Abstract
For identity and trace preserving one-parameter semigroups {T t} t≧0 on then×n-matricesM n we obtain a complete description of their “essentially commutative” dilations, i.e., dilations, which can be constructed on a tensor product ofM n by a commutativeW*-algebra.
We show that the existence of an essentially commutative dilation forT t is equivalent to the existence of a convolution semigroup of probability measures ρ t on the group Aut(M n) of automorphisms onM n such that\(T_t = \smallint _{Aut\left( {M_n } \right)} \alpha d\rho _t \left( \alpha \right)\), and this condition is then characterised in terms of the generator ofT t. There is a one-to-one correspondence between essentially commutative Markov dilations, weak*-continuous convolution semigroups of probability measures and certain forms of the generator ofT t. In particular, certain dynamical semigroups which do not satisfy the detailed balance condition are shown to admit a dilation. This provides the first example of a dilation for such a semigroup.
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References
Albeverio, S., Hoegh-Krohn, R., Olsen, G.: Dynamical semigroups and Markov processes onC*-algebras. J. Reine Angew. Math.319, 25–37 (1980)
Albeverio, S., Hoegh-Krohn, R.: A remark on dynamical semigroups in terms of diffusion processes. In: Quantum probability and applications II. Proceedings, Heidelberg 1984, Lecture Notes in Mathematics Vol.1136, pp. 40–45. Berlin, Heidelberg, New York: Springer 1985
Alicki, R., Fannes, M.: Dilations of quantum dynamical semigroups with classical Brownian motion. Commun. Math. Phys. (in press)
Davies, E. B.: Dilations of completely positive maps. J. Lond. Math. Soc. (2)17, 330–338 (1978)
Emch, G. G., Albeverio, S., Eckmann, J.-P.: Quasi-free generalizedK-flows. Rep. Math. Phys.13, 73–85 (1978)
Evans, D. E.: Positive linear maps on operator algebras. Commun. Math. Phys.48, 15–22 (1976)
Evans, D. E.: Completely positive quasi-free maps on the CAR algebra. Commun. Math. Phys.70, 53–68 (1979)
Evans, D. E., Lewis, J. T.: Dilations of dynamical semi-groups. Commun. Math. Phys.50, 219–227 (1976)
Evans, D. E., Lewis, J. T.: Dilations of irreversible evolutions in algebraic quantum theory. Commun. Dublin Inst. Adv. Stud. Ser.A24 (1977)
Frigerio, A., Gorini, V.: On stationary Markov dilations of quantum dynamical semigroups; Frigerio, A, Gorini, V.: Markov dilations and quantum detailed balance. Commun. Math. Phys.93, 517–532 (1984)
Frigerio, A.: Covariant Markov dilations of quantum dynamical semigroups. Preprint, Milano 1984
Hudson, R. L. Parthasarathy, K. R.: Quantum Ito's formula and stochastic evolutions. Commun. Math. Phys.93, 301–323 (1984)
Hunt, G. A.: Semi-groups of measures on Lie groups. Trans. Am. Math. Soc.81, 264–293 (1956)
Kossakowski, A., Frigerio, A., Gorini, V., Verri, M.: Quantum detailed balance and KMS condition. Commun. Math. Phys.57, 97–110 (1977)
Kümmerer, B.: A Dilation theory for completely positive operators onW*-algebras. Thesis, Tübingen 1982;
Kümmerer, B.: Markov dilations onW*-algebras. J. Funct. Anal.63, 139–177 (1985)
Kümmerer, B.: A non-commutative example of a continuous Markov dilation. Semesterbericht Funktionalanalysis, Tübingen, Wintersemester 1982/83, pp. 61–91;
Kümmerer, B.: Examples of Markov dilations over the 2 × 2 matrices. In: Quantum probability and applications to the quantum theory of irreversible processes. Proceedings, Villa Mondragone 1982, Lecture Notes in Mathematics Vol.1055, pp. 228–244. Berlin, Heidelberg, New York: Springer 1984
Kümmerer, B.: On the structure of Markov dilations onW*-algebras. In: Quantum probability and applications II. Proceedings, Heidelberg, 1984, Lecture Notes in Mathematics Vol.1136, pp. 332–347. Berlin, Heidelberg, New York: Springer 1985
Kümmerer, B., Schröder, W.: A new construction of unitary dilations: Singular coupling to white noise. In: Quantum probability and applications II. Proceedings, Heidelberg 1984, Lecture Notes in Mathematics Vol.1136, pp. 332–347. Berlin, Heidelberg, New York: Springer 1985
Lewis, J. T., Thomas, L. C.: How to make a heat bath. In: Functional integration, Proceedings Cumberland Lodge 1974, pp. 97–123, London: Oxford University Press (Clarendon) 1975
Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys.48, 119–130 (1976)
Maassen, H.: Quantum Markov processes on Fock space described by integral kernels. In: Quantum probability and applications II. Proceedings, Heidelberg 1984, Lecture Notes in Mathematics Vol.1136, pp. 361–374, Berlin, Heidelberg, New York: Springer 1985
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Communicated by R. Haag
Supported by the Deutsche Forschungsgemeinschaft
Supported by the Netherlands Organisation for the Advancement of pure research (ZWO)
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Kümmerer, B., Maassen, H. The essentially commutative dilations of dynamical semigroups onM n . Commun.Math. Phys. 109, 1–22 (1987). https://doi.org/10.1007/BF01205670
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DOI: https://doi.org/10.1007/BF01205670