# Connes Embedding Conjecture and Fusion Categories

I was recently introduced to Connes' Embedding Conjecture (CEC) which states:

Every separable type $II_{1}$ factor is embeddable into $R^{\omega}$. Where $\omega$ is a generic free ultrafilter on $\mathbb{N}$, $R$ is a hyperfinite type $II_{1}$ factor, and $R^{\omega}$ is the ultrapower of $R$ with respect to $\omega$.

I'm not well versed in the theory of factors and this conjecture is mysterious and difficult for me to understand. However, there do seem to be a lot of equivalent formulations of CEC. For some examples of this see: N. Ozawa, "About the Connes Embedding Conjecture---Algebraic Approaches---, arXiv:1212.1700 [math.QA]

My background is in fusion categories, and given the relationship between subfactors and fusion categories I was led to wonder:

Question: Are there statements in the theory of fusion categories that are equivalent to CEC, or does CEC have implications for fusion categories?

My understanding of the connections between fusion categories and subfactors, I'm sorry to say, is quite weak. So it very well could be that there is simply no relation between CEC and fusion categories, which would be a completely acceptable answer.

## 1 Answer

For fusion categories, there is a hyperfinite embedding result which does not require passing to ultraproducts.

If $\mathcal{C}$ is a unitary fusion category, then by results of Popa [MR1055708] (see also [MR3028581, Theorem 4.1]), there is a finite index, depth two subfactor, which we could reasonably denote $R\subset R\rtimes\mathcal{C}$, whose principal even half is $\mathcal{C}$. (Here, $R$ is the hyperfinite II$_1$-factor.) Moreover, by Popa's theorem, $R\subset R\rtimes \mathcal{C}$ is the unique hyperfinite subfactor up to subfactor isomorphism with this standard invariant.

As a corollary, every unitary fusion category has an essentially unique realization as a concrete category of bifinite bimodules over $R$. (One has to be careful to get the correct uniqueness statement here.)

I'm not really sure how to introduce ultraproducts or Connes' embedding conjecture into this story. I'll think it over, and if I have something reasonable to say, I'll edit this answer.

• I would have answered the same. One remark is that if one consider C${}^*$ tensor categories in more generality, one gets a lot of interesting analysis and CEC might be relevant. One definitely wants to drop the finiteness condition on fusion categories – Marcel Bischoff Jul 31 '15 at 0:10
• @David thanks, that is very helpful. – Paul Jul 31 '15 at 17:01
• @Marcel, if you have any other insights relating C* tensor categories and CED, then I'd be interested in hearing them. – Paul Jul 31 '15 at 17:03