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

TL;DR

This paper investigates the structure of Banach spaces of continuous functions on compact metric spaces, establishing conditions for complemented subspaces and quotients, and classifying these spaces up to isomorphism.

Contribution

It extends classical results by characterizing when certain C(K,X) spaces contain complemented copies or are quotients of others, providing a complete classification for specific cases.

Findings

C(βN×[1,ω],X) contains a complemented C(ω^ω) only if X contains c_0

C(ω^ω) is not a quotient of C(βN×[1,ω],l_p) for 1<p<∞

Separable C(K) spaces are classified as complemented subspaces or quotients of C(βN×[1,α],l_p)

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.