There is an exciting new development on Connes’ embedding problem. The recent preprint MIP*=RE by Ji, Natarajan, Vidick, Wright, Yuen claims to have solved the problem to the negative via a negative answer to Tsirelson’s problem via the relation to decision problems on the class MIP* of languages that can be decided by a classical verifier interacting with multiple all powerful quantum provers. I have to say that I don’t really understand what all this is about – but in any case there is quite some excitement about this and there seems to be a good chance that Connes’ problem might have a negative solution. To get some idea about the excitement around this, you might have look on the blogs of Scott Aaronson or of Gil Kalai. At the operator front I have not yet seen much discussion, but it might be that we still have to get over our bafflement.
Anyhow, there is now a realistic chance that there are type II factors which are not embeddable and this raises the question (among many others) what this means for free probability. I was asked this by a couple of people and as I did not have a really satisfying answer I want to think a bit more seriously about this. At the moment my answer is just: Okay, we have our two different approaches to free entropy and a negative solution to Connes embedding problem means that they cannot always agree. This is because we always have for the non-microstates free entropy that , if are free semicircular variables which are free from . The same property for the microstates free entropy , however, would imply that have microstates, i.e., the von Neumann algebra generated by is embeddable; see these notes of Shlyakhtenko.
But does this mean more then just saying that there are some von Neumann algebras for which we don’t have microstates but for which the non-microstates approach give some more interesting information, or is there more to it? I don’t know, but hopefully I will come back with more thoughts on this soon.
Of course, everybody is invited to share more information or thoughts on this!
When preparing my lectures for the asymptotic freeness of various random matrix ensembles I stumbled about the situation concerning Wigner matrices. We all know that Wigner matrices and deterministic matrices are asymptotically free, but all the proofs I am aware of are annoyingly complicated. Shouldn’t there be a nice and simple proof without too many technicalities?
As was said in the section “Open problems and possible future directions” of the report of the 2008 Banff workshop “Free Probability, Extensions, and Applications”: “Whereas engineers have no problems in applying asymptotic freeness results for unitarily invariant ensembles it has become apparent that they do not have the same confidence in the analogous results for Wigner matrices. The main reason for this is the lack of precise statements on this in the literature. This has to be remedied in the future.”
Actually, at that time there existed already some results in this direction in the paper On Certain Free Product Factors via an Extended Matrix Model from 1993 of Ken Dykema. There the asymptotic freeness between independent Wigner matrices and diagonal (or even more general: block diagonal) deterministic matrices had been shown. But the case of general deterministic matrices remained open. Taking on the challenge by the engineers, we were determined to write down nice and accessible proofs, also including the full deterministic case.
The result of this was that such statements and proofs were included in the book An Introduction to Random Matrices of Greg Anderson, Alice Guionnet, and Ofer Zeitouni on one side and in my book with Jamie on the other side. However, I have to admit what looked like an easy exercise to Jamie and me at the beginning turned out to me much more complicated. An intermediate outcome of this was my paper with Jamie on Sharp Bounds for Sums Associated to Graphs of Matrices, but even with this as a nice black box the final proof still required a couple of technical pages in our book.
So I would like to come back to the original challenge and want to see what we really know about the relation between Wigner matrices and deterministic matrices. What are the clean statements and what are nice proofs. The situation for Wigner matrices is also more complicated than for Gaussian matrices, as the real and imaginary part for Wigner matrices do not have to be independent, hence the complex situation cannot be directly reduced to the real one, and questions about the *-freeness of non-selfadjoint Wigner matrices is also not exactly the same as the freeness of selfadjoint ones. Of course, all is related and similar, but if one asks a concrete question, usually it is hard to find the answer exactly for this in the literature.
I hope to collect here information about what is out there in the literature on that problem, with the final goal of coming up with the cleanest statements and the simplest proofs. If you have any information or ideas in this context, please let me know.