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PreprintarXiv licencearXiv

Extreme points of the unit balls and isometries of noncommutative quasi-Banach Marcinkiewicz spaces

  • Kai Fang
  • Yi Gao
  • Jinghao Huang
  • Fedor Sukochev
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Abstract

We characterize extreme points of the (positive) unit balls of Marcinkiewicz spaces with respect to their natural quasi-norms. Having these results at hand, we show that every positive surjective isometry on a noncommutative Marcinkiewicz space is of elementary form. In particular, a linear mapping on a noncommutative Marcinkiewicz space over an atomic von Neumann algebra with all atoms having the same trace is a positive surjective isometry if and only if it is the restriction of a Jordan *-isomorphism which preserves the singular value functions. We also study (not necessarily positive) surjective isometries on weak Lp-spaces affiliated a σ-finite non-atomic von Neumann algebra for 1<p<∞, Marcinkiewicz sequence spaces, and weak ℓp operator ideals, p>0, as well as one-parameter groups of isometries on the latter two.

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Cite

BibTeXRIS
BibTeX
@misc{fang2026extreme,
  title = {{Extreme points of the unit balls and isometries of noncommutative quasi-Banach Marcinkiewicz spaces}},
  author = {Kai Fang and Yi Gao and Jinghao Huang and Fedor Sukochev},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01043},
  eprint = {2610.01043},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01043},
}
RIS
TY  - UNPB
TI  - Extreme points of the unit balls and isometries of noncommutative quasi-Banach Marcinkiewicz spaces
AU  - Kai Fang
AU  - Yi Gao
AU  - Jinghao Huang
AU  - Fedor Sukochev
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.01043
UR  - https://arxiv.org/abs/2610.01043
AB  - We characterize extreme points of the (positive) unit balls of Marcinkiewicz spaces with respect to their natural quasi-norms. Having these results at hand, we show that every positive surjective isometry on a noncommutative Marcinkiewicz space is of elementary form. In particular, a linear mapping on a noncommutative Marcinkiewicz space over an atomic von Neumann algebra with all atoms having the same trace is a positive surjective isometry if and only if it is the restriction of a Jordan *-isomorphism which preserves the singular value functions. We also study (not necessarily positive) surjective isometries on weak L_p-spaces affiliated a σ-finite non-atomic von Neumann algebra for 1<p<∞, Marcinkiewicz sequence spaces, and weak ℓ_p operator ideals, p>0, as well as one-parameter groups of isometries on the latter two.
ER  -