AI 中文总结
该研究为非交换德·菲内蒂理论构建动力学框架，确立相关定律和条件，通过构造条件期望等方法，识别出弱可扩展性这一最小分布对称性，建立了一般的非交换德·菲内蒂定理。
AI 中文摘要
我们为非交换德·菲内蒂理论建立了一个动力学框架。首先确立了量子随机过程的非交换休伊特 - 萨维奇 0 - 1 律，识别了可扩展过程分布的因式分解条件以表征尾平凡性，还从动力学角度通过分布的某种吸引性来表征。这三个等价的遍历条件确定了非交换遍历性质层次中的一个特殊层级，在经典概率设置下都归结为遍历性。从遍历情形推广到一般情形时，在过程规范双边扩展分布的 GNS 表示中，为可扩展过程构造了到尾代数的条件期望。在此表示中确立了非交换奥尔申定理，识别了尾代数与平稳代数，以及过程可交换时与可交换代数的关系。由此得到的条件期望继承了与满足休伊特 - 萨维奇 0 - 1 律的分布相同类型的因式分解性质。这种因式分解强化了先前文献中出现的条件独立性条件，在经典情形下归结为相同概念。这种动力学观点导致识别出非交换德·菲内蒂理论背后的最小分布对称性，即弱可扩展性，在经典设置下它等同于可交换性。我们证明了一个平稳过程是弱可扩展的当且仅当其尾代数允许一个满足休伊特 - 萨维奇型因式分解 的唯一正规条件期望，从而建立了一个一般的非交换德·菲内蒂定理。
英文摘要
We develop a dynamical framework for non-commutative de Finetti theory. We first establish the non-commutative Hewitt-Savage 0-1 law for quantum stochastic processes. We identify the factorization condition of the distribution of a spreadable process which characterizes tail-triviality, which is also characterized dynamically in terms of a certain attractivity property of the distribution. These three equivalent ergodic conditions identify a distinguished level in the non-commutative hierarchy of ergodic properties, which all collapse to ergodicity in the classical probability setting. To pass from the ergodic to the general case, we construct the conditional expectation onto the tail algebra for a spreadable process, in the GNS representation of the distribution of the canonical bilateral extension of the process. In this representation we establish the non-commutative Olshen Theorem identifying the tail algebra with the stationary algebra, and with the exchangeable algebra when the process is exchangeable. The resulting conditional expectation in particular inherits the same type of factorization property as distributions satisfying the Hewitt-Savage 0-1 law. This factorization strengthens conditional independence conditions arising in previous literature, while collapsing to the same notion in the classical case. This dynamical viewpoint leads to the identification of the minimal distributional symmetry underlying non-commutative de Finetti Theory, which we call weak spreadability, and which in the classical setting is equivalent to exchangeability. We prove that a stationary process is weakly spreadable if and only if its tail algebra admits a unique normal conditional expectation satisfying Hewitt-Savage type factorization, thereby establishing a general non-commutative de Finetti Theorem.
