Publication Date:
2002-10-14
Description:
We investigate in this paper the complementation of copies of $c_0(I)$ in some classes of Banach spaces (in the class of weakly compactly generated (WCG) Banach spaces, in the larger class $mathcal{V}$ of Banach spaces which are subspaces of some $C(K)$ space with $K$ a Valdivia compact, and in the Banach spaces $C([1, alpha ])$, where $alpha$ is an ordinal) and the embedding of $c_0(I)$ in the elements of the class $mathcal{C}$ of complemented subspaces of $C(K)$ spaces. Two of our results are as follows:(i) in a Banach space $X in mathcal{V}$ every copy of $c_0(I)$ with $# I 〈 aleph _{omega}$ is complemented;(ii) if $alpha _0 = aleph _0$, $alpha _{n+1} = 2^{alpha _n}$, $n geq 0$, and $alpha = sup {alpha _n : n geq 0}$ there exists a WCG Banach space with an uncomplemented copy of $c_0(alpha )$.So, under the generalized continuum hypothesis (GCH), $aleph _{omega}$ is the greatest cardinal $au$ such that every copy of $c_0(I)$ with $# I 〈 au$ is complemented in the class $mathcal{V}$. If $T : c_0(I) o C([1,alpha ])$ is an isomorphism into its image, we prove that:(i) $c_0(I)$ is complemented, whenever $| T | ,| T^{-1} | 〈 (3/2)^{frac 12}$;(ii) there is a finite partition ${I_1, dots , I_k}$ of $I$ such that each copy $T(c_0(I_k))$ is complemented.Concerning the class $mathcal{C}$, we prove that an already known property of $C(K)$ spaces is still true for this class, namely, if $X in mathcal{C}$, the following are equivalent:(i) there is a weakly compact subset $W subset X$ with ${
m Dens}(W) = au$;(ii) $c_0(au )$ is isomorphically embedded into $X$.This yields a new characterization of a class of injective Banach spaces.2000 Mathematical Subject Classification: 46B20, 46B26.
Print ISSN:
0024-6115
Electronic ISSN:
1460-244X
Topics:
Mathematics
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