论文标题
GNS对称量子马尔可夫半群的发电机的差分结构
The Differential Structure of Generators of GNS-symmetric Quantum Markov Semigroups
论文作者
论文摘要
我们表明，GNS对称量子Markov Semigroup的发电机可以写为派生的平方。这概括了Cipriani和Sauvageot的结果。与奇特的对称情况相比，一般情况下的衍生物满足了扭曲的产品规则，反映了其模块化组的非平凡性。我们介绍的tomita bimodules的新概念捕捉到了这种转折。如果Markov Semigroup满足一定的额外规律性条件，则可以在更大的von Neumann代数的$ l^2 $空间内实现相关的tomita bimodule，其构造是免费的Araki-Woods因子的操作员值。
We show that the generator of a GNS-symmetric quantum Markov semigroup can be written as the square of a derivation. This generalizes a result of Cipriani and Sauvageot for tracially symmetric semigroups. Compared to the tracially symmetric case, the derivations in the general case satisfy a twisted product rule, reflecting the non-triviality of their modular group. This twist is captured by the new concept of Tomita bimodules we introduce. If the quantum Markov semigroup satisfies a certain additional regularity condition, the associated Tomita bimodule can be realized inside the $L^2$ space of a bigger von Neumann algebra, whose construction is an operator-valued version of free Araki-Woods factors.
