# How big are the ultrapowers of the hyperfinite $II_1$-factor?

Let $R$ be the hyperfinite $II_1$-factor. It is well-known that it is the smallest $II_1$-factor, in the sense that every $II_1$-factor contains a copy of $R$.

Now, let $\omega$ be a free ultrafilter on the natural numbers and construct the tracial ultrapower $R^\omega$. It is well-known, that this is a quite big factor. Indeed

1) Its predual is not separable. Even, just to make a bit of self-advertisement, its unitary group contains an orthogonal copy of the free group on uncountable many generators (see Capraro V. and Paunescu L. Product of ultrafilters and applications to Connes Embedding Problem, to appear in J. Oper. Theory)

2) A well-known conjecture, Connes' Embedding Conjecture, states that every $II_1$-factor with separable predual should be embeddable into $R^\omega$. Namely, all known $II_1$-factors with separable predual are known to embed into $R^\omega$ (or we do not know if they embed)

On the other hand, $R^\omega$ is constructed starting from the smallest $II_1$-factor using a quite natural construction that should not increase to much. So my question is:

** Question: ** Is it true that any $II_1$-factor with non-separable predual contains a copy of (some) $R^\omega$?

Actually I don't think this is true, since for instance it is not clear how $R^\omega$ could be contained in the group factor of the free group on uncountable many generators. In some sense, the latter has very few relations and $R^\omega$ has lots of relations, but I don't understand quite well how this difference can be reflected in the group factors.

My interest in this problem comes from the following result by Taka Ozawa. He showed that there is indeed no way to find such a universal factor among the ones with separable predual. See N.Ozawa,* There is no separable universal $II_1$-factor*, Proc. Amer. Math. Soc., vol. 132, iss. 2, pp. 487-490, 2004.

So, a way to reformulate my question is: Is Connes' embedding conjecture the best conjecture one can formulate?

Maybe, it's not the best conjecture, but it's the best one can hope for.

Thanks in advance for any comment.

Valerio

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I think a more interesting problem to consider in light of Ozawa's above mentioned result is whether Connes' imbedding problem is equivalent to the existence of a separable II$_1$ factor $N$ and a free ultrafilter $\omega$ such that every separable II$_1$ factor embeds into $N^\omega$? Perhaps this should be asked as a separate problem. –  Jesse Peterson Sep 27 '11 at 15:39
Are you saying that might be enough to find another separable factor? Instead, I was thinking to the following problem: could it be true that if a $II_1$ factor contains all separable $II_1$ factors, then it contains $R^\omega$? Maybe this is the right way to say that Connes imbedding/embedding conjecture is the best we can hope for. P.s. how do you say imbedding or embedding?? –  Valerio Capraro Sep 27 '11 at 16:32
It seems likely to me that the techniques in Popa's paper ams.org.proxy/mathscinet-getitem?mr=703810 could be used to show without much effort that $R^\omega$ cannot embed into any free product of separable II$_1$ factors. Assuming this, then the free product of all isomorphism classes of separable II$_1$ factors would be a counter-example to your problem. –  Jesse Peterson Sep 27 '11 at 17:25
I think imbed and embed are both correct to use. I think I usually use embed for whatever that's worth. –  Jesse Peterson Sep 27 '11 at 17:25
Your link to Popa's paper has something wrong. Which paper is that? –  Valerio Capraro Sep 28 '11 at 7:32

This is not true. If we denote by $\mathbb F_{\mathbb R}$ the free group with generators indexed by $\mathbb R$ then any separable von Neumann subalgebra $N \subset L\mathbb F_{\mathbb R}$ must necessarily be contained in the von Neumann subalgebra $L\mathbb F_I$ for some countably infinite subset $I \subset \mathbb R$. (This is a direct consequence of applying Parseval's identity to vectors in $L^2N \subset \ell^2 \mathbb F_{\mathbb R}$.)
The separable $II_1$ factor $L ({\rm PSL}(3, \mathbb Z))$ embeds into $R^\omega$ since the group is residually finite, but does not embed into $L\mathbb F_I$ by a result of Connes and Jones since it has property (T). Hence, $L\mathbb F_{\mathbb R}$ cannot contain $R^\omega$.