A hierarchy of Banach spaces with $C(K)$ Calkin Algebras

TL;DR

This paper constructs a hierarchy of Banach spaces with Calkin algebras isomorphic to continuous function algebras on certain trees and compact spaces, revealing new connections between Banach space theory and function algebras.

Contribution

It introduces a method to realize Calkin algebras as $C(K)$ for various compact spaces, expanding the understanding of Banach space structures and their operator algebras.

Findings

Calkin algebra of $X_{\mathcal{T}}$ is homomorphic to $C(\mathcal{T})$

For every countable compact metric space $K$, there exists an $\mathcal{L}_\infty$-space with Calkin algebra isomorphic to $C(K)$

Establishes a hierarchy linking Banach spaces with function algebras on trees and compact spaces.

Abstract

For every well founded tree $\mathcal{T}$ having a unique root such that every non-maximal node of it has countable infinitely many immediate successors, we construct a $\mathcal{L}_\infty$-space $X_{\mathcal{T}}$. We prove that for each such tree $\mathcal{T}$, the Calkin algebra of $X_{\mathcal{T}}$ is homomorphic to $C(\mathcal{T})$, the algebra of continuous functions defined on $\mathcal{T}$, equipped with the usual topology. We use this fact to conclude that for every countable compact metric space $K$ there exists a $\mathcal{L}_\infty$-space whose Calkin algebra is isomorphic, as a Banach algebra, to $C(K)$.