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PreprintCC BY-NC-ND 4.0arXiv

Dual Banach spaces with AP but without BAP

  • Longfa Sun
  • Zhenguo Wei
  • Zhendong Xu
  • Hao Zhang
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Abstract

We give negative answers to the AP-to-BAP and BAP-to-MAP questions for dual Banach spaces. More precisely, there is a separable Banach space V with the metric approximation property (MAP) such that V* has the approximation property (AP) but no bounded approximation property (BAP). There is also a separable MAP space X, isomorphic to ℓ1, whose dual has BAP but no MAP. This resolves a longstanding open question explicitly stated in \cite{Oja2010}. The constructions rely on a finitely generated group without Property A and a quantitative estimate for optimal bounded approximation constants of projective tensor products.

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Cite

BibTeXRIS
BibTeX
@misc{sun2026dual,
  title = {{Dual Banach spaces with AP but without BAP}},
  author = {Longfa Sun and Zhenguo Wei and Zhendong Xu and Hao Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06579},
  eprint = {2611.06579},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06579},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Dual Banach spaces with AP but without BAP
AU  - Longfa Sun
AU  - Zhenguo Wei
AU  - Zhendong Xu
AU  - Hao Zhang
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06579
UR  - https://arxiv.org/abs/2611.06579
AB  - We give negative answers to the AP-to-BAP and BAP-to-MAP questions for dual Banach spaces. More precisely, there is a separable Banach space V with the metric approximation property (MAP) such that V^* has the approximation property (AP) but no bounded approximation property (BAP). There is also a separable MAP space X, isomorphic to ℓ_1, whose dual has BAP but no MAP. This resolves a longstanding open question explicitly stated in \cite{Oja2010}. The constructions rely on a finitely generated group without Property A and a quantitative estimate for optimal bounded approximation constants of projective tensor products.
ER  -