Operator Hilbert spaces without the operator approximation property

Arias, Alvaro

doi:10.1090/S0002-9939-02-06387-6

Operator Hilbert spaces without the operator approximation property
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Proc. Amer. Math. Soc. 130 (2002), 2669-2677 Request permission

Abstract:

We use a technique of Szankowski to construct operator Hilbert spaces that do not have the operator approximation property, including an example in a noncommutative $L_p$ space for $p\not =2$.

References

David P. Blecher and Vern I. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991), no. 2, 262–292. MR 1121615, DOI 10.1016/0022-1236(91)90042-4
Edward G. Effros and Zhong-Jin Ruan, A new approach to operator spaces, Canad. Math. Bull. 34 (1991), no. 3, 329–337. MR 1127754, DOI 10.4153/CMB-1991-053-x
Edward G. Effros and Zhong-Jin Ruan, On approximation properties for operator spaces, Internat. J. Math. 1 (1990), no. 2, 163–187. MR 1060634, DOI 10.1142/S0129167X90000113
E. Effros and Z.J. Ruan, Operator spaces, Oxford University Press, 2000.
Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. MR 402468, DOI 10.1007/BF02392270
M. Junge, Factorization theory for operator spaces, Habilitation thesis, Kiel University, 1996.
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
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
Robert F. Olin and James E. Thomson, Cellular-indecomposable subnormal operators. II, Integral Equations Operator Theory 9 (1986), no. 4, 600–609. MR 853632, DOI 10.1007/BF01204632
Françoise Lust-Piquard and Gilles Pisier, Noncommutative Khintchine and Paley inequalities, Ark. Mat. 29 (1991), no. 2, 241–260. MR 1150376, DOI 10.1007/BF02384340
Gilles Pisier, The operator Hilbert space $\textrm {OH}$, complex interpolation and tensor norms, Mem. Amer. Math. Soc. 122 (1996), no. 585, viii+103. MR 1342022, DOI 10.1090/memo/0585
Gilles Pisier, Non-commutative vector valued $L_p$-spaces and completely $p$-summing maps, Astérisque 247 (1998), vi+131 (English, with English and French summaries). MR 1648908
G. Pisier, An introduction to the theory of operator spaces (to appear).
A. Szankowski, Subspaces without the approximation property, Israel J. Math. 30 (1978), no. 1-2, 123–129. MR 508257, DOI 10.1007/BF02760833