In this paper, we construct an explicit quasi-isomorphism to study the cyclic cohomology of a def... more In this paper, we construct an explicit quasi-isomorphism to study the cyclic cohomology of a deformation quantization over a Riemannian étale groupoid. Such a quasi-isomorphism allows us to propose a general algebraic index problem for Riemannian étale groupoids. We discuss solutions to that index problem when the groupoid is proper or defined by a constant Dirac structure on a 3-dimensional torus.
In this paper we study the Hochschild cohomology ring of convolution algebras associated to orbif... more In this paper we study the Hochschild cohomology ring of convolution algebras associated to orbifolds, as well as their deformation quantizations. In the first case the ring structure is given in terms of a wedge product on twisted polyvectorfields on the inertia orbifold. After deformation quantization, the ring structure defines a product on the cohomology of the inertia orbifold. We
We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of... more We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $\Gamma\subset\R^p$, we construct a unique ``symbol valued trace'', which extends the $L^2$-trace on operators of small order. This allows to construct various trace functionals in a systematic way. Furthermore we study the higher-dimensional eta-invariants on algebras with parameter space $\R^{2k-1}$. Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over $\R^{2k-1}$. The eta-invariant of this family coincides with the spectral eta-invariant of the operator.
In this paper, we construct an explicit quasi-isomorphism to study the cyclic cohomology of a def... more In this paper, we construct an explicit quasi-isomorphism to study the cyclic cohomology of a deformation quantization over a Riemannian étale groupoid. Such a quasi-isomorphism allows us to propose a general algebraic index problem for Riemannian étale groupoids. We discuss solutions to that index problem when the groupoid is proper or defined by a constant Dirac structure on a 3-dimensional torus.
In this paper we study the Hochschild cohomology ring of convolution algebras associated to orbif... more In this paper we study the Hochschild cohomology ring of convolution algebras associated to orbifolds, as well as their deformation quantizations. In the first case the ring structure is given in terms of a wedge product on twisted polyvectorfields on the inertia orbifold. After deformation quantization, the ring structure defines a product on the cohomology of the inertia orbifold. We
We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of... more We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $\Gamma\subset\R^p$, we construct a unique ``symbol valued trace'', which extends the $L^2$-trace on operators of small order. This allows to construct various trace functionals in a systematic way. Furthermore we study the higher-dimensional eta-invariants on algebras with parameter space $\R^{2k-1}$. Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over $\R^{2k-1}$. The eta-invariant of this family coincides with the spectral eta-invariant of the operator.
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Papers by Markus Pflaum