I am a Ph.D student involved in topics like integrability of foliations arising from center stable bundles of partially hyperbolic dynamical systems. These are generally only continuous bundles so one either has to use the dynamics of the map, find some meaningful approximations or try to generalize the theorems of smooth integrability and foliation theory to continuous case (it is more doable in the case of foliation theory).
In studying these topics I have come to see that there are alot of interactions with geometry of foliations which I became interested with. Where smoothness fails and you can not use Frobenius theorem, you can make some intersting connections to foliation theory (for instance Novikov theorem) to see whether a system is integrable or not.
Following this I have been hearing the "new applications of C* algebras to dynamical systems" by finding "C* algebra of a dynamical system". and I have also seen in Connes book that C* algebras give a new way of studying foliation spaces. So I became interested in these, although I have only recently met them and did not have much chance to read about them.
My first question would, what direction (books, questions ideas) would you propose to someone who wants to learn this/these field. My second question is, what is vaguely the use of C* algebras about foliations and dynamical systems. What kind of their properties is studied through their use? Finally does C* algebras study the case of continuous foliatons :-)