5
$\begingroup$

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.

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ 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, 2012 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, 2012 at 17:45
  • 1
    $\begingroup$ 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, 2012 at 7:37
  • 1
    $\begingroup$ Also, to which paper of Effros are you referring? $\endgroup$
    – Jesse Peterson
    Jun 23, 2012 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, 2012 at 21:33

1 Answer 1

Reset to default
1
$\begingroup$

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.

Improve this answer
$\endgroup$
3
  • 1
    $\begingroup$ 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, 2012 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, 2012 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, 2012 at 1:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.