The Dixmier approximation theorem in algebras of measurable operators
HTML articles powered by AMS MathViewer
- by Guy H. Flint, Ben de Pagter and Fedor A. Sukochev PDF
- Proc. Amer. Math. Soc. 141 (2013), 909-918 Request permission
Abstract:
In this paper we are concerned with proving versions of the classical Dixmier approximation theorem in the setting of algebras of $\tau$-measurable operators $S\left ( \mathcal {M},\tau \right )$ and its $\mathcal {M}$-bimodules, where $\mathcal {M}$ is a semi-finite von Neumann algebra equipped with a semi-finite normal faithful trace $\tau$.References
- J. Dixmier, Les anneaux d’opérateurs de classe finie, Ann. Sci. École Norm. Sup. (3) 66 (1949), 209–261 (French). MR 0032940
- Peter G. Dodds, Theresa K.-Y. Dodds, and Ben de Pagter, Noncommutative Banach function spaces, Math. Z. 201 (1989), no. 4, 583–597. MR 1004176, DOI 10.1007/BF01215160
- Peter G. Dodds, Theresa K.-Y. Dodds, and Ben de Pagter, Noncommutative Köthe duality, Trans. Amer. Math. Soc. 339 (1993), no. 2, 717–750. MR 1113694, DOI 10.1090/S0002-9947-1993-1113694-3
- P. G. Dodds, T. K. Dodds, and F. A. Sukochev, Banach-Saks properties in symmetric spaces of measurable operators, Studia Math. 178 (2007), no. 2, 125–166. MR 2285436, DOI 10.4064/sm178-2-2
- P. G. Dodds, T. K. Dodds, F. A. Sukochev, and O. Ye. Tikhonov, A non-commutative Yosida-Hewitt theorem and convex sets of measurable operators closed locally in measure, Positivity 9 (2005), no. 3, 457–484. MR 2188531, DOI 10.1007/s11117-005-1384-0
- P.G. Dodds, B. de Pagter and F.A. Sukochev, Non-commutative Integration, in progress.
- Thierry Fack and Hideki Kosaki, Generalized $s$-numbers of $\tau$-measurable operators, Pacific J. Math. 123 (1986), no. 2, 269–300. MR 840845
- F. Hiai, Spectral majorization between normal operators in von Neumann algebras, Operator algebras and operator theory (Craiova, 1989) Pitman Res. Notes Math. Ser., vol. 271, Longman Sci. Tech., Harlow, 1992, pp. 78–115. MR 1189168
- Fumio Hiai and Yoshihiro Nakamura, Closed convex hulls of unitary orbits in von Neumann algebras, Trans. Amer. Math. Soc. 323 (1991), no. 1, 1–38. MR 984856, DOI 10.1090/S0002-9947-1991-0984856-9
- Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR 859186, DOI 10.1016/S0079-8169(08)60611-X
- N. J. Kalton and F. A. Sukochev, Symmetric norms and spaces of operators, J. Reine Angew. Math. 621 (2008), 81–121. MR 2431251, DOI 10.1515/CRELLE.2008.059
- S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR 649411
- Edward Nelson, Notes on non-commutative integration, J. Functional Analysis 15 (1974), 103–116. MR 0355628, DOI 10.1016/0022-1236(74)90014-7
- Adam Paszkiewicz, Convergence almost everywhere in $W^\ast$-algebras, Quantum probability and applications, II (Heidelberg, 1984) Lecture Notes in Math., vol. 1136, Springer, Berlin, 1985, pp. 420–427. MR 819522, DOI 10.1007/BFb0074490
- M. Terp, $L^{p}$-spaces associated with von Neumann algebras, Notes, Copenhagen University (1981).
Additional Information
- Guy H. Flint
- Affiliation: School of Mathematics and Statistics, University of New South Wales, Kensington 2052, NSW, Australia
- Email: guy.flint@gmail.com
- Ben de Pagter
- Affiliation: Delft Institute of Applied Mathematics, Delft University of Technology, Fac. EEMCS, Mekelweg 4, 2628CD Delft, The Netherlands
- Email: b.depagter@tudelft.nl
- Fedor A. Sukochev
- Affiliation: School of Mathematics and Statistics, University of New South Wales, Kensington 2052, NSW, Australia
- MR Author ID: 229620
- Email: f.sukochev@unsw.edu.au
- Received by editor(s): July 14, 2011
- Published electronically: July 2, 2012
- Additional Notes: The work of the third author was supported by the ARC
- Communicated by: Marius Junge
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 909-918
- MSC (2010): Primary 46L51, 47B10; Secondary 46L52
- DOI: https://doi.org/10.1090/S0002-9939-2012-11479-0
- MathSciNet review: 3003683