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

This manuscript presents a systematic study of Calkin algebras – the quotients $\mathcal{L}(X)\mathrm{/}\mathcal{K}(X)$ of bounded operators modulo compact operators on a Banach space $X$ – and establishes a framework for realizing commutative $C^{\ast }$-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 ${\phi }_T\text{\hspace{0.17em}⁣}:K\to \mathbb{C}$ of any bounded operator $T$ is Hölder continuous with optimal exponent $1\mathrm{/}2$, 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.