Three-space problems for the approximation properties

Godefroy, Gilles; Saphar, Pierre David

doi:10.1090/S0002-9939-1989-0930249-6

Three-space problems for the approximation properties
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by Gilles Godefroy and Pierre David Saphar

Proc. Amer. Math. Soc. 105 (1989), 70-75

DOI: https://doi.org/10.1090/S0002-9939-1989-0930249-6

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Abstract:

Let $M$ be a closed subspace of a Banach space $X$ . We suppose that $M$ has the B.A.P. and that ${M^ \bot }$ is complemented in ${X^*}$. Then, if $X/M$ has the B.A.P. (resp. the A.P.), the space $X$ has the same property. There are similar results if $M$ is an ${\mathcal {L}_\infty }$ space. If $X/M$ is an ${\mathcal {L}_1}$ space, then $X$ has the B.A.P. if and only if $M$ has the B.A.P. We notice that the quotient algebra $L(H)/K(H)$ ( $H$ infinite-dimensional Hilbert space) does not have the A.P.

References

Bernard Beauzamy, Introduction to Banach spaces and their geometry, Notas de Matemática [Mathematical Notes], vol. 86, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 670943
David W. Dean, Approximation and weak-star approximation in Banach spaces, Bull. Amer. Math. Soc. 79 (1973), 725–728. MR 326355, DOI 10.1090/S0002-9904-1973-13283-5
Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. MR 402468, DOI 10.1007/BF02392270
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
Stefan Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72–104. MR 552464, DOI 10.1515/crll.1980.313.72
W. B. Johnson and H. P. Rosenthal, On $\omega ^{\ast }$-basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77–92. MR 310598, DOI 10.4064/sm-43-1-77-92
W. B. Johnson, H. P. Rosenthal, and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488–506. MR 280983, DOI 10.1007/BF02771464
Åsvald Lima, Uniqueness of Hahn-Banach extensions and liftings of linear dependences, Math. Scand. 53 (1983), no. 1, 97–113. MR 733942, DOI 10.7146/math.scand.a-12019
Joram Lindenstrauss, On James’s paper “Separable conjugate spaces”, Israel J. Math. 9 (1971), 279–284. MR 279567, DOI 10.1007/BF02771677
J. Lindenstrauss and H. P. Rosenthal, The ${\cal L}_{p}$ spaces, Israel J. Math. 7 (1969), 325–349. MR 270119, DOI 10.1007/BF02788865
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
Wolfgang Lusky, A note on Banach spaces containing $c_0$ or $C_\infty$, J. Funct. Anal. 62 (1985), no. 1, 1–7. MR 790767, DOI 10.1016/0022-1236(85)90016-3
Andrzej Szankowski, $B({\cal H})$ does not have the approximation property, Acta Math. 147 (1981), no. 1-2, 89–108. MR 631090, DOI 10.1007/BF02392870
T. Ando, A theorem on nonempty intersection of convex sets and its application, J. Approximation Theory 13 (1975), 158–166. MR 385520, DOI 10.1016/0021-9045(75)90049-0