An infinite-dimensional index theorem and the Higson-Kasparov-Trout algebra

Takata, Doman

doi:10.48550/arxiv.1811.06811

Cited by 2 publications

(6 citation statements)

References 10 publications

(47 reference statements)

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“…(2) follows from the same argument of the proof of [T4,Proposition 5.11]. We have computed the spectrum of ∂ 2 in Lemma 5.26.…”

Section: Let Us Introduce An Alternative Description Ofmentioning

confidence: 97%

“…(1) is automatic from Proposition 5.9. For (2), see [T4,Theorem 4.18]. ( 3) is obvious by definition.…”

Section: Let Us Introduce An Alternative Description Ofmentioning

confidence: 99%

“…Proof. It is proved by the same argument of [T4,Proposition 4.24]. The necessary change is only the following: In the previous paper, we used the property that…”

Section: Let Us Introduce An Alternative Description Ofmentioning

confidence: 99%

“…∂, and then we define ∂ by a linear combination of them following [T4,Definition 5.2]. See also [Kos,Was,FHT2,Mei2] for details on algebraic Dirac operators.…”

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confidence: 99%

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Topological Aspects of the Equivariant Index Theory of Infinite-Dimensional LT-Manifolds

Takata

2020

Preprint

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Let T be the circle group and let LT be its loop group. We formulate and investigate several topological aspects of the LT -equivariant index theory for proper LT -spaces, where proper LT -spaces are infinite-dimensional manifolds equipped with "proper cocompact" LT -actions. Concretely, we introduce "RKK-theory for infinite-dimensional manifolds", and by using it, we formulate an infinite-dimensional version of the KK-theoretical Poincaré duality homomorphism, and an infinite-dimensional version of the RKK-theory counterpart of the assembly map, for proper LT -spaces.The left hand side of the Poincaré duality homomorphism is formulated by the "C * -algebra of a Hilbert manifold" introduced by Guoliang Yu. Thus, the result of this paper suggests that this construction carries some topological information of Hilbert manifolds. In order to formulate the assembly map in a classical way, we need crossed products, which require an invariant measure of a group. However, there is an alternative formula to define them using generalized fixed-point algebras. We will adopt it as the definition of "crossed products by LT ".

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“…(2) follows from the same argument of the proof of [T4,Proposition 5.11]. We have computed the spectrum of ∂ 2 in Lemma 5.26.…”

Section: Let Us Introduce An Alternative Description Ofmentioning

confidence: 97%

“…(1) is automatic from Proposition 5.9. For (2), see [T4,Theorem 4.18]. ( 3) is obvious by definition.…”

Section: Let Us Introduce An Alternative Description Ofmentioning

confidence: 99%

“…Proof. It is proved by the same argument of [T4,Proposition 4.24]. The necessary change is only the following: In the previous paper, we used the property that…”

Section: Let Us Introduce An Alternative Description Ofmentioning

confidence: 99%

“…∂, and then we define ∂ by a linear combination of them following [T4,Definition 5.2]. See also [Kos,Was,FHT2,Mei2] for details on algebraic Dirac operators.…”

mentioning

confidence: 99%

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Topological Aspects of the Equivariant Index Theory of Infinite-Dimensional LT-Manifolds

Takata

2020

Preprint

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“…In [25] an LS 1 -equivariant index is constructed as an element in the fusion ring from the view point of KK-theory and non-commutative geometry. He also developed an index theorem in infinite dimensional setting in [23] [24]. It would be also interesting to investigate how our construction is positioned in Takata's theory.…”

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confidence: 99%

Deformation of Dirac operators along orbits and quantization of non-compact Hamiltonian torus manifolds

Fujita¹

2020

Preprint

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We give a formulation of a deformation of Dirac operator along orbits of a group action on a possibly non-compact manifold to get an equivariant index and a K-homology cycle representing the index. We apply this framework to non-compact Hamiltonian torus manifolds to define geometric quantization from the view point of index theory. We give two applications.The first one is a proof of a [Q,R]=0 type theorem, which can be regarded as a proof of the Vergne conjecture for Abelian case. The other is a Danilov-type formula for toric case in the non-compact setting, which shows that this geometric quantization is independent of the choice of polarization. The proofs are based on the localization of index to lattice points. H. FUJITA 7.2. [Q,R]=0 for non-compact Hamiltonian torus manifolds 28 7.3. A Danilov-type formula for non-compact toric manifolds 29 8. Comments and further discussions 30 8.1. Application to quantization of Hamiltonian loop group spaces 30 8.2. Deformation as KK-products 31 References 31

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An infinite-dimensional index theorem and the Higson-Kasparov-Trout algebra

Cited by 2 publications

References 10 publications

Topological Aspects of the Equivariant Index Theory of Infinite-Dimensional LT-Manifolds

Topological Aspects of the Equivariant Index Theory of Infinite-Dimensional LT-Manifolds

Deformation of Dirac operators along orbits and quantization of non-compact Hamiltonian torus manifolds

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