Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T07:25:04.507Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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