Lifting of Kadec-Klee properties to symmetric spaces of measurable operators

Dodds, P.; Dodds, T.; Sukochev, F.

doi:10.1090/S0002-9939-97-03731-3

Lifting of Kadec-Klee properties to symmetric spaces of measurable operators
HTML articles powered by AMS MathViewer

by P. G. Dodds, T. K. Dodds and F. A. Sukochev PDF

Proc. Amer. Math. Soc. 125 (1997), 1457-1467 Request permission

Abstract:

We show that if $E$ is a separable symmetric Banach function space on the positive half-line, then $E$ has the Kadec-Klee property (respectively, uniform Kadec-Klee property) for local convergence in measure if and only if, for every semifinite von Neumann algebra $( \mathcal {M}, \tau )$, the associated space $E(\mathcal {M},\tau )$ of $\tau$-measurable operators has the same property.

References

Jonathan Arazy, More on convergence in unitary matrix spaces, Proc. Amer. Math. Soc. 83 (1981), no. 1, 44–48. MR 619978, DOI 10.1090/S0002-9939-1981-0619978-4
Mustafa A. Akcoglu and Louis Sucheston, La monotonicité uniforme des normes et théorèmes ergodiques, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 7, 359–360 (French, with English summary). MR 808627
Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
V. I. Chilin, P. G. Dodds and F. A. Sukochev, The Kadec-Klee property in symmetric spaces of measurable operators, Israel J. Math., (to appear).
V. I. Chilin, P. G. Dodds, F. A. Sukochev and A. A. Sedaev, Characterisations of Kadec-Klee properties in symmetric spaces of measurable functions, 1994.
F. A. Sukochev and V. I. Chilin, Convergence in measure in regular noncommutative symmetric spaces, Izv. Vyssh. Uchebn. Zaved. Mat. 9 (1990), 63–70 (Russian); English transl., Soviet Math. (Iz. VUZ) 34 (1990), no. 9, 78–87. MR 1107444
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
Vladimir I. Chilin, Andrei V. Krygin, and Pheodor A. Sukochev, Extreme points of convex fully symmetric sets of measurable operators, Integral Equations Operator Theory 15 (1992), no. 2, 186–226. MR 1147280, DOI 10.1007/BF01204237
P. G. Dodds, T. K. Dodds, P. N. Dowling, C. J. Lennard, and F. A. Sukochev, A uniform Kadec-Klee property for symmetric operator spaces, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 3, 487–502. MR 1342966, DOI 10.1017/S0305004100073813
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
D. van Dulst and V. de Valk, (KK)-properties, normal structure and fixed points of nonexpansive mappings in Orlicz sequence spaces, Canad. J. Math. 38 (1986), no. 3, 728–750. MR 845675, DOI 10.4153/CJM-1986-038-4
Thierry Fack and Hideki Kosaki, Generalized $s$-numbers of $\tau$-measurable operators, Pacific J. Math. 123 (1986), no. 2, 269–300. MR 840845
Yu-Ping Hsu, The lifting of the UKK property from $E$ to $C_E$, Proc. Amer. Math. Soc. 123 (1995), no. 9, 2695–2703. MR 1246527, DOI 10.1090/S0002-9939-1995-1246527-2
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
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
A. A. Sedaev, Weak and strong convergence in interpolation spaces, Proceedings of the Sixth Winter School on Mathematical Programming and Related Questions (Drogobych, 1973) Akad. Nauk SSSR Central. Èkonom.-Mat. Inst., Moscow, 1975, pp. 245–267 (Russian). MR 0493416
B. Simon, Convergence in trace ideals, Proc. Amer. Math. Soc. 83 (1981), no. 1, 39–43. MR 619977, DOI 10.1090/S0002-9939-1981-0619977-2
F. A. Sukochev, On the uniform Kadec-Klee property with respect to convergence in measure, J. Austral. Math. Soc. Ser. A 59 (1995), no. 3, 343–352. MR 1355225
M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, New York-Heidelberg, Berlin, 1979.