Extending multipliers of the Fourier algebra from a subgroup

Brannan, Michael; Forrest, Brian

doi:10.1090/S0002-9939-2014-11824-7

Extending multipliers of the Fourier algebra from a subgroup
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by Michael Brannan and Brian Forrest PDF

Proc. Amer. Math. Soc. 142 (2014), 1181-1191 Request permission

Abstract:

In this paper, we consider various extension problems associated with elements in the closure with respect to either the multiplier norm or the completely bounded multiplier norm of the Fourier algebra of a locally compact group. In particular, we show that it is not always possible to extend an element in the closure with respect to the multiplier norm of the Fourier algebra of the free group on two generators to a multiplier of the Fourier algebra of $SL(2,\mathbb {R})$.

References

Gilbert Arsac, Sur l’espace de Banach engendré par les coefficients d’une représentation unitaire, Publ. Dép. Math. (Lyon) 13 (1976), no. 2, 1–101 (French). MR 444833
M. Brannan, B. Forrest and C. Zwarich, Multipliers, completely bounded multipliers, and ideals in the Fourier algebra. Preprint, 2011.
M. Brannan, Operator spaces and ideals in Fourier algebras. Thesis, University of Waterloo, 2008.
Michael Cowling and Uffe Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), no. 3, 507–549. MR 996553, DOI 10.1007/BF01393695
Michael Cowling and Paul Rodway, Restrictions of certain function spaces to closed subgroups of locally compact groups, Pacific J. Math. 80 (1979), no. 1, 91–104. MR 534697
Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628
Brian Forrest, Some Banach algebras without discontinuous derivations, Proc. Amer. Math. Soc. 114 (1992), no. 4, 965–970. MR 1068120, DOI 10.1090/S0002-9939-1992-1068120-4
Brian E. Forrest, Volker Runde, and Nico Spronk, Operator amenability of the Fourier algebra in the cb-multiplier norm, Canad. J. Math. 59 (2007), no. 5, 966–980. MR 2354398, DOI 10.4153/CJM-2007-041-9
M. Ghandehari, Harmonic analysis of Rajchman algebras. Thesis, University of Waterloo, 2010.
Uffe Haagerup and Jon Kraus, Approximation properties for group $C^*$-algebras and group von Neumann algebras, Trans. Amer. Math. Soc. 344 (1994), no. 2, 667–699. MR 1220905, DOI 10.1090/S0002-9947-1994-1220905-3
Carl Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 91–123 (English, with French summary). MR 355482
Paul Jolissaint, A characterization of completely bounded multipliers of Fourier algebras, Colloq. Math. 63 (1992), no. 2, 311–313. MR 1180643, DOI 10.4064/cm-63-2-311-313
Michael Leinert, Faltungsoperatoren auf gewissen diskreten Gruppen, Studia Math. 52 (1974), 149–158 (German). MR 355480, DOI 10.4064/sm-52-2-149-158
Viktor Losert, Properties of the Fourier algebra that are equivalent to amenability, Proc. Amer. Math. Soc. 92 (1984), no. 3, 347–354. MR 759651, DOI 10.1090/S0002-9939-1984-0759651-8
V. Losert, On multipliers and completely bounded multipliers—the case $SL(2,\mathbb {R})$. Preprint, 2010.
Gilles Pisier, Multipliers and lacunary sets in non-amenable groups, Amer. J. Math. 117 (1995), no. 2, 337–376. MR 1323679, DOI 10.2307/2374918
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Nico Spronk, Measurable Schur multipliers and completely bounded multipliers of the Fourier algebras, Proc. London Math. Soc. (3) 89 (2004), no. 1, 161–192. MR 2063663, DOI 10.1112/S0024611504014650