A type $II_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x_{1}, x_{2},..., x_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $M$ with $\tau (u)=0$ and $||ux_{j}-x_{j}u||_{2}<\epsilon$ for all $1 \leq j \leq n$. (Here $||T||_2=(\tau(T^{*}T))^{1/2}$ for $T\in M$.)

A correspondence of von Neumann algebras $N$ and $M$ is a binormal Hilbert $N$-$M$ bimodule. One such bimodule $H$ is weakly contained in another $K$ if the first is in the closure of a countable (Hilbert) direct sum of copies of $K$ in the Fell topology defined in the preprint referred to below. In Popa's preprint, he considered a stronger notion of weak subequivalence which omits the direct sum in the above closure condition. Popa calls the latter condition weak containment, but in the literature that follows the above distinction is made. Before reading the following paragraph, choose and stick with one of these two notions and let $H\prec K$ denote ``$H$ weakly contained in/subequivalent to $K$''. I am interested in an answer to my question using either interpretation.

It seems that in Popa's 1986 Correspondence Preprint (cf. p. 45 of the preprint section 3.3 on "asymptotic commutativity") the correspondence characterization of property $\Gamma$ of a type $II_{1}$-factor is not quite right. There it is claimed that such an $N$ has $\Gamma$ if $L^{2}(N)\oplus L^{2}(N)\prec L^{2}(N)$. I think what was intended was the characterization found here on p. 27 in Theorem 3.9 part (2). That condition is more like ``$L^{2}(N)\prec L^{2}(N)\ominus \mathbb{C}1$'' which is not grammatical since it isn't properly a property of correspondences since the orthocomplement of the scalars is not an $N$-bimodule.

My question is the following:

Question: Can the Murray-von Neumann Property $\Gamma$ be characterized as Popa tried to via weak containment/weak subequivalence of certain correspondences?

Incidentally, this obviously would indirectly answer the question here pointed out to me by Yemon Choi.

  • $\begingroup$ Jon, are you sure that the definition of weak containment in Popa86 ("weak subequivalence") is the same as the definition of weak containment in your 2nd paragraph? $\endgroup$ – Yemon Choi Feb 14 '19 at 14:29
  • $\begingroup$ In fact it is not. In Popa's original preprint only closure was considered without multiplicity. With multiplicity, the first proposed characterization of Gamma holds for all N, even those without Gamma. I am interested in an answer to the question using either notion of weak containment. Thanks Yemon! $\endgroup$ – Jon Bannon Feb 14 '19 at 14:35
  • $\begingroup$ I have, perhaps inelegantly, incorporated the distinction Yemon points out in the body of the question. $\endgroup$ – Jon Bannon Feb 14 '19 at 16:44

A separable $\mathrm{II}_1$ factor $M$ is non-$\Gamma$ iff $L^2(M)\prec H$ and $H\prec L^2(M)$ imply $L^2(M)\subset H$.

Proof: Let $M$ be non-$\Gamma$ and take a finite critical set $F\subset M$. By $L^2(M)\prec H$, there is a unit vector $\xi\in H$ which is $(F,\epsilon)$-central. Assume for contrapositive that $L^2(M)\not\subset H$. Then, there is no central vector in $H$ and one can find $u\in U(M)$ such that $\|[u,\xi]\|\geq1/2$. This implies that $H\not\prec L^2(M)$ by non-$\Gamma$.

Conversely, suppose $M$ is $\Gamma$. Then take $\sigma\in\overline{\mathrm{Inn}}(M)\setminus\mathrm{Inn}(M)$ and put $H={}_ML^2(M)_{\sigma(M)}$. This $H$ violates the above condition.

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