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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
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