Musing on Kunen's compact $L$-space

TL;DR

This paper constructs a connected compact $L$-space extending Kunen's work under CH, revealing a Corson compact space with a Banach space of continuous functions not isomorphic to any on a zero-dimensional compactum.

Contribution

It introduces a connected version of Kunen's compact $L$-space, expanding understanding of the structure of Corson compact spaces and their associated Banach spaces.

Findings

Constructed a connected compact $L$-space under CH.

Identified a Corson compact space with unique Banach space properties.

Showed $C(K)$ is not isomorphic to any space of continuous functions on a zero-dimensional compactum.

Abstract

We present a connected version of the compact $L$-space constructed by Kenneth Kunen under CH. We show that this provides a Corson compact space $K$ such that the Banach space $C(K)$ is isomorphic to no space of continuous function on a zero-dimensional compactum.