A stability property of a class of Banach spaces not containing a complemented copy of 𝑙₁

Saab, Elias; Saab, Paulette

doi:10.1090/S0002-9939-1982-0633274-1

A stability property of a class of Banach spaces not containing a complemented copy of $l_{1}$
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by Elias Saab and Paulette Saab PDF

Proc. Amer. Math. Soc. 84 (1982), 44-46 Request permission

Abstract:

Let $E$ be a Banach space and $K$ be a compact Hausdorff space. The space $C(K,E)$ will stand for the Banach space of all continuous $E$-valued functions on $K$ equipped with the sup norm. It is shown that the space $E$ does not contain a complemented subspace isomorphic to ${l_1}$ if and only if $C(K,E)$ has the same property.

References

C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
J. Bourgain, An averaging result for $c_{0}$-sequences, Bull. Soc. Math. Belg. 30 (1978), no. 1, 83–87. MR 549653
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189

Sur les espaces de Banach contenant

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Gilles Pisier, Une propriété de stabilité de la classe des espaces ne contenant pas $l^{1}$, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 17, A747–A749 (French, with English summary). MR 511805
Paulette Saab, The Choquet integral representation in the affine vector-valued case, Aequationes Math. 20 (1980), no. 2-3, 252–262. MR 577491, DOI 10.1007/BF02190517