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Weak*- continuous homomorphisms of Fourier–Stieltjes algebras

Published online by Cambridge University Press:  01 July 2008

MONICA ILIE  and
ROSS STOKKE
MONICA ILIE
Affiliation:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON, Canada, P7B 5E1. e-mail: [email protected]
ROSS STOKKE
Affiliation:
Department of Mathematics and Statistics, University of Winnipeg, 515 Portage Avenue, Winnipeg, Canada, R3B 2E9. e-mail: [email protected]
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Abstract

For a locally compact group G, let B(G) denote its Fourier–Stieltjes algebra. Any continuous, piecewise affine map α: Y ⊂ H → G induces a completely bounded algebra homomorphism jα: B(G) → B(H) [14, 15] and we prove that jα is w* – w* continuous if and only if α is an open map. This extends one of the main results in [3], due to M.B. Bekka, E. Kaniuth, A.T. Lau and G. Schlichting. Several classical theorems regarding isomorphisms and extensions of homomorphisms on group algebras of abelian groups are extended to the setting of Fourier–Stieltjes algebras of amenable groups. As applications, when G is amenable we provide complete characterizations of those maps between Fourier–Stieltjes algebras that are either associated to a piecewise affine mapping, or are completely bounded and w* – w* continuous.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 1 , July 2008 , pp. 107 - 120
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Arsac, G.. Sur l'espace de Banach engendré par les coefficients d'une représentation unitaire. Publ. Dép. Math. (Lyon) 13 (1976), 1–101.Google Scholar
[2]Bekka, M. E. B., Lau, A. T. and Schlichting, G.. On invariant subalgebras of the Fourier–Stieltjes algebra of a locally compact group. Math. Ann. 294 (1992), 513–522.CrossRefGoogle Scholar
[3]Bekka, M. B., Kaniuth, E., Lau, A. T. and Schlichting, G.. Weak*-closedness of subspaces of Fourier–Stieltjes algebras and weak*-continuity of the restriction map. Trans. Amer. Math. Soc. 350 (1998), 2277–2296.CrossRefGoogle Scholar
[4]Blecher, D. P.. The standard dual of an operator space. Pacific Math. J. 153 (1992), 15–30.CrossRefGoogle Scholar
[5]Cohen, P. J.. On homomorphisms of group algebras. Amer. J. Math 82 (1960), 213–226.CrossRefGoogle Scholar
[6]Eymard, P.. L'algébre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 92 (1964), 181–236.CrossRefGoogle Scholar
[7]Effros, E. G. and Ruan, Z.-J.. A new approach to operator spaces. Canad. J. Math. 34 (1991), 329–337.CrossRefGoogle Scholar
[8]Effros, E. G. and Ruan, Z.-J.. Operator Spaces (Oxford University Press, 2000).Google Scholar
[9]Forrest, B., Kaniuth, E., Lau, A. T. and Spronk, N.. Ideals with bounded approximate identities in Fourier algebras. J. Funct. Anal. (1) 203 (2003), 286–304.CrossRefGoogle Scholar
[10]Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis (Springer–Verlag, 1963).Google Scholar
[11]Granirer, E. E. and Leinert, M.. On some topologies which coincide on the unit sphere of the Fourier–Stieltjes algebra B(G). Rocky Mountain J. Math. 11 (1981), 459–472.CrossRefGoogle Scholar
[12]Greenleaf, F. P.. Norm decreasing homomorphisms of group algebras. Pacific J. Math. 15 (1965), 1187–1219.CrossRefGoogle Scholar
[13]Host, B.. Le théorème des idempotents dans B(G). Bull. Soc. Math. France (2) 114 (1986), 215–223.CrossRefGoogle Scholar
[14]Ilie, M.. On Fourier algebra homomorphisms. J. Funct. Anal. 213 (2004), 88–110.CrossRefGoogle Scholar
[15]Ilie, M. and Spronk, N.. Completely bounded homomorphisms of the Fourier algebra. J. Funct. Anal. 225 (2)(2005), 480–499.CrossRefGoogle Scholar
[16]Rieffel, M. A.. Induced representations of C*-algebras. Advances in Math. 13 (1974), 176–257.CrossRefGoogle Scholar
[17]Kelley, J. T.. General Topology (D. Van Nostrand Company, Inc., 1955).Google Scholar
[18]Ruan, Z.-J.. The operator amenability of A(G). Amer. J. Math. 117 (1995), 1449–1474.CrossRefGoogle Scholar
[19]Rudin, W.. Fourier analysis on groups. Tracts in Pure and Applied Mathematics, No. 12 (Wiley Interscience, 1962).Google Scholar
[20]Runde, V.. Amenability for dual Banach algebras. Studia Math. (1) 148 (2001), 47–66.CrossRefGoogle Scholar
[21]Walter, M.. W*-algebras and nonabelian harmonic analysis. J. Funct. Anal. 11 (1972), 17–38.CrossRefGoogle Scholar