Connes embedding problem

In mathematics, Connes' embedding problem or embedding conjecture was formulated by A. Connes in the 1970's. The question asks whether any finite type von Neumann algebra can be embedded into the ultrapower of the hyperfinite factor. This classification question can be rephrased as a noncommutative moment problem; equivalently, it asks whether generators of any tracial von Neumann probability space admit a matrix model.

Many results of von Neumann algebras theory could be obtained assuming the conjecture. In addition, many striking equivalent statements have been obtained, and no example of non embeddable von Neumann algebra is known so far. While many tried to disprove this conjecture, many new techniques, using noncommutative algebra, real algebraic geometry, representation theory and random matrices have been introduced over the last years.

References

• Fields Workshop around Connes' Embedding Problem - University of Ottawa, May 16- 18, 2008