On Subspaces and Quotients of Banach Spaces C(K, X)
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We study the relation of \(\) to the subspaces and quotients of Banach spaces of continuous vector-valued functions \(\), where K is an arbitrary dispersed compact set. More precisely, we prove that every infinite dimensional closed subspace of \(\) totally incomparable to X contains a copy of \(\) complemented in \(\). This is a natural continuation of results of Cembranos-Freniche and Lotz-Peck-Porta. We also improve our result when K is homeomorphic to an interval of ordinals. Next we show that complemented subspaces (resp., quotients) of \(\) which contain no copy of \(\) are isomorphic to complemented subspaces (resp., quotients) of some finite sum of X. As a consequence, we prove that every infinite dimensional quotient of \(\) which is quotient incomparable to X, contains a complemented copy of \(\). Finally we present some more geometric properties of \(\) spaces.
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