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Uncomplemented copies of $$C(K)$$ inside $$C(K)$$. (English) Zbl 0899.46006
An old theorem of A. Sobczyk [Bull. Am. Math. Soc. 47, 938-947 (1941; Zbl 0027.40801)] implies that if $$K$$ is the one-point compactification of the natural numbers then $$C(K)$$ (the space of convergent sequences) is always complemented in any separable space $$X$$ that contains it. Later, A. Pelczynski [Stud. Math. 31, 513-522 (1968; Zbl 0169.15402)] showed that if $$K$$ is a compact metric space and if a separable Banach space $$X$$ contains a subspace $$Y$$ isomorphic to $$C(K)$$ then $$Y$$ contains a subspace $$Z$$ that is both isomorphic to $$C(K)$$ and complemented in $$X$$. Here the author proves an obversation to Pelczynski’s result as follows: If $$K$$ is a compact metric space with $$K^{(\omega)} \neq \emptyset$$ and if a Banach space $$X$$ contains a subspace $$Y$$ isomorphic to $$C(K)$$ then $$Y$$ contains a new subspace $$Z$$ that is both isomorphic to $$C(K)$$ and is not complemented in $$Y$$ and hence not in $$X$$. Note that $$K^{(1)}:= K'$$ (the accumulation points of $$K$$), $$K^{(n+1)}:= (K^{(n)})'$$ and $$K^{(\omega)}:= \bigcap K^{(n)}$$. The condition on $$K^{(\omega)}$$ is needed to avoid the Sobczyk result.
##### MSC:
 46B20 Geometry and structure of normed linear spaces 46B25 Classical Banach spaces in the general theory