Some Beurling–Fourier algebras on compact groups are operator algebras

Ghandehari, Mahya; Lee, Hun Hee; Samei, Ebrahim; Spronk, Nico

doi:10.1090/tran6653

Some Beurling-Fourier algebras on compact groups are operator algebras

Authors: Mahya Ghandehari, Hun Hee Lee, Ebrahim Samei and Nico Spronk
Journal: Trans. Amer. Math. Soc. 367 (2015), 7029-7059
MSC (2010): Primary 43A30, 47L30, 47L25; Secondary 43A75
DOI: https://doi.org/10.1090/tran6653
Published electronically: April 20, 2015
MathSciNet review: 3378823
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a compact connected Lie group. The question of when a weighted Fourier algebra on is completely isomorphic to an operator algebra will be investigated in this paper. We will demonstrate that the dimension of the group plays an important role in the question. More precisely, we will get a positive answer to the question when we consider a polynomial type weight coming from a length function on with the order of growth strictly bigger than half of the dimension of the group. The case of will be examined, focusing more on the details including negative results. The proof for the positive directions depends on a non-commutative version of the Littlewood multiplier theory, which we will develop in this paper, and the negative directions will be taken care of by restricting to a maximal torus.

[1] A. Beurling, Sur les intégrales de Fourier absolument convergentes et leur application à une transformation fonctionnelle, 9ème congrès des mathématiciens scandinaves, août 1938, Helsinki, 1939, pp. 345-366.
[2] Arne Beurling, Sur les spectres des fonctions, Analyse Harmonique., Colloques Internationaux du Centre National de la Recherche Scientifique, no. 15, Centre National de la Recherche Scientifique, Paris, 1949, pp. 9–29 (French). MR 0033367
[3] David P. Blecher, A completely bounded characterization of operator algebras, Math. Ann. 303 (1995), no. 2, 227–239. MR 1348798, https://doi.org/10.1007/BF01460988
[4] David P. Blecher and Christian Le Merdy, Operator algebras and their modules—an operator space approach, London Mathematical Society Monographs. New Series, vol. 30, The Clarendon Press, Oxford University Press, Oxford, 2004. Oxford Science Publications. MR 2111973
[5] David P. Blecher and Roger R. Smith, The dual of the Haagerup tensor product, J. London Math. Soc. (2) 45 (1992), no. 1, 126–144. MR 1157556, https://doi.org/10.1112/jlms/s2-45.1.126
[6] Marek Bożejko and Gero Fendler, Herz-Schur multipliers and uniformly bounded representations of discrete groups, Arch. Math. (Basel) 57 (1991), no. 3, 290–298. MR 1119903, https://doi.org/10.1007/BF01196860
[7] T. K. Carne, Not all 𝐻′-algebras are operator algebras, Math. Proc. Cambridge Philos. Soc. 86 (1979), no. 2, 243–249. MR 538746, https://doi.org/10.1017/S0305004100056061
[8] Benoît Collins, Hun Hee Lee, and Piotr Śniady, Dimensions of components of tensor products of representations of linear groups with applications to Beurling-Fourier algebras, Studia Math. 220 (2014), no. 3, 221–241. MR 3173046, https://doi.org/10.4064/sm220-3-2
[9] H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs. New Series, vol. 24, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. MR 1816726
[10] H. G. Dales and A. T.-M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005), no. 836, vi+191. MR 2155972, https://doi.org/10.1090/memo/0836
[11] Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
[12] Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 0228628
[13] Jacques Faraut, Analysis on Lie groups, Cambridge Studies in Advanced Mathematics, vol. 110, Cambridge University Press, Cambridge, 2008. An introduction. MR 2426516
[14] G. Fendler, K. Gröchenig, M. Leinert, J. Ludwig, and C. Molitor-Braun, Weighted group algebras on groups of polynomial growth, Math. Z. 245 (2003), no. 4, 791–821. MR 2020712, https://doi.org/10.1007/s00209-003-0571-6
[15] Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
[16] Brian Forrest, Arens regularity and discrete groups, Pacific J. Math. 151 (1991), no. 2, 217–227. MR 1132386
[17] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249
[18] Brian E. Forrest, Ebrahim Samei, and Nico Spronk, Convolutions on compact groups and Fourier algebras of coset spaces, Studia Math. 196 (2010), no. 3, 223–249. MR 2587297, https://doi.org/10.4064/sm196-3-2
[19] Brian E. Forrest, Ebrahim Samei, and Nico Spronk, Weak amenability of Fourier algebras on compact groups, Indiana Univ. Math. J. 58 (2009), no. 3, 1379–1393. MR 2542091, https://doi.org/10.1512/iumj.2009.58.3762
[20] A. Grothendieck, Une caractérisation vectorielle-métrique des espaces 𝐿¹, Canad. J. Math. 7 (1955), 552–561 (French). MR 0076301, https://doi.org/10.4153/CJM-1955-060-6
[21] Wilfried Hauenschild, Eberhard Kaniuth, and Ajay Kumar, Ideal structure of Beurling algebras on [𝐹𝐶]⁻ groups, J. Funct. Anal. 51 (1983), no. 2, 213–228. MR 701056, https://doi.org/10.1016/0022-1236(83)90026-5
[22] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, II. Springer Verlag, 1963.
[23] Barry Edward Johnson, Cohomology in Banach algebras, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 127. MR 0374934
[24] B. E. Johnson, Approximate diagonals and cohomology of certain annihilator Banach algebras, Amer. J. Math. 94 (1972), 685–698. MR 0317050, https://doi.org/10.2307/2373751
[25] B. E. Johnson, Non-amenability of the Fourier algebra of a compact group, J. London Math. Soc. (2) 50 (1994), no. 2, 361–374. MR 1291743, https://doi.org/10.1112/jlms/50.2.361
[26] Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR 719020
[27] Hun Hee Lee and Ebrahim Samei, Beurling-Fourier algebras, operator amenability and Arens regularity, J. Funct. Anal. 262 (2012), no. 1, 167–209. MR 2852259, https://doi.org/10.1016/j.jfa.2011.09.008
[28] H. H. Lee, E. Samei and N. Spronk, Some weighted group algebras are operator algebras, preprint, arXiv:1208.3791.
[29] Jean Ludwig, Nico Spronk, and Lyudmila Turowska, Beurling-Fourier algebras on compact groups: spectral theory, J. Funct. Anal. 262 (2012), no. 2, 463–499. MR 2854710, https://doi.org/10.1016/j.jfa.2011.09.017
[30] Gilles Pisier, Similarity problems and completely bounded maps, Second, expanded edition, Lecture Notes in Mathematics, vol. 1618, Springer-Verlag, Berlin, 2001. Includes the solution to “The Halmos problem”. MR 1818047
[31] Gilles Pisier, Grothendieck’s theorem, past and present, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 2, 237–323. MR 2888168, https://doi.org/10.1090/S0273-0979-2011-01348-9
[32] John F. Price, Lie groups and compact groups, Cambridge University Press, Cambridge-New York-Melbourne, 1977. London Mathematical Society Lecture Note Series, No. 25. MR 0450449
[33] Hans Reiter and Jan D. Stegeman, Classical harmonic analysis and locally compact groups, 2nd ed., London Mathematical Society Monographs. New Series, vol. 22, The Clarendon Press, Oxford University Press, New York, 2000. MR 1802924
[34] 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, https://doi.org/10.1112/S0024611504014650
[35] Nicholas Th. Varopoulos, Some remarks on 𝑄-algebras, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 4, 1–11 (English, with French summary). MR 0338780
[36] Nicholas Th. Varopoulos, Sur les quotients des algèbres uniformes, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A1344–A1346 (French). MR 0293403
[37] Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, No. 19. MR 0498996
[38] N. J. Young, The irregularity of multiplication in group algebras, Quart J. Math. Oxford Ser. (2) 24 (1973), 59–62. MR 0320756, https://doi.org/10.1093/qmath/24.1.59