Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T09:43:54.076Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  20 November 2018

Hun Hee Lee  and
Sang-gyun Youn
Hun Hee Lee
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, San56-1 Shinrim-dong Kwanak-gu, Seoul 151-747, Republic of Korea e-mail: [email protected], [email protected]
Sang-gyun Youn
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, San56-1 Shinrim-dong Kwanak-gu, Seoul 151-747, Republic of Korea e-mail: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

43A2043A3047L3047L25Fourier algebraconvolution algebraoperator algebraBeurling algebra
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 2 , 01 April 2017 , pp. 434 - 452
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Blecher, D. P., A completely bounded characterization of operator algebras. Math. Ann. 303(1995), no. 2, 227–239.http://dx.doi.org/10.1007/BF01460988 Google Scholar
[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] 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] 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] Eymard, P., L'algébre de Fourier d-un groupe localement compact. Bull. Soc. Math. France 92(1964), 181–236.Google Scholar
[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] Pack, T., Type and cotype inequalities for noncommutative Lp-spaces. J. Operator Theory 17(1987),no. 2, 255–279.Google Scholar
[8] Granbask, N., Amenability of weighted convolution algebras on locally compact groups. Trans. Amer. Math. Soc. 319(1990), no. 2, 765–775. http://dx.doi.org/10.1090/S0002-9947-1990-0962282-5 Google Scholar
[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.http://dx.doi.org/10.1090/tran6653 Google Scholar
[10] Helgason, S., Topologies of group algebras and a theorem of Littlewood. Trans. Amer. Math. Soc. 86(1957), 269–283.http://dx.doi.org/10.1090/S0002-9947-1957-0095428-5 Google Scholar
[11] Johnson, B. E., Cohomology in Banach algebras. Memoirs of the American Mathematical Society,127, American Mathematical Society, Providence, RI, 1972.Google Scholar
[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] 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] 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] 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] 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.http://dx.doi.org/10.1006/jfan.1997.3111 Google Scholar
[17] Price, J. F., On local central lacunary sets for compact Lie groups. Monatsh. Math. 80(1975), no. 3, 201–204.http://dx.doi.org/10.1007/BF01319915 Google Scholar
[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] 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] Vergnioux, R., The property of rapid decay for discrete quantum groups. J. Operator Theory 57(2007), no. 2, 303–324.Google Scholar
[21] Wallach, N. R., Harmonic analysis on homogeneous spaces. Pure and Applied Mathematics, 19, Marcel Dekker, Inc., New York, 1973.Google Scholar
[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] 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