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.

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