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The second conjugate algebra of the Fourier algebra of a locally compact group


Author: Anthony To Ming Lau
Journal: Trans. Amer. Math. Soc. 267 (1981), 53-63
MSC: Primary 43A30; Secondary 22D25
DOI: https://doi.org/10.1090/S0002-9947-1981-0621972-9
MathSciNet review: 621972
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Abstract: Let $ G$ be a locally compact group and let $ VN(G)$ denote the von Neumann algebra generated by the left translations of $ G$ on $ {L_2}(G)$. Then $ VN{(G)^{\ast}}$, when regarded as the second conjugate space of the Fourier algebra of $ G$, is a Banach algebra with the Arens product. We prove among other things that when $ G$ is amenable, $ VN{(G)^{\ast}}$ is neither commutative nor semisimple unless $ G$ is finite. We study in detail the class of maximal regular left ideals in $ VN{(G)^{\ast}}$. We also show that if $ {G_1}$ and $ {G_2}$ are discrete groups, then $ {G_1}$ and $ {G_2}$ are isomorphic if and only if $ VN{({G_1})^{\ast}}$ and $ VN{({G_2})^{\ast}}$ are isometric order isomorphic.


References [Enhancements On Off] (What's this?)

  • [1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839-849. MR 0045941 (13:659f)
  • [2] F. F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, Berlin and New York, 1973. MR 0423029 (54:11013)
  • [3] P. Civin, Ideals in the second conjugate algebra of a group algebra, Math. Scand. 11 (1962), 161-174. MR 0155200 (27:5139)
  • [4] P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847-870. MR 0143056 (26:622)
  • [5] N. Dunford and J. T. Schwartz, Linear operators. I, Interscience, New York, 1958.
  • [6] P. Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236. MR 0228628 (37:4208)
  • [7] E. E. Granirer, Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group, Trans. Amer. Math. Soc. 189 (1974), 371-382. MR 0336241 (49:1017)
  • [8] -, Density theorems for some linear subspaces and some $ {C^{\ast}}$-subalgebras of $ VN(G)$, Sympos. Mat., vol. XXII, Istit. Nazionale di Alta Mat., 1977, pp. 61-70.
  • [9] -, On group representations whose $ {C^{\ast}}$-algebra is an ideal in its von Neumann algebra (preprint).
  • [10] F. P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Math. Studies, no. 16, Van Nostrand Reinhold, New York, 1969. MR 0251549 (40:4776)
  • [11] N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R.I., 1956. MR 0081264 (18:373d)
  • [12] R. V. Kadison, Isometries of operator algebras, Ann. of Math. (2) 54 (1951), 325-338. MR 0043392 (13:256a)
  • [13] A. T. Lau, Uniformly continuous functionals on the Fourier algebra of any locally compact group, Trans. Amer. Math. Soc. 251 (1979), 39-59. MR 531968 (80m:43009)
  • [14] H. Leptin, Sur l'algèbre de Fourier d'un groupe localement compact, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A1180-A1182. MR 0239002 (39:362)
  • [15] L. H. Loomis, An introduction to abstract harmonic analysis, Van Nostrand, Princeton, N.J., 1953. MR 0054173 (14:883c)
  • [16] P. F. Renaud, Invariant means on a class of von Neumann algebras, Trans. Amer. Math. Soc. 170 (1972), 285-291. MR 0304553 (46:3688)
  • [17] S. Sakai, $ {C^{\ast}}$-algebras and $ {W^{\ast}}$-algebras, Springer-Verlag, Berlin and New York, 1971. MR 0442701 (56:1082)
  • [18] M. E. Walter, $ {W^{\ast}}$-algebras and nonabelian harmonic analysis, J. Funct. Anal. 11 (1972), 17-38. MR 0352879 (50:5365)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0621972-9
Keywords: Arens product, second conjugate algebra, Fourier algebra, Fourier-Stieltjes algebra, left regular representation, amenable locally compact group, regular ideal, group $ {C^{\ast}}$-algebra, Tauberian property, radical, uniformly continuous functionals, topological invariant mean
Article copyright: © Copyright 1981 American Mathematical Society

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