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.