is uniformly Kadec-Klee

Author:
Chris Lennard

Journal:
Proc. Amer. Math. Soc. **109** (1990), 71-77

MSC:
Primary 46B20; Secondary 47D25

DOI:
https://doi.org/10.1090/S0002-9939-1990-0943795-4

MathSciNet review:
943795

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Abstract: A dual Banach space is Kadec-Klee in the weak * topology if weak * and norm convergence of sequences coincide in the unit sphere of . We shall consider a stronger, uniform version of this property. A dual Banach space is uniformly Kadec-Klee in the weak * topology (UKK*) if for each we can find a in such that every weak -compact, convex subset of the unit ball of whose measure of norm compactness exceeds must meet the -ball of . We show in this paper that , the space of trace class operators on an arbitrary infinite-dimensional Hilbert space is UKK*. Consequently has weak -normal structure. This answers affirmatively a question of A. T. Lau and P. F. Mah. From this it follows that has the weak -fixed point property.

**[1]**Jonathan Arazy,*More on convergence in unitary matrix spaces*, Proc. Amer. Math. Soc.**83**(1981), no. 1, 44–48. MR**619978**, https://doi.org/10.1090/S0002-9939-1981-0619978-4**[2]**D. van Dulst and Brailey Sims,*Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK)*, Banach space theory and its applications (Bucharest, 1981) Lecture Notes in Math., vol. 991, Springer, Berlin-New York, 1983, pp. 35–43. MR**714171****[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**, https://doi.org/10.4153/CJM-1986-038-4**[4]**I. C. Gohberg and M. G. Kreĭn,*Introduction to the theory of linear nonselfadjoint operators*, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR**0246142****[5]**R. Huff,*Banach spaces which are nearly uniformly convex*, Rocky Mountain J. Math.**10**(1980), no. 4, 743–749. MR**595102**, https://doi.org/10.1216/RMJ-1980-10-4-743**[6]**M. Ĭ. Kadec′,*On the connection between weak and strong convergence*, Dopovidi Akad. Nauk Ukraïn. RSR**1959**(1959), 949–952 (Ukrainian, with Russian and English summaries). MR**0112021****[7]**V. Klee,*Mappings into normed linear spaces*, Fund. Math.**49**(1960/1961), 25–34. MR**0126690**, https://doi.org/10.4064/fm-49-1-25-34**[8]**Anthony To Ming Lau and Peter F. Mah,*Quasinormal structures for certain spaces of operators on a Hilbert space*, Pacific J. Math.**121**(1986), no. 1, 109–118. MR**815037****[9]**Joram Lindenstrauss and Lior Tzafriri,*Classical Banach spaces. I*, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR**0500056****[10]**Robert Schatten,*Norm ideals of completely continuous operators*, Ergebnisse der Mathematik und ihrer Grenzgebiete. N. F., Heft 27, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR**0119112****[11]**B. Simon,*Convergence in trace ideals*, Proc. Amer. Math. Soc.**83**(1981), no. 1, 39–43. MR**619977**, https://doi.org/10.1090/S0002-9939-1981-0619977-2**[12]**Brailey Sims,*Fixed points of nonexpansive maps on weak and weak * compact sets*, Lecture Notes, Queen's Univ., Kingston, 1982.**[13]**-,*The existence question for fixed points of nonexpansive maps*, Lecture Notes, Kent State Univ., 1986.

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DOI:
https://doi.org/10.1090/S0002-9939-1990-0943795-4

Article copyright:
© Copyright 1990
American Mathematical Society