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Phillips in [4] proved that \(c_0\) is an uncomplemented subspace of \(l_\infty\). We do not find in the literature many classes of separable Hausdorff compact spaces \(K\) such that there exists a subspace \(X\) isomorphic to \(c_0\) and uncomplemented in \(C(K)\). Except \(\beta \mathbb N\) appears essentially only the class of Mrówka spaces
(see [3]). The reason is simple, usually it is quite hard to show the uncomplementability.
There is one general method to do it, it is a modification of the Whitley proof of the Phillips theorem (see [6]). The method based on the facts that any \(C(K)\) space, when \(K\) is separable and compact, does not contain any isomorphic copy of \(c_0(\Gamma)\) for any uncountable set \(\Gamma\) but the quotient space \(C(K)/X\) contains such a copy (see
[2], [6], [1]).
We construct for every \(n\geqslant 2\) a subspace \(X_n\) isometric to \(c_0\) and complemented in \(C(\mathbb L^n)\), the \(n\)-fold
product of two arrows space \(\mathbb L\), such that
\(
\inf\{\|P\|:P:C(\mathbb L^n)\to X_n\text{ is a projection}\}\geqslant n+2
\)
and the quotient space \(C(\mathbb L^n)/X_n\) has a \((3+ 4\sqrt{2})\) norming sequence of norm one functionals.
The inequality together with the last fact enables us to find an isometric to \(c_0\) and uncomplemented
subspace \(Y\) of \(C(\mathbb L^\mathbb N)\)
such that the quotient space
\(C(\mathbb L^\mathbb N)/Y\) is isomorphic to a subspace of \(l_\infty\).
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