The non-commutative Yosida-Hewitt decomposition revisited
HTML articles powered by AMS MathViewer
- by P. G. Dodds and B. de Pagter PDF
- Trans. Amer. Math. Soc. 364 (2012), 6425-6457 Request permission
Abstract:
In this paper, a new approach to the non-commutative Yosida-Hewitt decomposition is presented in the general setting of non-commutative symmetric spaces of $\tau$-measurable operators affiliated with semi-finite von Neumann algebras. The principal theorem permits the systematic study of the spaces of normal and singular functionals in this general setting. These results are used to study the properties of elements of order continuous norm and of absolutely continuous norm.References
- T. Andô, On fundamental properties of a Banach space with a cone, Pacific J. Math. 12 (1962), 1163–1169. MR 150572
- A. M. Bikchentaev, The continuity of multiplication for two topologies associated with a semifinite trace on von Neumann algebra, Lobachevskii J. Math. 14 (2004), 17–24. MR 2034258
- V. I. Chilin and F. A. Sukochev, Weak convergence in non-commutative symmetric spaces, J. Operator Theory 31 (1994), no. 1, 35–65. MR 1316983
- Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
- J. Dixmier, Les fonctionnelles linéaires sur l’ensemble des opérateurs bornés d’un espace de Hilbert, Ann. of Math. (2) 51 (1950), 387–408 (French). MR 33445, DOI 10.2307/1969331
- Jacques Dixmier, von Neumann algebras, North-Holland Mathematical Library, vol. 27, North-Holland Publishing Co., Amsterdam-New York, 1981. With a preface by E. C. Lance; Translated from the second French edition by F. Jellett. MR 641217
- 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, A general Markus inequality, Miniconference on Operators in Analysis (Sydney, 1989) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 24, Austral. Nat. Univ., Canberra, 1990, pp. 47–57. MR 1060110
- Peter G. Dodds, Theresa K. Dodds, and Ben de Pagter, Weakly compact subsets of symmetric operator spaces, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 1, 169–182. MR 1104612, DOI 10.1017/S0305004100070225
- 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, 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 and C. J. Lennard, Normality in trace ideals, J. Operator Theory 16 (1986), no. 1, 127–145. MR 847335
- Peter G. Dodds and Ben de Pagter, Non-commutative Yosida-Hewitt theorems and singular functionals in symmetric spaces of $\tau$-measurable operators, Vector measures, integration and related topics, Oper. Theory Adv. Appl., vol. 201, Birkhäuser Verlag, Basel, 2010, pp. 183–198. MR 2743986
- Thierry Fack and Hideki Kosaki, Generalized $s$-numbers of $\tau$-measurable operators, Pacific J. Math. 123 (1986), no. 2, 269–300. MR 840845
- J. Grosberg and M. Krein, Sur la décomposition des fonctionnelles en composantes positives, C. R. (Doklady) Acad. Sci. URSS (N.S.) 25 (1939), 723–726 (French). MR 0002019
- Graham Jameson, Ordered linear spaces, Lecture Notes in Mathematics, Vol. 141, Springer-Verlag, Berlin-New York, 1970. MR 0438077
- Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR 719020
- 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
- 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
- Wilhelmus Anthonius Josephus Luxemburg, Banach function spaces, Technische Hogeschool te Delft, Delft, 1955. Thesis. MR 0072440
- Ben de Pagter, Non-commutative Banach function spaces, Positivity, Trends Math., Birkhäuser, Basel, 2007, pp. 197–227. MR 2382219, DOI 10.1007/978-3-7643-8478-4_{7}
- Narcisse Randrianantoanina, Sequences in non-commutative $L^p$-spaces, J. Operator Theory 48 (2002), no. 2, 255–272. MR 1938797
- Şerban Strătilă and László Zsidó, Lectures on von Neumann algebras, Editura Academiei, Bucharest; Abacus Press, Tunbridge Wells, 1979. Revision of the 1975 original; Translated from the Romanian by Silviu Teleman. MR 526399
- Masamichi Takesaki, On the conjugate space of operator algebra, Tohoku Math. J. (2) 10 (1958), 194–203. MR 100799, DOI 10.2748/tmj/1178244713
- Masamichi Takesaki, On the singularity of a positive linear functional on operator algebra, Proc. Japan Acad. 35 (1959), 365–366. MR 113153
- Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728
- M. Takesaki, Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, vol. 125, Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 6. MR 1943006, DOI 10.1007/978-3-662-10451-4
- M. Terp, $L^{p}$ spaces associated with von Neumann algebras, Notes, Copenhagen Univ., 1981.
- Kôsaku Yosida and Edwin Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952), 46–66. MR 45194, DOI 10.1090/S0002-9947-1952-0045194-X
- Adriaan Cornelis Zaanen, Integration, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1967. Completely revised edition of An introduction to the theory of integration. MR 0222234
- A. C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, North-Holland Publishing Co., Amsterdam, 1983. MR 704021
Additional Information
- P. G. Dodds
- Affiliation: School of Computer Science, Engineering and Mathematics, Flinders University, GPO Box 2100, Adelaide 5001, Australia
- Email: peter@csem.flinders.edu.au
- B. de Pagter
- Affiliation: Delft Institute of Applied Mathematics, Faculty EEMCS, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
- Email: b.depagter@tudelft.nl
- Received by editor(s): May 3, 2010
- Received by editor(s) in revised form: February 14, 2011
- Published electronically: June 26, 2012
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 364 (2012), 6425-6457
- MSC (2010): Primary 46L52, 46L51; Secondary 46E30
- DOI: https://doi.org/10.1090/S0002-9947-2012-05569-3
- MathSciNet review: 2958942