发表机构
School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入有界检测包络作为 Calkin 代数，通过构造可分 Banach 空间实现任意此类包络，并刻画其结构，应用于多种算子代数与卷积代数。
AI 中文摘要
对于单位 Banach 代数 $A$（满足 $\u007c\u007c1_A\u007c\u007c=1$）以及由有限余维的真闭双边理想构成的递减序列 $(I_r)$，我们引入有界检测包络 $$\u005Cwidehat A:=\u005Coperatorname{Env}^{b}_{(I_r)}(A)=\u005Cvarprojlim^{b}(A/I_r,\u005Cpi_r^s),\u0020\u0020\u0020\u0020\u007c\u007c(a_r)_r\u007c\u007c=\u005Csup_r\u007c\u007ca_r\u007c\u007c.$$ 结合向量值 Bourgain--Delbaen 扩张、oracle 图灵机递归以及 Argyros--Haydon 算子分析，我们构造了一个可分 Banach 空间 $X$，使得作为单位 Banach 代数（等距同构）有 $$\u005Coperatorname{Cal}(X)=\u005Cmathcal{B}(X)/\u005Cmathcal{K}(X)\u005Csimeq\u005Cwidehat A. $$ 我们还将这些包络刻画为具有范数一单位、允许等距预对偶以及可数个联合忠实的弱星到范数连续的有限维单位表示的单位 Banach 代数。应用包括任意具有范数一单位的有限维单位 Banach 代数及其可数 $\u005Cell_\u005Cinfty$-和，以及关于归一化 $1$-无条件有界完备 Schauder 基的单位化坐标代数。这些包括关联代数、$\u005Cell_\u005Cinfty$ 以及 Tsirelson 原始空间的单位化。进一步的应用涉及解析算子代数、可数离散群的 Fourier--Stieltjes 和 Fourier 乘子代数，以及经典和量子卷积代数。特别地，我们将 $H^\u005Cinfty(\u005Cmathbb D)$、$\u005Coperatorname{Mult}(H_2^2)$、$\u005Cmathcal L_2$、$M_{\u005Cmathrm{cb}}A(\u005Cmathbb F_2)$ 和 $C^u(O_3^+)^*$ 实现为 Calkin 代数，并保持其典范范数和乘积。
英文摘要
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 $$\widehat A:=\operatorname{Env}^{b}_{(I_r)}(A)=\varprojlim^{b}(A/I_r,π_r^s),\qquad \|(a_r)_r\|=\sup_r\|a_r\|.$$ 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.
