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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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##### References:
 [1] Accardi, L., Parthasarathy, K.R.: Stochastic calculus on local algebras. (Lect. Notes Math., vol. 1136, pp. 9–23) Berlin Heidelberg New York: Springer 1985 · Zbl 0584.60075 [2] Appelbaum, D., Hudson, R.L.: Fermion Itô’s formula and stochastic evolutions. Commun. Math. Phys.96, 473–496 (1984) · Zbl 0572.60052 [3] Araki, H.: Factorizable representations of current algebra. Publ. RIMS Kyoto Univ.5, 361–442 (1970) · Zbl 0238.22014 [4] Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II. Berlin Heidelberg New York: springer 1981 · Zbl 0463.46052 [5] Barnett, C., Streater, R.F., Wilde, I.F.: The Itô-Clifford integral. J. Funct. Anal.48, 172–212 (1982) · Zbl 0492.46051 [6] Davies, E.B.: A model of heat conduction. J. Stat. Phys.16, 161–170 (1978) [7] Dümcke, R.: The low-density limit for anN-level system interacting with a free Bose or Fermi gas. Commun. Math. Phys.97, 331–359 (1985) · Zbl 0614.46069 [8] Evans, M., Hudson, R.L.: Multidimensional quantum diffusions. (Lect. Notes. Math., vol. 1303, pp. 69–88) Berlin Heidelberg New York: Springer 1988 · Zbl 0648.46056 [9] Frigerio, A., Gorini, V.: Markov dilations and quantum detailed balance. Commun. Math. Phys.93, 517 (1984) · Zbl 0553.46044 [10] Frigerio, A.: Covariant Markov dilations of quantum dynamical semigroups. Publ. RIMS Kyoto Univ.21, 657–675 (1985) · Zbl 0593.60092 [11] Frigerio, A.: Construction of stationary quantum Markov process through quantum stochastic calculus. (Lect. Notes Math., vol. 1136, pp. 207–222) Berlin Heidelberg New York: Springer 1985 [12] Frigerio, A.: Quantum Poisson processes: physical motivations and applications. (Lect. Notes Math., vol. 1303, pp. 107–127) Berlin Heidelberg New York: Springer 1988 · Zbl 0635.60095 [13] Hudson, R.L., Lindsay, J.M.: Use of non-Fock quantum Brownian motion and a quantum martingale representation theorem. (Lect. Notes Math., vol. 1136, pp. 276–305) Berlin Heidelberg New York: Springer 1985 · Zbl 0569.60055 [14] Hudson, R.L., Parthasarathy, K.R.: Quantum Itô’s formula and stochastic evolutions. Commun. Math. Phys.93, 301–323 (1984) · Zbl 0546.60058 [15] Hudson, R.L., Parthasarathy, K.R.: Stochastic dilations of uniformly continuous completely positive semigroups. Acta Appl. Math.2, 353–378 (1984) · Zbl 0541.60066 [16] Kossakowski, A., Frigerio, A., Gorini, V., Verri, M.: Quantum detailed balance and KMS condition. Commun. Math. Phys.57, 97–110 (1977) · Zbl 0374.46060 [17] Kümmerer, B.: Markov dilations onW *-algebras. J. Funct. Anal.63, 139–177 (1985) · Zbl 0601.46062 [18] Kümmerer, B.: Markov dilations and non-commutative Poisson processes; preprint, Tübingen 1986. See also: Kümmerer, B.: Survey on a Theory of non-commutative stationary Markov processes. (Lect. Notes Math., vol. 1303, pp. 154–182) Berlin Heidelberg New York: Springer 1988 [19] Kümmerer, B.: On the structure of Markov dilations onW *-algebras. (Lect. Notes Math., vol. 1136, pp. 318–331) Berlin Heidelberg New York: Springer 1985 [20] Kümmerer, B., Maassen, H. The essentially commutative dilations of dynamical semigroups onM n . Commun. Math. Phys.109, 1–22 (1987) · Zbl 0627.60015 [21] Lindsay, J.M., Maassen, H.: The stochastic calculus of Bose noise. Preprint, Nijmegen 1988 · Zbl 0652.60068 [22] Lindsay, J.M.: Ph.D. thesis, Nottingham 1985 [23] Maassen, H.: Quantum Markov processes on Fock space described by integral kernels. (Lect. Notes Math., vol. 1136, pp. 361–374) Berlin Heidelberg New York: Springer 1985 [24] Streater, R.F., Wulfsohn, A.: Continuous tensor products of Hilbert space and generalized random fields, Il Nuovo Cimento57B, 330–339 (1968) · Zbl 0162.21302 [25] Takesaki, M.: Conditional expectations in von Neumann algebras. J. Funct. Anal.9, 306–321 (1971) · Zbl 0245.46089 [26] Takesaki, M.: Tomita’s theory of modular Hilbert algebras and its applications. (Lect. Notes Math., vol. 128) Berlin Heidelberg New York: Springer 1970 · Zbl 0193.42502
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