On operators on $C_0(\alpha\times L)$ under the Ostaszewski's $\clubsuit$-principle

TL;DR

This paper studies the structure of operators on certain function spaces constructed under the Ostaszewski's $\u2663$-principle, revealing their simplicity and classifying their complemented subspaces.

Contribution

It demonstrates that all operators on $C_0(eta imes L)$ are trivial and classifies all complemented subspaces of these spaces, under specific set-theoretic assumptions.

Findings

All operators on $C_0(eta imes L)$ are trivial.

The space $C_0(eta imes L)$ has a well-understood geometric structure.

All complemented subspaces are classified up to isomorphism.

Abstract

For an exotic locally compact Hausdorff space $L$, constructed under the assumption of the Ostaszewski's $\clubsuit$-principle, and a countable ordinal space $\alpha$, we prove that all operators defined on $C_0(\alpha\times L)$ are as simple as possible. We also investigate the geometry of such space $C_0(\alpha\times L)$ and we classify up to isomorphisms all its complemented subspaces.