量子概率中耗散系统动力学的存在性问题
The existence problem for dynamics of dissipative systems in quantum probability
摘要：
出于存在性问题的耗散系统自然产生的晶格模型从量子统计力学，我们考虑以下$C^{\ast}$ -代数设置：一个给定的厄米耗散映射$δ$密集定义在一个单位$C^{\ast}$ -代数$\mathfrak{A}$。${\frak A}$中的同一元素也在$δ$的结构域中。完全耗散映射$δ$的定义是：对于所有$n$，导出映射$(a_{ij})\to (δ(a_{ij}))$在$n$上通过${\frak A}$上的$n$复矩阵耗散。我们建立了完全耗散映射的不同类型的极大扩张的存在性。如果${\frak A}$的包络von Neumann代数是内射的，我们证明了$δ$的扩张是von Neumann代数中完全正映射的量子动力学半群的无穷小生成元.如果$δ$是一个给定的行为良好的 *-导子，那么我们证明了每个映射$δ$和$-δ$是完全耗散的。
英文摘要：
Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $δ$ is densely defined in a unital $C^{\ast}$-algebra $\mathfrak{A}$. The identity element in ${\frak A}$ is also in the domain of $δ$. Completely dissipative maps $δ$ are defined by the requirement that the induced maps, $(a_{ij})\to (δ(a_{ij}))$, are dissipative on the $n$ by $n$ complex matrices over ${\frak A}$ for all $n$. We establish the existence of different types of maximal extensions of completely dissipative maps. If the enveloping von Neumann algebra of ${\frak A}$ is injective, we show the existence of an extension of $δ$ which is the infinitesimal generator of a quantum dynamical semigroup of completely positive maps in the von Neumann algebra. If $δ$ is a given well-behaved *-derivation, then we show that each of the maps $δ$ and $-δ$ is completely dissipative.
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