A continuous field of C∗-algebras and the tangent groupoid for manifolds with boundary

For a smooth manifold X with boundary we construct a semi-groupoid T−X and a continuous field of C∗-algebras which extend Connes' construction of the tangent groupoid. We show the asymptotic multiplicativity of ℏ-scaled truncated pseudodifferential operators with smoothing symbols and compute the K-theory of the associated symbol algebra.