Hilbert Modules and Stochastic Dilation of a Quantum Dynamical Semigroup on a von Neumann Algebra
Hilbert Modules and Stochastic Dilation of a Quantum Dynamical Semigroup on a von Neumann Algebra
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冯诺依曼代数上量子动力学半群的希尔伯特模和随机扩张
DOI：
10.1007/s002200050682
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发表时间：
1999
期刊：
影响因子：
--
通讯作者：
K. Sinha
中科院分区：
文献类型：
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作者：
Debashish Goswami;K. Sinha
在von Neumann代数上构造一致连续量子动力学半群的弱Markov扩张的一般理论？的Fock过滤的开发与援助的坐标自由量子随机演算。从Tt的生成元的结构出发，导出了Evans和哈德逊意义下的正则结构映射的存在性，并通过求解右Fock模上映射的正则流方程，得到了Tt的量子随机膨胀。其中k 0是由？’。这就产生了一个 *-同态mjtof？。此外，它表明，每一个这样的流是由一个部分等距值的过程。这导致了一个自然的弱马尔可夫过程（在[B-P]意义下）的构造。
A general theory for constructing a weak Markov dilation of a uniformly continuous quantum dynamical semigroupTton a von Neumann algebra ? with respect to the Fock filtration is developed with the aid of a coordinate-free quantum stochastic calculus. Starting with the structure of the generator ofTt, existence of canonical structure maps (in the sense of Evans and Hudson) is deduced and a quantum stochastic dilation ofTtis obtained through solving a canonical flow equation for maps on the right Fock module ?⊗Γ(L2(ℝ+,k0)), wherek0is some Hilbert space arising from a representation of ?′. This gives rise to a *-homomorphismjtof ?. Moreover, it is shown that every such flow is implemented by a partial isometry-valued process. This leads to a natural construction of a weak Markov process (in the sense of [B-P]) with respect to Fock filtration.