  This is an expanded lecture note for "Masterclass on sofic groups and
applications to operator algebras" (University of Copenhagen, 5-9 November
2012). It is about algebraic aspects of the Connes Embedding Conjecture. It
contains new proofs of equivalence of the Connes Embedding Conjecture,
Positivstellensatze for trace positive polynomials, Kirchberg's Conjecture, and
Tsirelson's Problem.
