Skip to main content Accessibility help
×
Home

Norm One Idempotent cb-Multipliers with Applications to the Fourier Algebra in the cb-Multiplier Norm

  • Brian E. Forrest (a1) and Volker Runde (a2)

Abstract

For a locally compact group $G$ , let $A(G)$ be its Fourier algebra, let ${{M}_{cb}}A(G)$ denote the completely bounded multipliers of $A(G)$ , and let ${{A}_{Mcb}}\,(G)$ stand for the closure of $A(G)$ in ${{M}_{cb}}A(G)$ . We characterize the norm one idempotents in ${{M}_{cb}}A(G)$ : the indicator function of a set $E\,\subset \,G$ is a norm one idempotent in ${{M}_{cb}}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$ . As applications, we describe the closed ideals of ${{A}_{Mcb}}\,(G)$ with an approximate identity bounded by 1, and we characterize those $G$ for which ${{A}_{Mcb}}\,(G)$ is 1-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.)

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Norm One Idempotent cb-Multipliers with Applications to the Fourier Algebra in the cb-Multiplier Norm
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Norm One Idempotent cb-Multipliers with Applications to the Fourier Algebra in the cb-Multiplier Norm
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Norm One Idempotent cb-Multipliers with Applications to the Fourier Algebra in the cb-Multiplier Norm
      Available formats
      ×

Copyright

References

Hide All
[1] Bożejko, M., Remark on Herz-Schur multipliers on free groups. Math. Ann. 258(1981/82), no. 1, 1115. doi:10.1007/BF01450343
[2] Bożejko, M. and Fendler, G., Herz–Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group. Boll. Un. Mat. Ital. A (6) 3(1984), no. 2, 297302.
[3] Brannan, M., Forrest, B. E., and Zwarich, C., Multipliers and complemented ideals in the Fourier algebra. Preprint.
[4] Cohen, P. J., On a conjecture of Littlewood and idempotent measures.. Amer. J. Math. 82(1960), 191212. doi:10.2307/2372731
[5] Cowling, M. and Haagerup, U., Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one. Invent. Math. 96(1989), no. 3, 507549. doi:10.1007/BF01393695
[6] de Cannière, J. and Haagerup, U., Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups. Amer. J. Math. 107(1985), no. 2, 455500. doi:10.2307/2374423
[7] Effros, E. G. and Ruan, Z.-J., Operator Spaces. London Mathematical Society Monographs 23, The Clarendon Press, New York, 2000.
[8] Eymard, P., L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. France 92(1964), 181236.
[9] Forrest, B. E., Amenability and bounded approximate identities in ideals of A(G). Illinois J. Math. 34(1990), no. 1, 125.
[10] Forrest, B. E., Completely bounded multipliers and ideals in A(G) vanishing on closed subgroups. In: Banach Algebras and Their Applications. Contemp. Math. American Mathematical Society, Providence, RI, 2004, pp. 8994.
[11] Forrest, B. E. and Wood, P. J., Cohomology and the operator space structure of the Fourier algebra and its second dual. Indiana Univ. Math. J. 50(2001), no. 3, 12171240.
[12] Forrest, B. E., Kaniuth, E., Lau, A. T.-M., and Spronk, N., Ideals with bounded approximate identities in Fourier algebras. J. Funct. Anal. 203(2003), no. 1, 286304. doi:10.1016/S0022-1236(02)00121-0
[13] Forrest, B. E. and Runde, V., Amenability and weak amenability of the Fourier algebra. Math. Z. 250(2005), no. 4, 731744. doi:10.1007/s00209-005-0772-2
[14] Forrest, B. E., Runde, V., and Spronk, N., Nico Operator amenability of the Fourier algebra in the cb-multiplier norm. Canad. J. Math. 59(2007), no. 5, 966980. doi:10.4153/CJM-2007-041-9
[15] Gilbert, J. E., Lp-convolution opeators and tensor products of Banach spaces, I, II, and III. Unpublished manuscripts.
[16] Host, B., Le théorème des idempotents dans B(G) . Bull. Soc. Math. France 114(1986), no. 2, 215223.
[17] Ilie, M. and Spronk, N., Completely bounded homomorphisms of the Fourier algebras. J. Funct. Anal. 225(2005), no. 2, 480499. doi:10.1016/j.jfa.2004.11.011
[18] Johnson, B. E., Cohomology in Banach Algebras. Memoirs of the American Mathematical Society 127, American Mathematical Society, Providence, RI, 1972.
[19] Johnson, B. E., Approximate diagonals and cohomology of certain annihilator Banach algebras.. Amer. J. Math. 94(1972), 685698. doi:10.2307/2373751
[20] Johnson, B. E., Non-amenability of the Fourier algebra of a compact group. J. London Math. Soc. 50(1994), no. 2, 361374.
[21] Jolissaint, P., A characterization of completely bounded multipliers of Fourier algebras. Colloq. Math. 63(1992), no. 2, 311313.
[22] Katavolos, A. and Paulsen, V. I., On the ranges of bimodule projections. Canad. Math. Bull. 48(2005), no. 1, 97111. doi:10.4153/CMB-2005-009-4
[23] Leinert, M., Abschätzung von Normen gewisser Matrizen und eine Anwendung. Math. Ann. 240(1979), no. 1, 1319. doi:10.1007/BF01428295
[24] Leptin, H., Sur l’algèbre de Fourier d’un groupe localement compact. C. R. Acad. Sci. Paris Sér. A-B 266(1968), A1180A1182.
[25] Livshits, L., A note on 01 Schur multipliers. Linear Algebra Appl. 222(1995), 1522. doi:10.1016/0024-3795(93)00268-5
[26] Losert, V., Properties of the Fourier algebra that are equivalent to amenability. Proc. Amer. Math. Soc. 92(1984), no. 3, 347354.
[27] Pisier, G., Similarity Problems and Completely Bounded Maps. Lecture Notes in Mathematics 1618, Springer-Verlag, Berlin, 1996.
[28] Ruan, Z.-J., The operator amenability of A(G) . Amer. J. Math. 117(1995), no. 6, 14491474. doi:10.2307/2375026
[29] Runde, V., Lectures on Amenability. Lecture Notes in Mathematics 1774, Springer-Verlag, Berlin, 2002.
[30] Runde, V., The amenability constant of the Fourier algebra. Proc. Amer. Math. Soc. 134(2006), no. 5, 14731481. doi:10.1090/S0002-9939-05-08164-5
[31] Spronk, N., Measurable Schur multipliers and completely bounded multipliers of the Fourier algebras.. Proc. London Math. Soc. 89(2004), 161192. doi:10.1112/S0024611504014650
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed