局部可测算子代数上的等距映射
Isometries on algebras of locally measurable operators
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- Institute for Advances Study in Mathematics, Harbin Institute of Technologies（哈尔滨工业大学数学前沿研究院）
- Suzhou Research Institute of Harbin Institute of Technology（哈尔滨工业大学苏州研究院）
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中文总结 AI 辅助
该研究证明了局部可测算子代数上双射线性等距映射的典范表示，明确了$F$-范数与概率测度、维数函数对的对应关系，推广了经典算子代数等距理论。
中文摘要 AI 辅助
设$LS(\textit{M})$是附属于冯·诺依曼代数$\textit{M}$的局部可测算子代数，配备由维数函数和概率测度定义的$F$-范数。我们证明，$LS(\textit{M})$之间的任意双射线性等距映射都具有$\boldsymbol{\textit{Φ}(x)=wJ(x)}$的典范表示，其中$w$是酉元，$J$是约旦$^*$-同构，该结果推广了巴拿赫-斯通定理、卡迪逊定理等经典结论。在基础冯·诺依曼代数满足若干结构假设（包括所有$\text{II}_\text{∞}$型、$\text{III}$型代数，所有因子，以及中心无原子的代数）的条件下，我们证明$F$-范数与概率测度$\boldsymbol{\textit{μ}}$和维数函数$\boldsymbol{\textit{D}}$的对之间存在一一对应关系，而中心有原子的代数不满足该对应关系。
英文摘要
Let $LS(\mathcal{M})$ be the algebra of locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$, equipped with an $F$-norm defined via a dimension function and a probability measure. We prove that every bijective linear isometry between $LS(\mathcal{M})$ 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 $\mathrm{II}_\infty$ and type $\mathrm{III}$ 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.
