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THE ASYMPTOTIC LIFT OF A COMPLETELY POSITIVE MAP
 

Summary: THE ASYMPTOTIC LIFT OF A
COMPLETELY POSITIVE MAP
WILLIAM ARVESON
Abstract. Starting with a unit-preserving normal completely positive
map L : M M acting on a von Neumann algebra - or more generally a
dual operator system - we show that there is a unique reversible system
: N N (i.e., a complete order automorphism 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 N
.
In general, we show the action of the asymptotic lift is trivial iff L is
slowly oscillating in the sense that
lim

  

Source: Arveson, William - Department of Mathematics, University of California at Berkeley

 

Collections: Mathematics