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LIFTING ENDOMORPHISMS TO AUTOMORPHISMS WILLIAM ARVESON AND DENNIS COURTNEY
 

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 unit-preserving -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

  

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

 

Collections: Mathematics