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Background: In the $xy$-plane (or a 2-sphere if you are concerned about compactness), we can think about three different types of flow given by vector fields. (1) pushes everything towards the origin, (2) pushes everything away from the origin, (3) rotates about the origin. All these exponentiate to a unitary flow $u(t)$ on the $L^2$ Hilbert space (multiply by appropriate divergences to preserve the norm). This is very standard differential geometry or whatever, and the asymptotic behaviour is fairly obvious.

Problem: Suppose we are only given an abstract unitary operator $u(t)$ on a Hilbert space with a $*$-representation of a $C^*$-algebra (the continuous functions in the commutative case). How could we distinguish these cases (or cases similar to these) simply by looking at the unitary? How could we determine the limiting ($t\to +\infty$) state (if it exists?) on the $C^*$-algebra given an initial ($t=0$) state (a weight in the Hilbert space).

Thanks: I know that studying these flows is likely to be a completely standard problem for many readers of these pages. However, this is the first time I have been faced with this problem, and I would appreciate some assistance for where to look. This question arose from studying vector fields in noncommutative geometry.

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  • $\begingroup$ Could you please elaborate your question: What is that unitary $U(t)$ associated to a flow? In your problem do you mean: How can we extract the sink, source or center feature of the singularity based on non commutative abstract dynamical system? Your question is very interesting since I have an old interest in these areas but I really need your more explanation $\endgroup$ Commented Sep 6 at 11:39
  • $\begingroup$ to pass from xy plane to S^2 the Poincare compactification looks very relevant however why just S^2 and not "torus" mathoverflow.net/questions/178077/… $\endgroup$ Commented Sep 6 at 11:42
  • $\begingroup$ Is $U(t)f(x)=f(\phi_t(x))$ where $\phi$ is the flow of our vector field? $\endgroup$ Commented Sep 6 at 11:53

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