When one studies automorphisms of $II_1$ factors, one usually looks at the point norm topology - It is a well known result of Effros that if a $II_1$ factor $\mathcal{M}$ does not have property $\Gamma$, then Inn($\mathcal{M}$) is closed in Aut($\mathcal{M})$. The converse is also true : If Inn($\mathcal{M}$) is closed in Aut($\mathcal{M})$ in the point norm topology, then $\mathcal{M}$ does not have property $\Gamma$.

Has anyone studied the topology of pointwise SOT(equivalent to pointwise 2-norm) convergence? Formally, a net of automorphisms $\alpha_{\beta}$ converges to the automorphism $\alpha$ if for every $x$ in $\mathcal{M}$, $||\alpha(x) - \alpha_{\beta}(x)||_2 \rightarrow 0$.

Is is known, for instance, whether the inner automorphisms are always dense in Aut($\mathcal{M}$) in this topology?

Edit: Jesse Peterson is right - I was confusing topologies. Also, the statement that Inner automorphisms are closed in the point 2 - norm topology on Aut(M) $\Leftrightarrow $ The $II_1$ factor does not have property $\Gamma$ is theorem XIX.3.8 in Takesaki III. I thought it was due to Effros, but Takesaki does not give a reference.

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    $\begingroup$ Actually the point wise $\|\cdot\|_2$ topology is really the appropriate topology for studying Aut(M). Because in this topology you can use the Hilbert space structure. For example, Popa's notion of malleable deformation is a one-parameter family that converges to the identity in point wise $\|\cdot\|_2$ $\endgroup$ – Owen Sizemore Jun 22 '12 at 15:35
  • $\begingroup$ 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? $\endgroup$ – mohanravi Jun 22 '12 at 17:45
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    $\begingroup$ I think that perhaps you are confusing topologies. For II$_1$ factors it is much more common to study the topology of pointwise converges in the 2-norm. (Which is equivalent to pointwise convergence in $M_*$.) The characterization of property $\Gamma$ that I know uses this topology and not the topology of pointwise norm convergence in $M$. $\endgroup$ – Jesse Peterson Jun 23 '12 at 7:37
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    $\begingroup$ Also, to which paper of Effros are you referring? $\endgroup$ – Jesse Peterson Jun 23 '12 at 7:38
  • $\begingroup$ The equivalence between non-property $\Gamma$, and the subgroup of inner-automorphisms being closed is due independently to Sakai, "On automorphism groups of II$_1$-factors", 1974 (ams.org/mathscinet-getitem?mr=380443), and Connes, "Almost periodic states and factors of type III$_1$, 1974 (ams.org/mathscinet-getitem?mr=358374). See for instance the MathSciNet reviewers remarks on Sakai's paper. $\endgroup$ – Jesse Peterson Jul 4 '12 at 21:33

In regards to your original question. It is known that this is not always true. For example, Connes proof that a property (T) factor has countable fundamental group and Out(M)=Aut(M)/Inn(M), is done by taking the quotient as topological groups with both have the pointwise $\|\cdot\|_2$ topology.

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    $\begingroup$ 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. $\endgroup$ – mohanravi Jun 22 '12 at 21:47
  • $\begingroup$ Hmm, it's been a while since I looked at that. It is certainly true that you can also do it with the $\|\cdot\|_2$ norm, since that is the appropriate norm for also considering the convergence of the bimodules associated to the automorphisms. $\endgroup$ – Owen Sizemore Jun 23 '12 at 0:27
  • $\begingroup$ 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. $\endgroup$ – mohanravi Jun 23 '12 at 1:23

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