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Compact left ideal groups in semigroup compactification of locally compact groups

Published online by Cambridge University Press:  24 October 2008

J. W. Baker  and
A. T. Lau
J. W. Baker
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, England
A. T. Lau
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, CanadaT6G-261
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Extract

Let G be a locally compact group and let UG denote the spectrum of the C*-algebra LUC(G) of bounded left uniformly continuous complex-valued functions on G, with the Gelfand topology. Then there is a multiplication on UG extending the multiplication on G (when naturally embedded in UG) such that UG is a semigroup and for each x ∈ UG, the map y ↦ yx from UG into UG is continuous, i.e. UG is a compact right topological semigroup. Consequently UG has a unique minimal ideal K which is the union of minimal (closed) left ideals UG. Furthermore K is the union of the set of maximal subgroups of K (see [3], theorem 3·ll).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 3 , May 1993 , pp. 507 - 517
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Baker, J. W. and Milnes, P.. The ideal structure of the Stone–Čech compactifloation of a group. Math. Proc. Cambridge Philos. Soc. 82 (1977), 401–409.CrossRefGoogle Scholar
[2]Berglund, J. F., Junghenn, H. D. and Milnes, P.. Compact Right Topological Semigroups and Generalizations of Almost Periodicity. Lecture Notes in Math. vol. 663 (Springer-Verlag, 1978).CrossRefGoogle Scholar
[3]Berglund, J. F., Junghenn, H. D. and Milnes, P.. Analysis on Semigroups: Function Spaces, Compactifications, Representations (Wiley, 1989).Google Scholar
[4]Granirer, E.. On the invariant mean on topological semigroups and topological groups. Pacific J. Math. 15 (1965), 107–140.CrossRefGoogle Scholar
[5]Granirer, E.. Exposed Points of Convex Sets and Weak Sequential Convergence. Memoirs Amer. Math. Soc. no. 123 (American Mathematical Society, 1972).CrossRefGoogle Scholar
[6]Granirer, E. and Lau, A. T.. Invariant means on locally compact groups. Illinois J. Math. 15 (1971), 249–257.CrossRefGoogle Scholar
[7]Helmer, D. and Isik, N.. Construction of right topological compactifications for discrete versions of subsemigroups of compact groups. Semigroup Forum 39 (1989), 65–73.CrossRefGoogle Scholar
[8]Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis, vol. 1 (Springer-Verlag, 1970).Google Scholar
[9]Lau, A. T., Milnes, P. and Pym, J.. Compactifications of locally compact groups and closed subgroups. Trans. Amer. Math. Soc. 329 (1992), 97–115.CrossRefGoogle Scholar
[10]Milnes, P. and Pym, J. S.. Haar measure for compact right topological groups. Proc. Amer. Math. Soc. 114 (1992), 387–393.CrossRefGoogle Scholar
[11]Palmer, T. W.. Classes of nonabelian, noncompact locally compact groups. Rocky Mountain J. Math. 8 (1973), 683–741.Google Scholar
[12]Pym, J. S.. Compact semigroups with one-sided continuity. In The Analytical and Topological Theory of Semigroups (De Gruyter, 1990), pp. 197–217.CrossRefGoogle Scholar
[13]Rudin, W.. Fourier Analysis on Groups (Interscience, 1960).Google Scholar
[14]Ruppert, W.. On semigroup compactifications of topological groups. Proc. Roy. Irish Acad. Sect. A 79 (1979), 17–200.Google Scholar
[15]Walker, R. C.. The Stone–Cech Compactification (Springer-Verlag, 1974).CrossRefGoogle Scholar