Semadeni-Pe{\l}czy\'nski derivative and functions on nonmetrizable cubes

Abstract

We study Banach spaces $C(K)$ of real-valued continuous functions from the finite product of compact lines. It turns out that the topological character of these compact lines can be used to distinguish whether two spaces of continuous functions on products are isomorphic or embeddable to each other. In particular, for compact lines $K_1, \dots, K_n, L_1, \dots, L_k$ of uncountable character and $k \neq n$, we claim that Banach spaces $C(\prod_{i=1}^n K_i)$ and $C(\prod_{j=1}^k L_j)$ are not isomorphic.