Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-30T15:08:28.305Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant subspace theorems for amenable groups

Published online by Cambridge University Press:  20 January 2009

A. T. Lau ,
A. L. T. Paterson  and
J. C. S. Wong
A. T. Lau
Affiliation:
University of AlbertaAlberta, Canada
A. L. T. Paterson
Affiliation:
University of CalgaryCalgary, Canada
J. C. S. Wong
Affiliation:
University of AberdeenAberdeen, Scotland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [5], Ky Fan proved the following remarkable amenability “invariant subspace” theorem:

Let G be an amenable group of continuous, invertible linear operators acting on a locally convex space E. Let H be a closed subspace of finite codimension n in E and X⊂E be such that:

(i) H and X are G-invariant;

(ii) (e + H) ∩X is compact convex for all e ∈ E;

(iii) X contains an n-dimensional subspace V of E. Then there exists an n-dimensional subspace of E contained in X and invariant under G.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 32 , Issue 3 , October 1989 , pp. 415 - 430
Copyright
Copyright © Edinburgh Mathematical Society 1989

References

REFERENCES

1.Dixon, W. G., Special Relativity (Cambridge University Press, 1978).Google Scholar
2.Eymard, P., Moyennes invariantes et représentations unitaires (Lecture Notes in Mathematics, 300, Springer-Verlag, Berlin, 1972).CrossRefGoogle Scholar
3.Fan, K., Invariant subspaces of certain linear operators, Bull. Amer. Math. Soc. 69 (1963), 773–777.CrossRefGoogle Scholar
4.Fan, K., Invariant cross-sections and invariant linear subspaces, Israel J. Math. 2 (1964), 19–26.CrossRefGoogle Scholar
5.Fan, K., Invariant subspaces for a semigroup of linear operators, Indag. Math. 27 (1965), 447–451.CrossRefGoogle Scholar
6.Greenleaf, F. P., Invariant means on topological groups (Van Nostrand Math. Stud., Princeton, N.J., 1969).Google Scholar
7.Iohvidov, I. S., Unitary operators in a space with indefinite metric, Zapiski Har'kov. Mat. Obšč. 21 (1949), 79–86.Google Scholar
8.Köthe, G., Topological vector spaces I (Springer-Verlag, Berlin-Heidelberg-New York, 1969).Google Scholar
9.Krein, M. G., On an application of the fixed-point principle in the theory of linear transformations of spaces with an indefinite metric, Amer. Math. Soc. Transl. Ser. 2, 1 (1955), 27–35.Google Scholar
10.Lau, A. T., Finite-dimensional invariant subspaces for a semigroup of linear operators, J. Math. Anal. Appl. 97 (1983), 374–379.CrossRefGoogle Scholar
11.Lau, A. T. and Wong, J. C. S., Invariant subspaces for algebras of linear operators and amenable locally compact groups, 1986, preprint.Google Scholar
12.Lau, A. T. and Wong, J. C. S., Finite-dimensional invariant subspaces for measurable semigroups of linear operators, J. Math. Anal. Appl. 127 (1987), 548–558.CrossRefGoogle Scholar
13.Naimark, M. A., Commutative unitary operators on a πk-space, Soviet Math. 4 (1963) 543–545.Google Scholar
14.Naimark, M. A., On commuting unitary operators in spaces with indefinite metric, Acta Sci. Math. (Szeged) 24 (1963), 177–189.Google Scholar
15.Paterson, A. L. T., Amenability (Mathematical Surveys and Monographs, American Mathematical Society, 1988).CrossRefGoogle Scholar
16.Pier, J.-P., Amenable locally compact groups (John Wiley and Sons, New York, 1984).Google Scholar
17.Pontrjagin, L. S., Hermitian operators in spaces with indefinite metric, Izv. Akad. Nauk SSSR. Ser. Mat. 149 (1963), 1261–1263.Google Scholar