Geometry of the Banach spaces C(beta mathbb N times K, X) for compact metric spaces K

Abstract

A classical result of Cembranos and Freniche states that the C(K, X) spaces contains a complemented copy of c_0 whenever K is an infinite compact Hausdorff space and X is an infinite dimensional Banach space. This paper takes this result as a starting point and begins a study of the conditions under which the spaces C(alpha), alpha<omega_1 are quotients of or complemented in spaces C(K,X). In contrast to the c_0 result, we prove that if C(beta mathbb N times [1,omega], X) contains a complemented copy of C(omega^omega) then X contains a copy of c_0. Moreover, we show that C(omega^omega) is not even a quotient of C(beta mathbb N times [1,omega], l_p), 1<p< infinity. We then completely determine the separable C(K) spaces which are isomorphic to a complemented subspace or a quotient of the C(beta mathbb N times [1,alpha], l_p) spaces for countable ordinals \alpha and 1 <= p< infinity. As a consequence, we obtain the isomorphic classification of the C(beta mathbb N times K, l_p) spaces for infinite compact metric spaces K and 1 <= p < infinity. Indeed, we establish the following more general cancellation law. Suppose that the Banach space X contains no copy of c_0 and K_1 and K_2 are infinite compact metric spaces, then the following statements are equivalent: (1) C(beta mathbb N times K_1, X) is isomorphic to C(beta mathbb N times K_2, X) (2) C(K_1) is isomorphic to C(K_2). These results are applied to the isomorphic classification of some spaces of compact operators.