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We introduce higher order derivations on C*-algebras which are anoncommutative analogue of differential operators for commutative C*-algebras.We show that the domain of higher derivations is closed under smooth functionalcalculus. As a corollary, we show that Schwartz functions on Connes tangentgroupoid are closed under smooth functional calculus.

Higher order derivations on C*-algebras and applications to smooth functional calculus and Schwartz functions on the tangent groupoid