Differential operators on C*-algebras and applications to smooth functional calculus and Schwartz functions on the tangent groupoid

We introduce the notion of a differential operator on C*-algebras. This is a noncommutative analogue of a differential operator on a smooth manifold. We show that the common closed domain of all differential operators is closed under smooth functional calculus. As a corollary, we show that Schwartz functions on Connes tangent groupoid are closed under smooth functional calculus.