# What are some results that assume the Connes' embedding conjecture or any of its reformulations?

As you all (may) know, the Connes embedding conjecture was disproven last year. Also, as its Wikipedia page shows, there are multiple reformulations (but it is definitely not an exhaustive list):

• Kirchberg's QWEP conjecture in C*-algebra theory
• Tsirelson's problem in quantum information theory
• The predual of any (separable) von Neumann algebra is finitely representable in the trace class

So, are there any results that depend on the conjecture or any of the above (or any other) reformulation?

• From the Wikipedia page about Connes embeddng conjecture: "Positive solution to the problem would imply that invariant subspaces exist for a large class of operators in II-1-factors (Uffe Haagerup); all countable discrete groups are hyperlinear. " Jan 20 at 2:14
• The paper arxiv.org/abs/2006.05629 also addresses some consequences (both in logic and in operator algebras), such as the Blackadar-Kirchberg problem (or MF problem), that asks whether every stably finite C*-algebra embeds into an ultrapower of the universal UHF algebra. Jan 20 at 8:56
• @DiegoMartínez I could only find the MF problem, where are the other problems listed? Jan 20 at 14:27
• @DUOLabs the computability of the universal theory of the hyperfinite II$_1$ factor is another. I am not an expert on these topics, and it's been a while since I read it, but there may be more listed (or mentioned in passing). Jan 20 at 14:35