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Given a manifold B with conical singularities, we consider the cone algebra with discrete asymptotics, introduced by Schulze, on a suitable scale of Lp-Sobolev spaces. Ellipticity is proven to be equivalent to the Fredholm property in these spaces; it turns out to be independent of the choice of p. We then show that the cone algebra is closed under… (More)
Adiabatic vacuum states are a well-known class of physical states for linear quantum fields on Robertson-Walker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting field theories. Hadamard states form a special subclass of… (More)
We consider the norm closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a manifold X with boundary ∂X. We first describe the image and the kernel of the continuous extension of the boundary principal symbol homomorphism to A. If X is connected and ∂X is not empty, we then show that the K-groups of A are… (More)
Let M be a closed manifold. We show that the Kontsevich-Vishik trace, which is defined on the set of all classical pseudodifferential operators on M , whose (complex) order is not an integer greater than or equal to − dim M , is the unique functional which (i) is linear on its domain, (ii) has the trace property and (iii) coincides with the L 2-operator… (More)
For a smooth manifold X with boundary we construct a semigroupoid T − X and a continuous field C * r (T − X) of C *-algebras which extend Connes' construction of the tangent groupoid. We show the asymptotic multiplicativity of-scaled truncated pseudodifferential operators with smoothing symbols and compute the K-theory of the associated symbol algebra.
Can Boutet de Monvel's algebra on a compact manifold with boundary be obtained as the algebra Ψ 0 (G) of pseudodifferential operators on some Lie groupoid G? If it could, the kernel G of the principal symbol homomorphism would be isomorphic to the groupoid C *-algebra C * (G). While the answer to the above question remains open, we exhibit in this paper a… (More)
We study the C *-closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a compact connected manifold X with boundary ∂X = ∅. We find short exact sequences in K-theory which split, so that Ki(A/K) ∼ = Ki(C(X))⊕K1−i(C0(T * X •)). Using only simple K-theoretic arguments and the Atiyah-Singer Index Theorem, we show… (More)
For a pseudodifferential boundary operator A of order ν ∈ Z and class 0 (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Tr(AB −s), where B is an auxiliary system formed of the Dirichlet realization of a second order strongly elliptic differential operator and an elliptic operator on the… (More)
For manifolds with boundary, we define an extension of Wodzicki's noncommutative residue to boundary value problems in Boutet de Monvel's calculus. We show that this residue can be recovered with the help of heat kernel expansions and explore its relation to Dixmier's trace.