underlying groups by examining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similarly, we investigate complete representability as an operator algebra of deformed Fourier algebras on some finitely generated discrete groups with connection to the growth rate of the group.

Détail

DOI : 10.4153/CJM-2016-027-7

Mots-clés : 43A20, 43A30, 47L30, 47L25, Fourier algebra, convolution algebra, operator algebra, Beurling algebra

Bibliographie

[1] [1] Blecher, D. P., A completely bounded characterization of operator algebras. Math. Ann. 303(1995), no. 2, 227–239. Google Scholar | DOI

[2] [2] Blecher, D. P. and Smith, R. R., The dual of the Haagerup tensor product. J. Lond. Math. Soc. (2) 45(1992), no. 1, 126–144. http://dx.doi.Org/10.1112/jlms/s2-45.1.126 Google Scholar

[3] [3] Dales, H. G., A. Lau, T.-M., The second duals of Beurling algebras. Mem. Amer. Math. Soc. 177(2005), no. 836. http://dx.doi.Org/10.1090/memo/0836 Google Scholar

[4] [4] Dasgupta, A. and Ruzhansky, M., Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces. Bull. Sci. Math. 138(2014), no. 6, 756–782. http://dx.doi.Org/10.1016/j.bulsci.2O13.12.001 Google Scholar

[5] [5] Eymard, P., L'algébre de Fourier d-un groupe localement compact. Bull. Soc. Math. France 92(1964), 181–236. Google Scholar

[6] [6] Effros, E. G. and Ruan, Z.-J., Operator spaces. London Mathematical Society Monographs, New Series, 23, The Clarendopn Press, Oxford University Press, New York, 2000. Google Scholar

[7] [7] Pack, T., Type and cotype inequalities for noncommutative L-spaces. J. Operator Theory 17(1987),no. 2, 255–279. Google Scholar

[8] [8] Granbask, N., Amenability of weighted convolution algebras on locally compact groups. Trans. Amer. Math. Soc. 319(1990), no. 2, 765–775. Google Scholar | DOI

[9] [9] Ghandehari, M., Lee, H. H., Samei, E., and Spronk, N., Some Beurling-Fourier algebras are operator algebras. Trans. Amer. Math. Soc. 367(2015), no. 10, 7029–7059. Google Scholar | DOI

[10] [10] Helgason, S., Topologies of group algebras and a theorem of Littlewood. Trans. Amer. Math. Soc. 86(1957), 269–283. Google Scholar | DOI

[11] [11] Johnson, B. E., Cohomology in Banach algebras. Memoirs of the American Mathematical Society,127, American Mathematical Society, Providence, RI, 1972. Google Scholar

[12] [12] Kaniuth, E., A course in commutative Banach algebras. Graduate Texts in Mathematics, 246, Springer, New York, 2009. http://dx.doi.Org/10.1007/978-0-387-72476-8 Google Scholar

[13] [13] Lee, H. H. and Samei, E., Beurling-Fourier algebras, operator amenability and Arens regularity. J. Funct. Anal. 262(2012), no. 1, 167–209. http://dx.doi.Org/10.1016/j.jfa.2O11.09.008 Google Scholar

[14] [14] Lee, H. H., Samei, E., and Spronk, N., Some weighted group algebras are operator algebras. Proc. Edinb. Math. Soc. (2) 58(2015), no. 2, 499–519. http://dx.doi.Org/10.1017/S0013091514000212 Google Scholar

[15] [15] Ludwig, J., Spronk, N., and Turowska, L., Beurling-Fourier algebras of compact groups. J. Funct.Anal. 262(2012), no. 2, 463–499. http://dx.doi.Org/10.1016/j.jfa.2O11.09.017 Google Scholar

[16] [16] Lust-Piquard, F., On the coefficient problem: a version of the Kahane-Katznelson-de Leeuw theorem for spaces of matrices. J. Funct. Anal. 149(1997), no. 2, 352–376. Google Scholar | DOI

[17] [17] Price, J. F., On local central lacunary sets for compact Lie groups. Monatsh. Math. 80(1975), no. 3, 201–204. Google Scholar | DOI

[18] [18] Ruan, Z.-J.. The operator amenability of A(G). Amer. J. Math. 117(1995), no. 6,1449–1474. http://dx.doi.Org/10.2307/2375026 Google Scholar

[19] [19] Spronk, N., Measurable Schur multipliers and completely bounded multipliers of the Fourier algebras. Proc. London Math. Soc. 89(2004), no. 1,161–192. http://dx.doi.Org/10.1112/S0024611504014650 Google Scholar

[20] [20] Vergnioux, R., The property of rapid decay for discrete quantum groups. J. Operator Theory 57(2007), no. 2, 303–324. Google Scholar

[21] [21] Wallach, N. R., Harmonic analysis on homogeneous spaces. Pure and Applied Mathematics, 19, Marcel Dekker, Inc., New York, 1973. Google Scholar

[22] [22] Walter, M. E., W*-algebras and nonabelian harmonic analysis. J. Functional Analysis 11(1972), 17–38. http://dx.doi.Org/10.1016/0022-1236(72)90077-8 Google Scholar

[23] [23] Wendel, J. G., On isometric isomorphism of group algebras. Pacific J. Math. 1(1951), 305–311. http://dx.doi.ord/10.2140/nim.1951.1.305 Google Scholar