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

TL;DR

This paper introduces differential operators on C*-algebras, demonstrating their properties and applications, including the closure of Schwartz functions under smooth functional calculus on the tangent groupoid.

Contribution

It defines differential operators on C*-algebras and proves their domain is closed under smooth functional calculus, extending classical analysis to noncommutative geometry.

Findings

The domain of all differential operators on C*-algebras is closed under smooth functional calculus.

Schwartz functions on Connes tangent groupoid are closed under smooth functional calculus.

Provides a noncommutative analogue of differential operators on smooth manifolds.

Abstract

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.