The asymptotic lift of a completely positive map |
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Authors: | William Arveson |
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Institution: | Department of Mathematics, University of California, Berkeley, CA 94720, USA |
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Abstract: | Starting with a unit-preserving normal completely positive map acting on a von Neumann algebra—or more generally a dual operator system—we show that there is a unique reversible system (i.e., a complete order automorphism α of a dual operator system N) that captures all of the asymptotic behavior of L, called the asymptotic lift of L. This provides a noncommutative generalization of the Frobenius theorems that describe the asymptotic behavior of the sequence of powers of a stochastic n×n matrix. In cases where M is a von Neumann algebra, the asymptotic lift is shown to be a W∗-dynamical system (N,Z), and we identify (N,Z) as the tail flow of the minimal dilation of L. We are also able to identify the Poisson boundary of L as the fixed algebra Nα. In general, we show the action of the asymptotic lift is trivial iff L is slowly oscillating in the sense that |
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Keywords: | Completely positive map von Neumann algebra Asymptotics |
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