There is presently quite some activity around the q-Gaussians, about which I talked in my last post. Tomorrow (i.e., on Monday, April 25) there will be another talk in the UC Berkeley Probabilistic Operator Algebra Seminar on this topic. Mario Klisse from TU Delft will speak on his joint paper On the isomorphism class of q-Gaussian C∗-algebras for infinite variables with Matthijs Borst, Martijn Caspers and Mateusz Wasilewski. Whereas my paper with Akihiro deals only with the finite-dimensional case (and I see not how to extend this to infinite d) they deal with the infinite-dimensional case, and, quite surprisingly, they have a non-isomorphism result: namely that the C*-algebras for q=0 and for other q are not isomorphic. This makes the question for the von Neumann algebras even more interesting. It still could be that the von Neumann algebras are isomorphic, but then by a reason which does not work for the C*-algebras – this would be in contrast to the isomorphism results of Alice and Dima, which show the isomorphism of the von Neumann algebras (for finite d and for small q) by actually showing that the C*-algebras are isomorphic.
I am looking forward to the talk and hope that afterwards I have a better idea what is going on – so stay tuned for further updates.