Isometries on algebras of locally measurable operators
Abstract
Let LS(ℳ) be the algebra of locally measurable operators affiliated with a von Neumann algebra ℳ, equipped with an F-norm defined via a dimension function and a probability measure. We prove that every bijective linear isometry between LS(ℳ) admits a canonical representation of the form Φ(x)=wJ(x), where w is a unitary element and J is a Jordan *-isomorphism, which extends classical results such as the Banach--Stone theorem and Kadison's theorem. Under several structural assumptions on the underlying von Neumann algebras (including all type II∞ and type III algebras, and all factors, and algebras with atomless centers), we prove the one-to-one correspondence between the F-norm and the pair $(μ, D)$ of a probability measure and a dimension function, which fails for algebras with atomic centers.
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BibTeX
@misc{huang2026isometries,
title = {{Isometries on algebras of locally measurable operators}},
author = {Jinghao Huang and Karimbergen Kudaybergenov and Bing Yan},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28710},
eprint = {2608.28710},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28710},
}
RIS
TY - UNPB
TI - Isometries on algebras of locally measurable operators
AU - Jinghao Huang
AU - Karimbergen Kudaybergenov
AU - Bing Yan
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28710
UR - https://arxiv.org/abs/2608.28710
AB - Let LS(ℳ) be the algebra of locally measurable operators affiliated with a von Neumann algebra ℳ, equipped with an F-norm defined via a dimension function and a probability measure. We prove that every bijective linear isometry between LS(ℳ) admits a canonical representation of the form Φ(x)=wJ(x), where w is a unitary element and J is a Jordan ^*-isomorphism, which extends classical results such as the Banach--Stone theorem and Kadison's theorem. Under several structural assumptions on the underlying von Neumann algebras (including all type II_∞ and type III algebras, and all factors, and algebras with atomless centers), we prove the one-to-one correspondence between the F-norm and the pair $(μ, D)$ of a probability measure and a dimension function, which fails for algebras with atomic centers.
ER -