# Canonical Time Evolution for Type $II_{1}$-Factors?

This question was spurred by the answer of Steve Huntsman to the MO question here. The Tomita-Takesaki modular automorphism group gives rise to a canonical time evolution on a type $III$ factor (discussed in a blog post of Connes, here). I've heard Alain Connes suggest that there ought to be a canonical time evolution for type $II$ factors. I'd really like to know what experts think this thing should look like, if it should exist for certain classes of $II_1$ factors. In his talk, Connes mentioned a particular case that was suggestive, but I can't remember what it was.

Question: What should be the canonical time evolution for a type $II_{1}$ factor?

Of course, this should be some sort of "heat" semigroup of c.p. maps. I'm asking which of these should be canonical? Also, having something like this would give us a powerful invariant for finite von Neumann algebras, which explains why I have written should.

If this is possible for certain classes of factors, I'd be interested in references for this.

I'm also interested in arguments that prove/suggest that certain classes of factors cannot admit a canonical time evolution.

For example, in the class of group von Neumann algebras of finitely generated groups one can consider the c.p. semigroup that comes from a length function on the group. In any useful sense is this independent of the choice of length function?

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If the length function is not conditionally negative definite, why should this semigroup be c.p.? –  Alain Valette May 1 '11 at 9:25
Right. I think care must be taken to restrict to those groups for which this is a cnd function. (This kind of indicates what I've suggested may not be canonical...) –  Jon Bannon May 1 '11 at 12:31
@Jiang: The best known results in this direction follow the line of investigation started in Ioana, Peterson and Popa, which contains the first explicit computation of outer automorphism groups of factors. This work involves free products, but one cannot access free groups directly with these techniques. –  Jon Bannon Dec 10 at 0:01
@Jon, thanks, do you know that the factor, denote it by A, which has trivial outer morphism group, constructed in the above mentioned paper has unique Cartan masa up to unitary conjugacy? Since there are lots of results in the past years on the existence of unique Cartan masa up to unitary conjugacy, I have not checked whether the factor A has been covered in the latest work. If not, it seems that we still only know only one II_1 factor, i.e., the injective one, which has unique Cartan up to any automorphism of the factor. –  Jiang Dec 10 at 2:37
@Jon, also I want to say that Ioana's latest paper has reduced the calculation of the outer automorphism group of some equivalence relation associated to some action of the free group to the calculation of normalizer of some free subgroup inside SO(3), it is interesting. –  Jiang Dec 10 at 2:40