Question

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.

Answer 1

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.