Inverse-Limit Detection Envelopes as Calkin Algebras
Abstract
For a unital Banach algebra A with |1A|=1 and a decreasing sequence (Ir) of proper closed two-sided ideals of finite codimension, we introduce the bounded detection envelope Â:=Envb(Ir)(A)=varprojlimb(A/Ir,πrs), |(ar)r|=supr|ar|. 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, Cal(X)=ℬ(X)/𝒦(X)≃Â. 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 ℓ∞-sums, as well as unitized coordinatewise algebras over normalized $1$-unconditional boundedly complete Schauder bases. These include incidence algebras, ℓ∞, 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∞(𝔻), Mult(H22), ℒ2, McbA(𝔽2), and Cu(O3+)* as Calkin algebras, with their canonical norms and products.
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BibTeX
@misc{liu2026inverselimit,
title = {{Inverse-Limit Detection Envelopes as Calkin Algebras}},
author = {Rui Liu and Jie Shen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.32330},
eprint = {2609.32330},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32330},
}
RIS
TY - UNPB
TI - Inverse-Limit Detection Envelopes as Calkin Algebras
AU - Rui Liu
AU - Jie Shen
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2609.32330
UR - https://arxiv.org/abs/2609.32330
AB - 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 Â:=Env^b_(I_r)(A)=varprojlim^b(A/I_r,π_r^s), |(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, Cal(X)=ℬ(X)/𝒦(X)≃Â. 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 ℓ_∞-sums, as well as unitized coordinatewise algebras over normalized $1$-unconditional boundedly complete Schauder bases. These include incidence algebras, ℓ_∞, 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^∞(𝔻), Mult(H_2^2), ℒ_2, M_cbA(𝔽_2), and C^u(O_3^+)^* as Calkin algebras, with their canonical norms and products.
ER -