Appears in collection : Operator Algebras and Quantum Information Theory

In joint work with Haagerup, we established in 2015 a reformulation of the Connes embedding problem in terms of an asymptotic property of quantum channels possessing a certain factorizability property, introduced by Anantharaman-Delaroche. I will discuss new results concerning the class of channels which exactly factor through matrix algebras, and a number of open problems. I will also discuss ongoing work, joint with Brown and Rørdam, related to the remarkable recent breakthrough of Slofstra.