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Quantum Poisson processes and dilations of dynamical semigroups. (English) Zbl 0684.60081
Summary: The notion of a quantum Poisson process over a quantum measure space is introduced. This process is used to construct new quantum Markov processes on the matrix algebra \(M_ n\) with stationary faithful state \(\phi\). If (\({\mathcal M},\mu)\) is the quantum measure space in question (\({\mathcal M}^ a \)von Neumann algebra and \(\mu\) a faithful normal weight), then the semigroup \(e^{tL}\) of transition operators on \((M_ n,\phi)\) has generator \[ L: M_ n\to M_ n: a\to i[h,a]+(id\otimes \mu)(u^*(a\otimes\mathbf{1})u-a\otimes\text\textbf{1}), \] where u is an arbitrary unitary element of the centraliser of \((M_ n\otimes {\mathcal M},\phi \otimes \mu)\).

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
81P20 Stochastic mechanics (including stochastic electrodynamics)
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