# Number of II${}_1$ factors

McDuff proved that there exist continuum many non-isomorphic (separable) II${}_1$ factors. I would like to politely ask whether it is known/open if one can find $2^{\mathfrak{c}}$ (or at least $\mathfrak{c}^+$) many such factors.

My feeling is that this is not possible to construct more than $\mathfrak{c}$ separable von Neumann algebras by a simple cardinality argument. The ball of $B(H)$ for $H$ separable, is metrisable under the ultraweak topology, so it has at most $\mathfrak{c}$ ultraweakly closed subsets. So we cannot have more than $\mathfrak{c}$ different balls, and consequently, have more than $\mathfrak{c}$ non-isomorphic algebras. Is this correct?

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## 1 Answer

Your argument is correct. An alternate and more "intrinsic" argument is to look at the predual, which is a separable Banach space. There are only continuum many separable Banach spaces, since they are determined by the metric on a countable dense subset that is a $\mathbb Q[i]$-vector space. Thus there are only continuum many separable von Neumann algebras, when considered just with their structure of dual Banach spaces. As Nik points out in the comments, the dual Banach space structure may not determine the algebra structure. However, given any such dual Banach space, you can fix a countable weak*-dense subset, and then any algebra structure will be determined by what it does on that countable set, so you again have only continuum many possibilities.

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A von Neumann algebra is determined as a dual Banach space by its predual, but I don't think its algebra structure is determined --- there are von Neumann algebras which aren't *-isomorphic to their opposite algebra, though certainly they are isometric. Actually I thought the original argument was okay: the ball of any von Neumann algebra sitting in $B(H)$ is an ultraweakly closed subset of the ball of $B(H)$, and there are only continuum many of the latter, so ... –  Nik Weaver Nov 29 '12 at 22:55
Ah, I see, I misread the original argument. I've never seen a proof, but I've been told that von Neumann algebras are determined with all their structure by just the predual, by a theorem of Sakai. –  Eric Wofsey Nov 30 '12 at 0:02
I think the theorem of Sakai you're thinking of is about uniqueness of the predual. There's only one predual up to isometric isomorphism (not true of dual Banach spaces in general). The paper at arxiv.org/pdf/1001.3890 gives several references to von Neumann algebras not *-isomorphic to their opposite algebras. –  Nik Weaver Nov 30 '12 at 0:43
Ah, you're right. Thanks! –  Eric Wofsey Nov 30 '12 at 0:46
No problem, nice to see you still have some interest in these things. –  Nik Weaver Nov 30 '12 at 3:25
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