We have realized that that assertion in our paper ' Asymptotically Optimal Multi-Paving' that the existence of $(O(\epsilon^{-2}),\epsilon)$ pavings for zero diagonal matrices implies a polylogarithmic estimate for the quantitative commutator problem is incorrect. We have only been able to very slightly improve the Johnson-Ozawa-Schechtman result: We are able to show that given any zero diagonal $A \in M_m(\mathbb{C})$, there is a representation $A = [B,C]$ such that $||B|| ||C|| \leq C \operatorname{exp}(D\sqrt{\operatorname{log}(m)}) ||A||$, where $C, D$ are universal constants.

Owen, thanks for pointing out that point $||\cdot||_2$ convergence gives the convergence of the associated correspondences. I now see why Inn($\mathcal{M}$) is closed in Aut($\mathcal{M}$) in this topology for property T factors.

I just looked at the MathScinet review of Connes' paper - He uses the topology of pointwise norm convergence for the predual. He shows that if am ICC group has property T then the group of inner automorphisms is closed in Aut($L\Gamma$) in this aforementioned topology. I'll look through his paper more carefully to see if the same thing also holds for the point $||\cdot||_2$ topology.

Thinking about Owen Sizemore's comment, I just realised that it is not clear to me whether the point $||\cdot||_2$ limit of a net of automorphisms of a $II_1$ factor is an automorphism. It is an unital *endomorphism certainly, but why should it be onto?

Naive question perhaps - Why are the two questions equivalent? I don't see why the range of an unital idempotent is weakly closed.

Thanks to Matt for pointing out Takesaki's result. As Jon pointed out, I was wondering if there is a usable criterion that allows us to conclude that a given masa inside a type III factor admits a normal conditional expectation. Incidentally, Cartan masas, by definition, admit normal conditional expectations and there are many type III factors with Cartan masas.(Thanks to Stuart White for this comment).