 
Summary: ASYMPTOTIC LIFTS OF POSITIVE LINEAR MAPS
WILLIAM ARVESON AND ERLING STØRMER
Abstract. We show that the notion of asymptotic lift generalizes nat
urally to normal positive maps : M M acting on von Neumann
algebras M. We focus on cases in which the domain of the asymptotic
lift can be embedded as an operator subsystem M M, and charac
terize when M is a Jordan subalgebra of M in terms of the asymptotic
multiplicative properties of .
1. Introduction
Let : M M be a normal unit preserving positive linear map acting
on a dual operator system M; we refer to such a pair (M, ) as a UP map.
While we are primarily interested in UP maps that act on von Neumann
algebras M, it is useful to broaden the context as above. The powers of
form an irreversible dynamical semigroup {n : n = 0, 1, 2, . . . } acting on
M. In this paper we generalize work begun in [Arv04] and [Arv06], together
with complementary results in [Stø06], to further develop the asymptotic
theory of such semigroups.
One may view UP maps as the objects of a category, in which a homo
morphism from 1 : M1 M1 to 2 : M2 M2 is a UP map E : M1 M2
such that E 1 = 2 E. There is a natural notion of isomorphism in
