The asymptotic lift of a completely positive map
| dc.creator | Arveson, William | |
| dc.date | 2006-06-21 | |
| dc.date | 2006-07-15 | |
| dc.date.accessioned | 2026-07-07T07:17:32Z | |
| dc.date.available | 2026-07-07T07:17:32Z | |
| dc.description | 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 α: N --> N (i.e., a complete order automorphism αof a dual operator system N) that captures all of the asymptotic behavior of L, called the {\em 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 x n matrix. In cases where M is a von Neumann algebra, the asymptotic lift is shown to be a W*-dynamical system (N,\mathbb Z), whick we identify as the tail flow of the minimal dilation of L. We are also able to identify the Poisson boundary of L as the fixed point algebra of (N,\mathbb Z). In general, we show the action of the asymptotic lift is trivial iff L is {\em slowly oscillating} in the sense that $$ \lim_{n\to\infty}\|ρ\circ L^{n+1}-ρ\circ L^n\|=0,\qquad ρ\in M_* . $$ Hence αis often a nontrivial automorphism of N. | |
| dc.description | New section added with an applicaton to the noncommutative Poisson boundary. Clarification of Sections 3 and 4. Additional references. 23 pp | |
| dc.identifier | https://arxiv.org/abs/math/0606541 | |
| dc.identifier | http://arxiv.org/abs/math/0606541 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/113976 | |
| dc.subject | Operator Algebras | |
| dc.subject | Functional Analysis | |
| dc.subject | 46L55; 46L09 | |
| dc.title | The asymptotic lift of a completely positive map | |
| dc.type | text |