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

  • Rui Liu
  • Jie Shen
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Abstract

For 𝕂=ℝ or β„‚ 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 XV such that Cal(XV)=ℬ(XV)/𝒦(XV)≃ 𝕂&0 V*&𝕂 as Banach algebras. The identification of V* 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 π’₯/𝒦(X) for a closed two-sided ideal π’₯ of ℬ(X), and the subspace--ideal correspondence preserves the canonical order.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026duals,
  title = {{Duals of separable Banach Spaces as Calkin Algebras and Universal Ideal Quotients}},
  author = {Rui Liu and Jie Shen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.32334},
  eprint = {2609.32334},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32334},
}
RIS
TY  - UNPB
TI  - Duals of separable Banach Spaces as Calkin Algebras and Universal Ideal Quotients
AU  - Rui Liu
AU  - Jie Shen
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.32334
UR  - https://arxiv.org/abs/2609.32334
AB  - For 𝕂=ℝ or β„‚ 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 Cal(X_V)=ℬ(X_V)/𝒦(X_V)≃ 𝕂&0 V^*&𝕂 as Banach algebras. The identification of V^* 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 π’₯/𝒦(X) for a closed two-sided ideal π’₯ of ℬ(X), and the subspace--ideal correspondence preserves the canonical order.
ER  -