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Isomorphisms and multipliers on second dual algebras of Banach algebras

Published online by Cambridge University Press:  24 October 2008

Fereidoun Ghahramani  and
Anthony To-Ming Lau
Fereidoun Ghahramani
Affiliation:
Department of Mathematics and Astronomy, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
Anthony To-Ming Lau
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2GI
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Extract

Suppose that A is a Banach algebra and let A be the second dual algebra of A equipped with the first Arens product 3. In this paper we characterize compact and weakly compact multipliers of A, when A possesses a bounded approximate identity and is a two sided ideal in A. We use this to study the isomorphisms between second duals of various classes of Banach algebras satisfying the above properties.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 1 , January 1992 , pp. 161 - 168
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

1Akemann, C.. Some mapping properties of the group algebras of a compact group. Pacific J. Math. 22 (1967), 18.CrossRefGoogle Scholar
2Akemann, C. and Wright, S.. Compact actions on C*-algebra. Glasgow Math. J. 21 (1980), 143149.CrossRefGoogle Scholar
3Arens, R.. The adjoint of a bilinear operation. Proc. Amer. Math. Soc. 2 (1951), 939948.CrossRefGoogle Scholar
4Bade, W. G. and Dales, H. G.. Norms and ideals in radical convolution algebras. J. Funct. Anal. 41 (1981), 77109.CrossRefGoogle Scholar
5Bergland, M. C. F.. Ideal C*-algebras. Duke Math. J. 40 (1973), 241257.Google Scholar
6Bonsall, F. F. and Duncan, J.. Complete Normed Algebras (Springer-Verlag, 1973).CrossRefGoogle Scholar
7Derighetti, A.. Some results on the Fourier Stieltjes algebra of a locally compact group. Comment. Math. Helv. 45 (1970), 219228.CrossRefGoogle Scholar
8Duncan, J. and Hosseiniun, S. A. R.. The second dual of a Banach algebra. Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 309325.CrossRefGoogle Scholar
9Eymard, E.. L'algbre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 92 (1964), 181236.CrossRefGoogle Scholar
10Ghahramani, F.. Isomorphisms between radical weighted convolution algebras. Proc. Edinburgh Math. Soc. 26 (1983), 343351.CrossRefGoogle Scholar
11Ghahramani, F.. The connectedness of the group of automorphisms of L 1(0, 1). Trans. Amer. Math. Soc. 302 (1987), 646659.Google Scholar
12Ghahramani, F.. The group of automorphisms of L 1{0, 1) is connected. Trans. Amer. Math. Soc. 314 (1989), 851859.Google Scholar
13Ghahramani, F. and Lau, A. T.. Isometric isomorphisms between the second conjugate algebra of the group algebra. Bull. London Math. Soc. 20 (1988), 342344.CrossRefGoogle Scholar
14Ghahramani, F., Lau, A. T. and Losert, V.. Isometric isomorphisms between Banach algebras related to locally compact groups. Trans. Amer. Math. Soc. 321 (1990), 273283.CrossRefGoogle Scholar
15Granirer, E. E.. On group representations whose C*-algebra is an ideal in its Von Neumann algebra. Ann. Inst. Fourier (Grenoble) 29 (1979), 3752.CrossRefGoogle Scholar
16Grosser, M.. L 1(G) as an ideal in its second dual space. Proc. Amer. Math. Soc. 73 (1979), 363364.Google Scholar
17Grosser, M. and Losert, V.. The norm-strict bidual of a Banach algebra and the dual of C u(G). Manuscripta Math. 45 (1984), 127146.CrossRefGoogle Scholar
18Isik, N., Pym, J. S. and lger, A.. The second dual of the group algebra of a compact group. J. London Math. Soc. (2) 35 (1987), 135148.CrossRefGoogle Scholar
19Kaplansky, I.. Normed algebras. Duke Math. J. 16 (1949), 399418.CrossRefGoogle Scholar
20Lau, A. T.. The second conjugate algebra of the Fourier algebra of a locally compact group. Trans. Amer. Math. Soc. 267 (1981), 5363.CrossRefGoogle Scholar
21Lau, A. T. and Paterson, A.. The exact cardinality of the set of topological left invariant means on an amenable locally compact group. Proc. Amer. Math. Soc. 98 (1986), 7580.CrossRefGoogle Scholar
22Lau, A. T. and Losert, V.. On the second conjugate algebra of L 1(G) of a locally compact group. J. London Math. Soc. (2) 37 (1988), 464470.CrossRefGoogle Scholar
23Lau, A. T. and Pym, J.. Concerning the second dual of the group algebra of a locally compact group. J. London Math. Soc. (2) 41 (1990) 445460.CrossRefGoogle Scholar
24Palmer, T. W.. Arens multiplication and a characterization of W*-algebras. Proc. Amer. Math. Soc. 44 (1974), 8187.Google Scholar
25Paterson, A.. Amenability. Math. Surveys and Monographs no. 29 (American Mathematical Society, 1988).CrossRefGoogle Scholar
26Renauld, P. F.. Centralizers of the Fourier algebra of an amenable group. Proc. Amer. Math. Soc. 32 (1972), 539542.Google Scholar
27Takesaki, M.. Theory of Operator Algebras, vol. 1 (Springer-Verlag, 1979).CrossRefGoogle Scholar
28Tomiuk, B. J.. Arens regularity and the algebra of double multipliers. Proc. Amer. Math. Soc. 81 (1981), 293298.CrossRefGoogle Scholar
29lger, A.. Arens regularity sometimes implies the RNP. Pacific J. Math. 143 (1990), 377399.Google Scholar
30Walter, M. E.. W*-algebras and non-abelian harmonic analysis. J. Fund. Anal. 11 (1972), 1138.CrossRefGoogle Scholar
31Watanabe, S.. A Banach algebra which is an ideal in the second dual. Sci. Rep. Niigata Univ. Ser. A 2 (1974), 95101.Google Scholar