The three space problem for smooth partitions of unity and \(C(K)\) spaces (Q913194)

We show that a Banach space \(X\) admits \(C^ k\)-smooth partitions of unity provided there is a subspace \(Y\subset X\) such that \(Y\) is an isomorphic copy of some \(c_ 0(\Gamma)\) and \(X/Y\) admits \(C^ k\)-smooth partitions of unity. A \(C(K)\) space admits \(C^{\infty}\)-smooth partitions of unity and is Lipschitz equivalent to a \(c_ 0(\Gamma)\) space provided \(K^{(\omega_ 0)}=\emptyset\).