I know only a little bit about C$^*$-algebras and I want a to know if you know a nice apllication or the influence of them in the field of partial differential equations (it is better that it is understandable for graduate students), or maybe can explain me why they are important for pseudo-differential-operators. A similar question i found here https://math.stackexchange.com/questions/798342/application-of-calgebras but there are not enough answers to convince me. I'm not sure if it is ok to ask it here, but maybe anyone could know more about C$^*$-algebras and partial differential equations. Regards
I think the canonical connection between C*-algebra and differential operators is Connes' index theorem for foliated manifolds. I don't know if that counts as PDEs but it's certainly related. Every foliated manifold $M$ has an associated C*-algebra $A$ which is noncommutative (except in trivial cases) but in some way embodies the idea of "the continuous functions on $M$ that are constant on leaves". Any pseudodifferential operator $D$ on $M$ which is elliptic on each leaf has an "index" which belongs to the $K_0$ group of $A$. There is an analytic definition and a topological definition of the index, and Connes' index theorem says that they agree. It is a profound generalization of the Atiyah-Singer index theorem. Connes' notes on the subject can be found here.