Analysis seminar
Speaker: Mikael Rørdam
Title: Around the Connes Embedding Problem: from operator algebras to groups and quantum information theory
Abstract: In his famous classification paper from 1976, Alain Connes suggested that it ought to be true that all finite von Neumann algebras admit an approximate embedding into the so-called hyperfinite type II_1 factor R. One can replace the hyperfinite II_1 factor by a matrix algebra, and so the Connes Embedding Problem asks if any von Neumann algebra (or C*-algebra) with a tracial state can be approximated by matrix algebras, with respect to the norm arising from the trace. The depth of this problem is witnessed by its many reformulations (and applications) in different areas of mathematics, including in group theory (Are all groups hyperlinear?) and later, via deep theorems of Kirchberg, in quantum information theory (in the form of Tsirelson’s conjecture). Very recently, a negative answer to the Connes Embedding Problem has been announced by Ji, Natarajan, Vidick, Wright and Yuen in their 245 pages-long paper titled MIP*=RE, using quantum complexity theory.
I will explain several facets of the Connes Embedding Problem, with particular emphasis on the related interplay between operator algebras and quantum information theory.