Magdalena Musat (University of Copenhagen)
Factorizable completely positive maps and the Connes embedding problem
Abstract: In recent work with Uffe Haagerup, we study factorization and dilation properties of Markov maps between von Neumann algebras, motivated by 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 M_n(C) for all n at least 3, as well as an example of a one-parameter semigroup (T(t)) for t\geq 0, of Markov maps on M_4(C) such that T(t) fails to be factorizable for all small values of t > 0. The existence of non-factorizable Markov maps turned out to have an interesting application to an open problem in quantum information theory, known as the asymptotic quantum Birkhoff conjecture (AQBP). We solve the conjecture in the negative by showing that every non-factorizable Markov map on M_n(C) for all n at least 3, provides a counterexample. We also discuss very recent developments concerning the question whether every factorizable map does satisfy the AQBP, and establish connections to the Connes embedding problem. |
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