Duals of separable Banach Spaces as Calkin Algebras and Universal Ideal Quotients

Abstract

For $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and every separable Banach space $V$, we use an oracle-relative two-sorted finite-extension construction, whose oracle records the local rational structure of $V$, to construct a separable Banach space $X_V$ such that $$\mathrm{Cal}(X_V)=\mathcal{B}(X_V)/\mathcal{K}(X_V)\simeq \left(\begin{array}{cc}\mathbb{K}& 0\\ V^{\ast }& \mathbb{K}\end{array}\right)$$ as Banach algebras. The identification of $V^{\ast }$ with the Jacobson radical is isometric. Consequently, every nonzero dual Banach space with a separable predual admits, after an equivalent renorming, a unital Banach-algebra structure isomorphic to the Calkin algebra of a separable Banach space. Moreover, there is a separable Banach space $X$ such that every separable Banach space is isometric to $\mathcal{J}/\mathcal{K}(X)$ for a closed two-sided ideal $\mathcal{J}$ of $\mathcal{B}(X)$, and the subspace--ideal correspondence preserves the canonical order.