A Banach space $C(K)$ reading the dimension of $K$

TL;DR

Under Jensen's diamond principle, the paper constructs specific compact Hausdorff spaces $K$ such that the isomorphic Banach spaces $C(K)$ uniquely determine the dimension of any space $L$ with an isomorphic $C(L)$, revealing deep links between topology and Banach space structure.

Contribution

The paper introduces a method to construct compact spaces $K$ with $C(K)$ spaces that encode the dimension of other spaces via isomorphism, under set-theoretic assumptions.

Findings

Constructed spaces $K$ are separable and connected.

$C(K)$ spaces have few operators, with a specific form.

Isomorphism of $C(K)$ spaces implies equal dimension of underlying spaces.

Abstract

Assuming Jensen's diamond principle ($\diamondsuit$) we construct for every natural number $n>0$ a compact Hausdorff space $K$ such that whenever the Banach spaces $C(K)$ and $C(L)$ are isomorphic for some compact Hausdorff $L$, then the covering dimension of $L$ is equal to $n$. The constructed space $K$ is separable and connected, and the Banach space $C(K)$ has few operators i.e. every bounded linear operator $T:C(K)\rightarrow C(K)$ is of the form $T(f)=fg+S(f)$, where $g\in C(K)$ and $S$ is weakly compact.