Operator Algebras of Bourgain Delbaen Spaces: Realization, Rigidity, and Ideal Structure

TL;DR

This paper constructs reflexive Banach spaces with Calkin algebras isomorphic to continuous functions on compact spaces, revealing deep links between operator algebras, space geometry, and topology.

Contribution

It advances the realization of specific commutative C*-algebras as Calkin algebras, including stability under products and a classification of ideals, extending previous results.

Findings

Constructed reflexive Banach spaces with Calkin algebra isomorphic to C(K)

Established stability under finite products for Calkin algebras

Classified all closed two-sided and prime ideals in the operator algebra

Abstract

This manuscript presents a systematic study of Calkin algebras -- the quotients $\mathcal{L}(X)/\mathcal{K}(X)$ of bounded operators modulo compact operators on a Banach space $X$ -- and establishes a framework for realizing commutative $C^*$-algebras as such quotients while preserving geometric and topological information. Building on Motakis's reflexive version of the Bourgain--Delbaen construction, we prove that for every compact metric space $K$, there exists a reflexive Banach space $\mathfrak{X}_{C(K)}$ whose Calkin algebra is isomorphic to $C(K)$ as a Banach algebra. Our contributions advance this result in several directions: we establish stability under finite products, enabling the realization of finite direct sums of $C(K)$ spaces and matrix algebras $M_m(C(K))$ as Calkin algebras; we prove a localization principle showing compact operators on $\mathfrak{X}_{C(K)}$ can be approximated by finite-rank operators whose support respects the metric structure of $K$; we demonstrate that the diagonal function $\varphi_T\colon K\to\mathbb{C}$ of any bounded operator $T$ is H\"older continuous with optimal exponent $1/2$, revealing a deep analytic constraint; we prove a rigidity theorem showing the Banach algebra structure of $\mathcal{L}(\mathfrak{X}_{C(K)})$ completely determines the topology of $K$, extending the classical Banach--Stone theorem; we classify all closed two-sided ideals and prime ideals in $\mathcal{L}(\mathfrak{X}_{C(K)})$ in terms of open subsets and points of $K$; and we resolve longstanding problems, notably by constructing the first reflexive Banach spaces with infinite-dimensional reflexive Calkin algebras. These results forge a deep connection between Banach space geometry, operator algebras, and topological invariants, revealing how Calkin algebras can be precisely engineered through the geometry of their underlying spaces.