论文标题
通过运输计划的von Neumann代数的二次Wasserstein指标
Quadratic Wasserstein metrics for von Neumann algebras via transport plans
论文作者
论文摘要
我们展示了如何通过运输计划而不是通过动态方法来获得一类二次的沃斯坦斯坦指标，也就是说，在von Neumann代数$ a $的一组忠实的正常状态下，瓦斯林指标是订单2的瓦斯林指标。进行这项工作的两个关键点是对tomita-takesaki理论和双模型的相对张量产物产生的运输成本的适当表述（或Connes意义上的对应关系）。三角形不平等，对称性和$ w_ {2}（μ，μ）= 0 $均已一般起作用，但是要表明$ w_ {2}（μ，ν）= 0 $ $ use $μ=ν$，我们需要假设$ a $ a $是有限生成的。
We show how one can obtain a class of quadratic Wasserstein metrics, that is to say, Wasserstein metrics of order 2, on the set of faithful normal states of a von Neumann algebra $A$, via transport plans, rather than through a dynamical approach. Two key points to make this work, are a suitable formulation of the cost of transport arising from Tomita-Takesaki theory and relative tensor products of bimodules (or correspondences in the sense of Connes). The triangle inequality, symmetry and $W_{2}(μ,μ)=0$ all work quite generally, but to show that $W_{2}(μ,ν)=0$ implies $μ=ν$, we need to assume that $A$ is finitely generated.
