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 (i.e. within this class no "canonical" semigroup can serve as an isomorphism invariant).