Why was Connes' embedding conjecture refuted and there are still no known non-hyperlinear groups?
In [MIP*=RE by JNVWY] the authors construct a non-local game that resolves Tsirelson's problem to the negative and by that refute Connes' embedding conjecture (CEC). The game *-algebra (see e.g. [KPS]) enables one to construct a finitely presented *-algebra from a given (synchronous) game. The *-algebra of the game from [JNVWY] is representable, has a trace, and its von Neumann closure is not embeddable in an ultraproduct of the hyperfinite II1II1-factor, i.e., it is not hyperlinear. If one could manage to generate a game for which the *-algebra both refutes CEC and is the complex group ring of some group GG, then one would have found a non-hyperlinear group. There is a known way of constructing games whose *-algebra is the group ring of some group GG. These are called "Linear constraint system games" (See e.g. [Slofstra, KPS]). In this talk I am going to:
- Go through the definitions of non-local games, game values and Tsirelson's problem.
- Introduce the game *-algebra.
- Study representations of the game *-algebra and their connection to various game values.
- Define linear constraint system games and discuss some properties of their *-algebra (namely, their game group).
- Explain how far the game in [JNVWY] is from being a linear constraint system game.