Complementations in and \ell _\infty (X)
Volume 172 / 2023
Abstract
We investigate the geometry of C(K,X) and \ell _{\infty }(X) spaces through complemented subspaces of the form (\bigoplus _{i\in \varGamma }X_i)_{c_0}. For Banach spaces X and Y, we prove that if \ell _{\infty }(X) has a complemented subspace isomorphic to c_0(Y), then, for some n \in \mathbb N , X^n has a subspace isomorphic to c_0(Y). If K and L are Hausdorff compact spaces and X and Y are Banach spaces having no subspace isomorphic to c_0 we further prove the following:
(1) If C(K)\sim c_0(C(K)) and C(L)\sim c_0(C(L)) and \ell _{\infty }(C(K,X))\sim \ell _{\infty }(C(L,Y)), then K and L have the same cardinality.
(2) If K and L are infinite and metrizable and \ell _{\infty }(C(K,X))\sim \ell _{\infty }(C(L,Y)), then C(K) is isomorphic to C(L).