Equivalent Forms of the Connes Embedding Problem

A most influential conjecture of Alain Connes in his seminal paper of 1976 has become, perhaps, a cornerstone problem in many areas including operator algebras. This conjecture, called the Connes Embedding Problem, states that every separable II_1 von Neumann algebra is embeddable into the ultraproduct of the hyperfinite II_1-factor von Neumann algebra. Though this conjecture is still an open problem, many equivalences have been found and we shall develop and discuss many of these.  In this talk, the notions of hypertraces, injective modules and completely positive maps are developed and will be the main tools for characterizing the Connes’ Embedding Problem.