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Structure of block quantum dynamical semigroups and their product systems

机译：块量子动态半群及其产品系统的结构

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Paschke's version of Stinespring's theorem associates a Hilbert C*-module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a C*-algebra A one may associate an inclusion system E = (E-t) of Hilbert A-A-modules with a generating unit xi = (xi(t)). Suppose B is a von Neumann algebra, consider M-2(B), the von Neumann algebra of 2 x 2 matrices with entries from B. Suppose (Phi(t))(t >= 0) with Phi(t) = (psi(t)* phi(2)(t) phi(1)(t) psi(t)), is a QDS on M-2(B) which acts block-wise and let (E-t(i))(t >= 0) be the inclusion system associated to the diagonal QDS (phi(i)(t))(t >= 0) with the generating unit (xi(i)(t))(t >= 0), i = 1, 2. It is shown that there is a contractive (bilinear) morphism T = (T-t)(t >= 0) from (E-t(2))(t >= 0) to (E-t(1))(t >= 0) such that psi(t)(a) = for all a is an element of B. We also prove that any contractive morphism between inclusion systems of von Neumann B-B-modules can be lifted as a morphism between the product systems generated by them. We observe that the E-0-dilation of a block quantum Markov semigroup (QMS) on a unital C*-algebra is again a semigroup of block maps.
机译：Paschke的符号的定理版本将Hilbert C * -Module与生成向量相关联到每个完全正地图。在此构建于C * -Algebra上的每个量子动态半群（QDS），可以将Hilbert A-A模块的包含系统E =（E-T）与生成单元Xi =（Xi（T））相关联。假设B是一个von neumann代数，考虑m-2（b），von neumann代数为2 x 2矩阵，来自b的条目。假设（phi（t））（t> = 0）与phi（t）=（ psi（t）* phi（2）（t）phi（1）（t）psi（t））是M-2（b）上的QDS，其作用块，并让（等（i））（t > = 0）是与生成单元（xi（i）（t））（t> = 0）的对角线QDS（PHI（i）（t））（t> = 0）相关联的包含系统（t> = 0），i =如图1,2所示。从（等（2））（t> = 0）到（等（1））（t> = 0）使得PSI（a）= 用于所有a的是b的一个元素。我们还证明了任何收缩von Neumann BB模块的包涵体之间的态势可以作为它们产生的产品系统之间的态势被提升。我们观察到一个块量子Markov半群（QMS）的E-0扩展在一个UNITIT C * -Algebra上再次成为块地图的半群。

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