Inverse-Limit Detection Envelopes as Calkin Algebras

Abstract

For a unital Banach algebra $A$ with $\|1_A\|=1$ and a decreasing sequence $(I_r)$ of proper closed two-sided ideals of finite codimension, we introduce the bounded detection envelope $$\widehat A:=\operatorname{Env}^{b}_{(I_r)}(A)=\varprojlim^{b}(A/I_r,\pi_r^s),\qquad \|(a_r)_r\|=\sup_r\|a_r\|.$$ Combining vector-valued Bourgain--Delbaen extensions, oracle Turing-machine recursion, and Argyros--Haydon operator analysis, we construct a separable Banach space $X$ such that, isometrically as unital Banach algebras, $$\operatorname{Cal}(X)=\mathcal{B}(X)/\mathcal{K}(X)\simeq\widehat A. $$ We also characterize these envelopes as the unital Banach algebras with norm-one units admitting an isometric predual and countably many jointly faithful weak-star-to-norm continuous finite-dimensional unital representations. Applications include arbitrary finite-dimensional unital Banach algebras with norm-one units and their countable $\ell_\infty$-sums, as well as unitized coordinatewise algebras over normalized $1$-unconditional boundedly complete Schauder bases. These include incidence algebras, $\ell_\infty$, and the unitization of Tsirelson's original space. Further applications concern analytic operator algebras, Fourier--Stieltjes and Fourier multiplier algebras of countable discrete groups, and classical and quantum convolution algebras. In particular, we realize $H^\infty(\mathbb D)$, $\operatorname{Mult}(H_2^2)$, $\mathcal L_2$, $M_{\mathrm{cb}}A(\mathbb F_2)$, and $C^u(O_3^+)^*$ as Calkin algebras, with their canonical norms and products.