The lifting of the UKK property from 𝐸 to 𝐶_{𝐸}

Hsu, Yu-Ping

doi:10.1090/S0002-9939-1995-1246527-2

The lifting of the UKK property from $E$ to $C_ E$
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Proc. Amer. Math. Soc. 123 (1995), 2695-2703 Request permission

Abstract:

In this paper we show that ${C_E}$, the unitary matrix space associated with the symmetrically normed sequence space E, has the UKK property for the weak operator topology if E has the UKK property for the pointwise convergence topology. We also prove that the quasi-normed space ${C_p} = {C_{{l_p}}}$, for $0 < p < 1$, has the UKK property for the weak operator topology.

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

A uniform Kadec-Klee property for symmetric operator space

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
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142, DOI 10.1090/mmono/018
I. C. Gohberg and A. S. Markus, Some relations between eigenvalues and matrix elements of linear operators, Mat. Sb. (N.S.) 64 (106) (1964), 481–496 (Russian). MR 0170218
Robert C. James, Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), 542–550. MR 173932, DOI 10.2307/1970663
Chris Lennard, ${\scr C}_1$ is uniformly Kadec-Klee, Proc. Amer. Math. Soc. 109 (1990), no. 1, 71–77. MR 943795, DOI 10.1090/S0002-9939-1990-0943795-4
Chris Lennard, A new convexity property that implies a fixed point property for $L_1$, Studia Math. 100 (1991), no. 2, 95–108. MR 1121710, DOI 10.4064/sm-100-2-95-108
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
Charles A. McCarthy, $c_{p}$, Israel J. Math. 5 (1967), 249–271. MR 225140, DOI 10.1007/BF02771613
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, DOI 10.1007/978-3-642-87652-3
Barry Simon, Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge-New York, 1979. MR 541149, DOI 10.1007/BFb0064579
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