# Infinite number of non-isomorphic von Neumann algebras with property Gamma?

A II$_1$ factor $\mathcal M$ with trace $\tau$ has property Gamma if for every $\epsilon > 0$ and finite set $\{x_1,\cdots, x_n\} \subset \mathcal M$ there exists a trace 0 unitary element $y\in\mathcal M$ such that $\sum_{i=1}^n \|x_iy - yx_i\|_2 < \epsilon$, where the 2-norm is $\|a\|_2 = \tau(a^*a)^{1/2}$. Murray and von Neumann defined this property to give the very first example of non-isomorphic II$_1$-factors, showing that the hyperfinite II$_1$-factor has Gamma and $L(\mathbb F_2)$ does not

Property Gamma can seem like a technical assumption but it is in fact a very natural and beautiful condition. An equivalent form is that $\mathcal M$ does not have property Gamma if and only if $\mathcal M' \cap \mathcal M^\mathcal U = \mathbb C$.

In 1969, McDuff published a lovely paper proving that there are an infinite number of non-isomorphic II$_1$-factors. Over the years there have been other such examples due to McDuff, Sakai, Connes and Popa (that I know of). However, I believe that none of these answered the following question:

Is there an infinite number of non-isomorphic II$_1$-factors with property Gamma?

This should be true and of some interest.

• Hi Chris! It is a fact that if you tensor a $II_1$ factor with a factor with property $\Gamma$, the resulting thing has property $\Gamma$. Therefore all of the McDuff factors (in the 1969 paper) have property $\Gamma$, as they tensorially absorb the hyperfinite II_1 factor (which has \Gamma). – Jon Bannon Sep 17 '15 at 0:43
• @JonBannon You are absolutely correct! I was misreading a later paper. Is it easy to see that these are McDuff factors? Perhaps you could give an answer outlining this or giving a reference. Thanks. – Chris Ramsey Sep 17 '15 at 14:42
• A paywall is keeping me away from the paper, but in McDuff's second paragraph it is outlined: plms.oxfordjournals.org/content/s3-21/3/443.extract If I have a moment I can try to write an answer, but this might take a while. – Jon Bannon Sep 18 '15 at 11:23
• @Jiang: That really is a nuclear bomb to kill a fly, though, isn't it? – Jon Bannon Sep 19 '15 at 5:51
• @JonBannon, Sometimes, big theorems suffer from the same fate as nuclear bombs: once invented, never used. I like this theorem very much. One reason is that it involves three fundamental objects in II$_1$ factors: the hyperfinite II$_1$ factor $R$, the free group factor $L(F_n)$ and tensor product. And many questions on $R\otimes L(F_n)$ is unanswered, e.g., I think it is not known whether there exists a MASA $A$ in $R\otimes L(F_n)$ such that $A$ is a maximal injective subalgebra. I tend to believe no. – Jiang Sep 19 '15 at 13:12