Abstract
For an arbitrary uniformly continuous completely positive semigroup (ℑ t : t≥ 0) on the space B(ɧ0) of bounded operators on a Hilbert space ɧ0, we construct a family (U(t): t ≥ 0) of unitary operators on a Hilbert space ℌ0 = ɧ0 ⊗ ℌ and a conditional expectation E0 from B(ℌ0) to B(ℌ0), such that, for arbitrary t ≥0, X ∈ B(ɧ0) ℑ t (X) = E0[U(t)X ⊗ IU(t)†]. The unitary operators U(t) satisfy a stochastic differential equation involving a noncommutative generalisation of infinite dimensional Brownian motion. They do not form a semigroup.
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Part of this work was completed when the first author was visiting research associate at the Center for Relativity, Physics Department, The University of Texas at Austin, Austin, TX 78712, U.S.A., supported in part by NSF PHY 81-01381.
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References
Bratteli, O. and Robinson, D.: Operator Algebras and Statistical Mechanics, Volume II, Springer-Verlag, Berlin, 1981.
Hudson, R. L. and Parthasarathy, K. R.: ‘Quantumn Itô’s Formula and Stochastic Evolutions’, Comm. Math. Phys. 93 (1984), 301-323.
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© 1984 D. Reidel Publishing Company
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Hudson, R.L., Parthasarathy, K.R. (1984). Stochastic Dilations of Uniformly Continuous Completely Positive Semigroups. In: Bratteli, O., Jørgensen, P.E.T. (eds) Positive Semigroups of Operators, and Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-6484-6_6
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DOI: https://doi.org/10.1007/978-94-009-6484-6_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-6486-0
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