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

TL;DR

This paper investigates Banach spaces of continuous functions on products of compact lines, showing how their topological properties determine isomorphism or embeddability, especially distinguishing spaces based on the number of factors.

Contribution

It introduces a method to distinguish Banach spaces of continuous functions on nonmetrizable cubes using topological characteristics of the underlying compact lines.

Findings

Spaces with different numbers of factors are not isomorphic.

Topological properties of compact lines influence the structure of function spaces.

The work provides criteria for non-isomorphism based on product dimensions.

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.