Inverse-Limit Detection Envelopes as Calkin Algebras

Abstract

For a unital Banach algebra $A$ with $\Vert 1_A\Vert =1$ and a decreasing sequence $(I_r)$ of proper closed two-sided ideals of finite codimension, we introduce the bounded detection envelope $$\hat{A}:={\mathrm{Env}}_{(I_r)}^b(A)=\stackrel{b}{\underleftarrow{\lim }}(A/I_r,{\pi }_r^s),\Vert (a_r{)}_r\Vert =\underset{r}{\sup }\Vert a_r\Vert .$$ 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, $$\mathrm{Cal}(X)=\mathcal{B}(X)/\mathcal{K}(X)\simeq \hat{A}.$$ 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 $H^{\infty }(\mathbb{D})$, $\mathrm{Mult}(H_2^2)$, ${\mathcal{L}}_2$, $M_{\mathrm{cb}}A({\mathbb{F}}_2)$, and $C^u(O_3^{+}{)}^{\ast }$ as Calkin algebras, with their canonical norms and products.