 
Summary: LIFTING ENDOMORPHISMS TO AUTOMORPHISMS
WILLIAM ARVESON AND DENNIS COURTNEY
Abstract. Normal endomorphisms of von Neumann algebras need not
be extendable to automorphisms of a larger von Neumann algebra, but
they always have asymptotic lifts. We describe the structure of endo
morphisms and their asymptotic lifts in some detail, and apply those
results to complete the identification of asymptotic lifts of unital com
pletely positive linear maps on von Neumann algebras in terms of their
minimal dilations to endomorphisms.
1. Introduction
We work in the category whose objects are pairs (M, ) consisting of a
normal unitpreserving endomorphism : M M of a von Neumann
algebra M, and whose maps are equivariant normal homomorphisms that
map unit to unit. The isomorphisms of this category are conjugacies, in
which 1 : M1 M1 is said to be conjugate to 2 : M2 M2 if there is a
isomorphism : M1 M2 satisfying 1 = 2 .
Consider the problem of extending an endomorphism : M M to
a automorphism of a larger von Neumann algebra, assuming that the
necessary condition ker = {0} is satisfied. In that case is an isometric
endomorphism of M, and a straightforward construction produces a unital
