Connes' embedding problem, formulated by Alain Connes in the 1970s, is a major problem in von Neumann algebra theory. During that time, the problem was reformulated in several different areas of mathematics. Dan Voiculescu developing his free entropy theory found that Connes’ embedding problem is related to the existence of microstates. Some results of von Neumann algebras theory can be obtained assuming positive solution to the problem. The problem is connected to some basic questions in quantum theory, which led to the realization that it also has important implications in computer science. The problem admits a number of equivalent formulations. Notably, it is equivalent to the following long standing problems:
The predual of any von Neumann algebra is finitely representable in the trace class.
In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen announced a result in quantum complexity theory that implies a negative answer to Connes' embedding problem.
Statement
Let be a free ultrafilter on the natural numbers and let R be the hyperfinite type II1 factor with trace. One can construct the ultrapower as follows: let be the von Neumann algebra of norm-bounded sequences and let. The quotient turns out to be a II1 factor with trace, where is any representative sequence of. Connes' embedding problem asks whether every type II1 factor on a separable Hilbert space can be embedded into some. Positive solution to the problem would imply that invariant subspaces exist for a large class of operators in II-1-factors ; all countable discrete groups are hyperlinear. A positive solution to the problem would be implied by equality between free entropy and free entropy defined by microstates. In January 2020, a group of researchers claimed to have resolved the problem in the negative, i.e., there exist type II1 von Neumann factors that do not embed in an ultrapower of the hyperfinite II1 factor. The isomorphism class of is independent of the ultrafilter if and only if the continuum hypothesis is true, but such an embedding property does not depend on the ultrafilter because von Neumann algebras acting on separable Hilbert spaces are, roughly speaking, very small. The problem admits a number of equivalent formulations.
Conferences dedicated to Connes' embedding problem
Connes' embedding problem and quantum information theory workshop; Vanderbilt University in Nashville Tennessee; May 1-7, 2020
The many faceted Connes' Embedding Problem; BIRS, Canada; July 14 -19, 2019
Winter school: Connes' embedding problem and quantum information theory; University of Oslo, January 07-11, 2019
Workshop on Sofic and Hyperlinear Groups and the Connes Embedding Conjecture; UFSC Florianopolis, Brazil; June 10-21 2018
