Hodge theory for elliptic complexes over unital $$C^*$$ C ∗ -algebras

Krýsl, Svatopluk

doi:10.1007/s10455-013-9394-9

Cited by 4 publications

(8 citation statements)

References 14 publications

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“…Let us denote the order of △ i by r i . In Krýsl [9] (Theorem 11), the following implication is proved. If for each i ∈ N 0 the continuous extension of the Laplacian △ i to W ri (E i ) has closed image, then the cohomology groups…”

Section: De Rham Complex Twisted By the Oscillatory Representationmentioning

confidence: 92%

“…In particular, their kernels are finitely generated projective Hilbert A-modules. In Krýsl [9], the results of [3] were transferred to A-elliptic complexes and conclusions for the cohomology groups of these complexes were made.…”

Section: Introductionmentioning

confidence: 99%

“…This the de Rham complex twisted by the oscillatory module. We prove that this complex is A-elliptic and use a result from [9] to get an information on the cohomology groups of this complex when M is compact. As far as we know, this is the first A-elliptic complex in infinite rank vector bundles described in the literature.…”

Section: Introductionmentioning

confidence: 99%

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Cohomology of the de Rham complex twisted by the oscillatory representation

Krýsl

2014

Differential Geometry and its Applications

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We introduce a Hilbert A-module structure on the higher oscillatory module, where A denotes the C * -algebra of bounded endomorphisms of the basic oscillatory module. We also define the notion of an exterior covariant derivative in an A-Hilbert bundle and use it for a construction of an A-elliptic complex of differential operators for certain symplectic manifolds equipped with a symplectic connection. * E-mail address: Svatopluk.Krysl@mff.cuni.cz, Tel./Fax: + 420 222 323 221/+ 420 222 323 394 † I thank to Mike Eastwood for introducing me into the world of representation theory of Lie groups and its applications in differential geometry, and for his numerous inspiring lectures.

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Section: De Rham Complex Twisted By the Oscillatory Representationmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

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Cohomology of the de Rham complex twisted by the oscillatory representation

Krýsl

2014

Differential Geometry and its Applications

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“…Proof. See Theorem 7 in Krýsl [13] for the first claim, and the formula (5) in [13] for the second one.…”

Section: Complexes Of Pseudodifferential Operators In C * -Hilbert Bumentioning

confidence: 99%

Elliptic complexes over C∗-algebras of compact operators

Krýsl

2016

Journal of Geometry and Physics

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For a C * -algebra A of compact operators and a compact manifold M, we prove that the Hodge theory holds for A-elliptic complexes of pseudodifferential operators acting on smooth sections of finitely generated projective A-Hilbert bundles over M. For these C * -algebras, we get also a topological isomorphism between the cohomology groups of an A-elliptic complex and the space of harmonic elements. Consequently, the cohomology groups appear to be finitely generated projective C * -Hilbert modules and especially, Banach spaces. We prove as well, that if the Hodge theory holds for a complex in the category of Hilbert A-modules and continuous adjointable Hilbert A-module homomorphisms, the complex is self-adjoint parametrix possessing.Math. Subj. Class. 2010: Primary 46M18; Secondary 46L08, 46M20

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“…For a recent approach of Hodge theory using Hilbert modules, we refer to the recent article [44]. See also [42,43].…”

Section: Introductionmentioning

confidence: 99%

On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C<sup>*</sup>-Dynamical Systems

Fathizadeh

Gabriel

2016

SIGMA

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Abstract. The analog of the Chern-Gauss-Bonnet theorem is studied for a C * -dynamical system consisting of a C * -algebra A equipped with an ergodic action of a compact Lie group G. The structure of the Lie algebra g of G is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra A ⊂ A as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique G-invariant state on A, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on A and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case.

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Hodge theory for elliptic complexes over unital $$C^*$$ C ∗ -algebras

Cited by 4 publications

References 14 publications

Cohomology of the de Rham complex twisted by the oscillatory representation

Cohomology of the de Rham complex twisted by the oscillatory representation

Elliptic complexes over C∗-algebras of compact operators

On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C<sup>*</sup>-Dynamical Systems

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