Factorization and dilation problems for completely positive maps on von Neumann algebras

Publikation: Bidrag til tidsskriftTidsskriftartikelfagfællebedømt

We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the question of existence of non-factorizable Markov maps, as formulated by C. Anantharaman-Delaroche.We provide simple examples of non-factorizable Markov maps on Mn(C) for all n ≥ 3, as well as an example of a one-parameter semigroup (T (t))t≥0 of Markov maps on M4(C) such that T (t) fails to be factorizable for all small values of t > 0. As applications, we solve in the negative an
open problem in quantum information theory concerning an asymptotic version of the
quantum Birkhoff conjecture, as well as we sharpen the existing lower bound estimate
for the best constant in the noncommutative little Grothendieck inequality.
OriginalsprogEngelsk
TidsskriftCommunications in Mathematical Physics
Vol/bind303
Sider (fra-til)555-594
Antal sider40
ISSN0010-3616
DOI
StatusUdgivet - 2011

ID: 33980586