Stochastic dilations of uniformly continuous completely positive semigroups |
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Authors: | R. L. Hudson K. R. Parthasarathy |
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Affiliation: | (1) Mathematics Department, University of Nottingham, University Park, NG7 2RD Nottingham, England;(2) Indian Statistical Institute, 7 SJS Sansanwal Marg, 110016 New Delhi, India |
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Abstract: | For an arbitrary uniformly continuous completely positive semigroup (t:t0) on the space of bounded operators on a Hilbert space, we construct a family (U(t)t0) of unitary operators on a Hilbert space and a conditional expectation from to, such that, for arbitraryt0,. The unitary operatorsU(t) satisfy a stochastic differential equation involving a noncommutative generalisation of infinite dimensional Brownian motion. They do not form a semigroup.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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Keywords: | 46L50 46L55 46L60 47D05 81H05 81C20 81C35 82A35 15A66 60E07 60G20 60H05 34F05 60H15 60J65 60K35 81D05 47A20 |
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