This question was spurred by the answer of Steve Huntsman to the MO question [here.][1] The Tomita-Takesaki modular automorphism group can be regarded as a canonical time evolution on a type $III$ factor (discussed in a blog post of Connes, [here][2]). 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 would a canonical time evolution for (certain classes
> of) type $II_{1}$ factors look like?

Apologies for the vague question, but I'm fishing for what's known about this topic in order to adjust my world-view.


  [1]: http://mathoverflow.net/questions/57820/is-there-a-mathematical-axiomatization-of-time-other-than-perhaps-entropy
  [2]: http://noncommutativegeometry.blogspot.com/2007/10/heart-bit-1.html