Magdalena Musat

Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras

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

The Effros–Ruan conjecture for bilinear forms on C*-algebras

{M_n(\mathbb{C})}

An Asymptotic Property of Factorizable Completely Positive Maps and the Connes Embedding Problem

for all n ≥ 3, as well as an example of a one-parameter semigroup (T(t))t≥0 of Markov maps on

On the best constants in noncommutative Khintchine-type inequalities

{M_4(\mathbb{C})}

Just-infinite

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.

C^*

We prove that for 1⩽p<q<∞ the analogue of the classical result [BMO,Lp]pq=Lq holds in the setting of a finite von Neumann algebra M, equipped with an increasing filtration (Mn)n⩾1 of von Neumann subalgebras. We also obtain the corresponding results for the real method of interpolation. We discuss the appropriate operator space matrix norms and show that these interpolation results hold in the category of operator spaces.