Large ideals in B(L1(0,1))

@misc{nasseri2026large,
  title = {{Large ideals in B(L1(0,1))}},
  author = {Amir Bahman Nasseri},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18348},
  eprint = {2609.18348},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18348},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Large ideals in B(L1(0,1))
AU  - Amir Bahman Nasseri
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18348
UR  - https://arxiv.org/abs/2609.18348
AB  - We show that the norm-closed linear span of products of two Dunford-Pettis operators on L^1(0,1) lies strictly between the representable and the Dunford-Pettis operators. This gives an additional proper large closed ideal in ℬ(L^1(0,1)), answering the existence question of Johnson, Pisier and Schechtman. The quotient of the Dunford-Pettis ideal by this square ideal contains a contractively complemented isometric copy of L^1(0,1). We also prove that the Dunford-Pettis ideal has no right approximate identity, answering a question of Johnson and Schechtman. More precisely, a positive norm-one convolution operator R satisfies |R−RT|≥1 and |R−TR|≥1 for every Dunford-Pettis operator T. The proofs combine translation averaging, Rajchman measures supported on a strongly independent set, and two atomless disintegrations of positive operators. The approximate-identity obstruction extends to every finite measure space with a nonzero atomless part. Finally, we prove that quantitative multiplication estimates for an ideal pass to all its closed power ideals.
ER  -