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Isomorphisms between the second duals of group algebras of locally compact groups

Published online by Cambridge University Press:  24 October 2008

Hamid-Reza Farhadi
Hamid-Reza Farhadi
Affiliation:
Department of Mathematics and Astronomy, University of Manitoba, Winnipeg, Manitoba, CanadaR3T 2N2
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Abstract

Let G be a locally compact group and L1(G) be the group algebra of G. We show that G is abelian or compact if every continuous automorphism of L1(G)** maps L1(G) onto L1(G) This characterizes all groups with this property and answers a question raised by F. Ghahramani and A. T. Lau in [7]. We also show that if G is a compact group and θ is any (algebra) isomorphism from L1(G)** onto L1(H)**, then H is compact and θ maps L1(G) onto L1(H).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 4 , May 1996 , pp. 657 - 663
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Akeman, C.. Some mapping properties of the group algebra of a compact group. Pacific J. Math. 22 (1967), 1–8.CrossRefGoogle Scholar
[2]Bonsall, F. F. and Duncan, J.. Complete normed algebras (Springer-Verlag, 1973).CrossRefGoogle Scholar
[3]Burckel, R. B.. Weakly almost periodic functions on semigroups (Gordon and Breach, 1970).Google Scholar
[4]Duncan, J. and Hosseiniun, S. A. R.. The second dual of a Banach algebra. Proc. Royal Soc. Edinburgh 84 A (1979), 309–325.CrossRefGoogle Scholar
[5]Filali, M.. The ideal structure of some Banach algebras. Math. Proc. Camb. Phil. Soc. 111 (1992), 567–576.CrossRefGoogle Scholar
[6]Ghahramani, F. and Lau, A. T.. Isometric isomorphisms between the second conjugate algebras of group algebras. Bull. London Math. Soc. 20 (1988), 342–344.CrossRefGoogle Scholar
[7]Ghahramani, F. and Lau, A. T.. Isomorphisms and multipliers on second dual algebras of Banach algebras. Math. Proc. Camb. Phil. Soc. 111 (1992), 161–168.CrossRefGoogle Scholar
[8]Ghahramani, F., Lau, A. T. and Losert, V.. Isometric isomorphisms between Banach algebras related to locally compact groups. Trans. Amer. Math. Soc. 321 (1990), 273–283.CrossRefGoogle Scholar
[9]Grosses, S. and Moskowitz, M.. Compactness conditions in topological groups. J. Reine Angew. Math. 246 (1971), 1–40.Google Scholar
[10]Hewitt, E. and Ross, K. A.. Abstract harmonic analysis, vol. I (Springer, 1963).Google Scholar
[11]Hewitt, E. and Ross, K. A.. Abstract harmonic analysis, vol. II (Springer, 1970).Google Scholar
[12]Hofmann, K. H. and Mostert, P.. Splitting in topological groups. Mem. Amer. Math. Soc. 43 (1963).Google Scholar
[13]Isik, N., Pym, J. S. and Ulger, A.. The second dual of the group algebra of a compact group. J. London Math. Soc. (2) 35 (1987), 135–148.CrossRefGoogle Scholar
[14]Lau, A. T. M.. Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups. Math. Proc. Camb. Phil. Soc. 99 (1986), 273–283.CrossRefGoogle Scholar
[15]Lau, A. T. M. and Losert, V.. On the second conjugate algebra of L 1(G) of a locally compact group. J. London Math. Soc. (2) 37 (1988), 464–470.CrossRefGoogle Scholar
[16]Lau, 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), 445–460.CrossRefGoogle Scholar
[17]Milnes, P.. Uniformity and uniformly continuous functions for locally compact groups. Proc. Amer. Math. Soc. (2) 109 (1990), 567–570.CrossRefGoogle Scholar
[18]Palmer, T. W.. Classes of nonabelian noncompact, locally compact groups. Rocky Mountain J. Math. 8 (1978), 683–741.CrossRefGoogle Scholar
[19]Pier, J.-P.. Amenable locally compact groups (Wiley, 1984).Google Scholar