A question about automorphisms of $II_1$ factors - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:57:34Z http://mathoverflow.net/feeds/question/100361 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100361/a-question-about-automorphisms-of-ii-1-factors A question about automorphisms of $II_1$ factors mohanravi 2012-06-22T14:35:51Z 2012-07-04T14:31:00Z <p>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 $\Gamma$. </p> <p>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 $\mathcal{M}$, $||\alpha(x) - \alpha_{\beta}(x)||_2 \rightarrow 0$.</p> <p>Is is known, for instance, whether the inner automorphisms are always dense in Aut($\mathcal{M}$) in this topology? </p> <p>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. </p> http://mathoverflow.net/questions/100361/a-question-about-automorphisms-of-ii-1-factors/100396#100396 Answer by Owen Sizemore for A question about automorphisms of $II_1$ factors Owen Sizemore 2012-06-22T21:17:31Z 2012-06-22T21:17:31Z <p>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. </p>