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By Spiros A. (gr-athn) Castillo, Jesús F. [castillo, Antonio S. [suárez Granero, Antonio (e-madc-ma, Mar [jiménez Sevilla, Mar (e-madc-ma) Moreno and José P. (e-madas


Complementation and embeddings of c0(I) in Banach spaces. Proc. London Math. Soc. (3) 85 (2002), no. 3, 742–768. In the paper under review, the authors consider some known properties of the space c0 with respect to the spaces c0(I), where I is an uncountable set. Let X be a Banach space and T: c0(I) → X be an isomorphic embedding. The authors prove several results connected with the solution of the problem: when is T (c0(I)) complemented in X? Besides the case X = C([1; k]), where k is an ordinal, they consider the case when X belongs to a rather wide class of spaces, namely the class of spaces isomorphic to subspaces of C(K), where K is a Valdivia compact. A. Pełczyński [Studia Math. 19 (1960), 209–228; MR0126145 (23 3441)] proved that every complemented subspace L ⊂ C(Q) contains a subspace E isomorphic to c0. Generalizing Pełczyński’s theorem, the authors prove a sufficient condition for a complemented subspace L ⊂ C(Q) to contain an isomorphic copy of c0(I). The results obtained are used for characterization of some class of nonseparable injective spaces. The results of the paper sharpen and complement some known results concerning the space c0(I). Several problems are also formulated. Reviewed by G. M. Ustino

Year: 2013
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