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Weakly compact subsets of symmetric operator spaces

Published online by Cambridge University Press:  24 October 2008

Peter G. Dodds ,
Theresa K. Dodds  and
Ben De Pagter
Peter G. Dodds
Affiliation:
School of Information Science & Technology, The Flinders University of South Australia, GPO Box 2100, Adelaide 5001, Australia
Theresa K. Dodds
Affiliation:
School of Information Science & Technology, The Flinders University of South Australia, GPO Box 2100, Adelaide 5001, Australia
Ben De Pagter
Affiliation:
Department of Mathematics, Delft University of Technology, Julianalaan 132, NL-2628 BL Delft, The Netherlands
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Abstract

Under natural conditions it is shown that the rearrangement invariant hull of a weakly compact subset of a properly symmetric Banach space of measurable operators affiliated with a semi-finite von Neumann algebra is again relatively weakly compact.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 1 , July 1991 , pp. 169 - 182
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Aliprantis, C. D. and Burkinshaw, O.. Positive Operators (Academic Press, 1985).Google Scholar
[2]Akemann, C. A.. The dual space of an operator algebra. Trans. Amer. Math. Soc. 126 (1967), 286–302.Google Scholar
[3]Ando, T.. On fundamental properties of a Banach space with a cone. Pacific J. Math. 12 (1962), 1163–1169.CrossRefGoogle Scholar
[4]Chong, K. M.. Spectral orders, uniform integrability and Lebesgue's dominated convergence theorem. Trans. Amer. Math. Soc. 191 (1974), 395–404.Google Scholar
[5]Dodds, P. G., Dodds, T. K. and Pagter, B. de. Non-commutative Banach function spaces. Math. Z. 201 (1989), 583–597.CrossRefGoogle Scholar
[6]Dodds, P. G., Dodds, T. K. and Pagter, B. de. A general Markus inequality. Proc. CMA (ANU), Miniconference on Operators in Analysis 24 (1989), 47–57.Google Scholar
[7]Dodds, P. G., Dodds, T. K. and Pagter, B. de. Non-commutative Köthe duality. TU Delft (1990).Google Scholar
[8]Diestel, J.. Sequences and Series in Banach Space. Graduate Texts in Math. no. 92 (Springer-Verlag, 1984).CrossRefGoogle Scholar
[9]Dodds, P. G. and Lennard, C. J.. Normality in trace ideals. J. Operator Theory 16 (1986), 127–145.Google Scholar
[10]Dunford, N. and Schwartz, J.. Linear Operators, Part I (Wiley-Interscience, 1964).Google Scholar
[11]Fack, T. and Kosaki, H.. Generalized s-numbers of τ-measurable operators. Pacific J. Math. 123 (1986), 269–300.CrossRefGoogle Scholar
[12]Fremlin, D. H.. Topological Riesz Spaces and Measure Theory (Cambridge University Press, 1974).CrossRefGoogle Scholar
[13]Fremlin, D. H.. Stable subspaces of L 1+L ∞. Proc. Cambridge Philos. Soc. 64 (1968), 625–643.Google Scholar
[14]Garling, D. J. H.. On symmetric sequence spaces. Proc. London Math. Soc. 16 (1966), 85–106.CrossRefGoogle Scholar
[15]Garling, D. J. H.. On ideals of operators in Hilbert space. Proc. London Math. Soc. 17 (1967), 115–138.Google Scholar
[16]Grothendieck, A.. Topological Vector Spaces (Gordon and Breach, 1973).Google Scholar
[17]Krein, S. G., Petunin, Ju. I. and Semenov, E. M.. Interpolation of linear operators. Translations of Mathematical Monographs. Amer. Math. Soc. 54 (1982).Google Scholar
[18]Luxemburg, W. A. J.. Rearrangement invariant Banach function spaces. Queen's Papers in Pure and Appl. Math. no. 10 (1967), 83–144.Google Scholar
[19]Nelson, E.. Notes on non-commutative integration. J. Fund. Anal. 15 (1974), 103–116.Google Scholar
[20]Ovčinnikov, V. I.. s-numbers of measurable operators. Funktsional. Anal. i Prilozhen. 4 (1970), 78–85.Google Scholar
[21]Ovčinnikov, V. I.. Symmetric spaces of measurable operators. Dokl. Akad. Nauk SSSR 191 (1970), 769–771 (Russian)Google Scholar
Ovčinnikov, V. I.. English Translation: Soviet Math. Dokl. 11 (1970), 448–451.Google Scholar
[22]Ryff, J. V.. Orbits of L 1-functions under doubly stochastic transformations. Trans. Amer. Math. Soc. 117 (1965), 92–100.Google Scholar
[23]Sakai, S.. Topological properties of W*-algebras. Proc. Japan Acad. 33 (1957), 439–444.Google Scholar
[24]Sukochew, F.. Construction of non-commutative symmetric spaces. Dokl. Akad. Nauk UzSSR 8 (1986), 4–6.Google Scholar
[25]Takesaki, M.. Theory of Operator Algebras I (Springer-Verlag, 1979).Google Scholar
[26]Terp, M.. L p-spaces associated with von Neumann algebras. Notes, Copenhagen University, (1981).Google Scholar
[27]Yeadon, F. J.. Ergodic theorems for semifinite von Neumann algebras: II. Math. Proc. Cambridge Philos. Soc. 88 (1980), 135–147.CrossRefGoogle Scholar
[28]Zaanen, A. C.. Integration (North-Holland, 1967).Google Scholar