$C(K)$-spaces with few operators relative to posets

TL;DR

This paper develops a method to construct $C(K)$-spaces with limited operators based on posets, enabling classification of operator ideals and addressing automatic continuity questions.

Contribution

It introduces a novel construction of $C(K)$-spaces relative to posets, allowing for classification of operator ideals and insights into automatic continuity.

Findings

Constructed $C(K)$-spaces with few operators relative to posets.

Classified closed operator ideals in these spaces.

Resolved questions on automatic continuity.

Abstract

Extending a method developed by Koszmider and Laustsen for constructing $C(K)$-spaces we produce families of $C(K)$-spaces with few operators relative to a partially ordered set $\mathcal{P}$. Using these spaces, we construct new $C(K)$-spaces whose closed operator ideals can be completely classified. Additionally, we use these spaces to resolve some questions regarding automatic continuity.