Nonunital spectral triples and metric completeness in unbounded KK-theory

Mesland, Bram; Rennie, Adam

doi:10.1016/j.jfa.2016.08.004

Cited by 37 publications

(70 citation statements)

References 42 publications

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“…In the paper , M. Hilsum introduces the Banach

*

‐algebra

Lip (δ)

associated to a symmetric operator using the norm

{false∥ \cdot false∥}_{δ}

defined in equation . Its structure as an operator

*

‐algebra has been first used in and later in , in the context of the unbounded Kasparov product.…”

Section: Operator ∗‐Algebrasmentioning

confidence: 99%

“…More generally, this fact presents an obstruction to a straightforward definition of the analogue of adjointable operators on operator

*

‐modules. When the operator

*

‐algebra

K (X)

has a bounded approximate unit, the use of multiplier algebras provides a solution to this problem, as has been employed in . In many situations, notably in case the Riemannian manifold

M

in Example above is not complete, such an approximate unit does not exist.…”

Section: Operator ∗‐Correspondencesmentioning

confidence: 99%

“…For example, there are cases where

K (X)

does not have a bounded approximate identity, but where its closure in the representation does. The reader is referred to [, Definition 3.9, Proposition 3.10, and Remark 3.11] for a detailed discussion of this situation.…”

Section: Operator ∗‐Correspondencesmentioning

confidence: 99%

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Operator ∗‐correspondences in analysis and geometry

Blecher

Kaad

Mesland

2018

Proc. London Math. Soc.

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An operator ∗‐algebra is a non‐self‐adjoint operator algebra with completely isometric involution. We show that any operator ∗‐algebra admits a faithful representation on a Hilbert space in such a way that the involution coincides with the operator adjoint up to conjugation by a symmetry. We introduce operator ∗‐correspondences as a general class of inner product modules over operator ∗‐algebras and prove a similar representation theorem for them. From this, we derive the existence of linking operator ∗‐algebras for operator ∗‐correspondences. We illustrate the relevance of this class of inner product modules by providing numerous examples arising from noncommutative geometry.

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“…In the paper , M. Hilsum introduces the Banach

*

‐algebra

Lip (δ)

associated to a symmetric operator using the norm

{false∥ \cdot false∥}_{δ}

defined in equation . Its structure as an operator

*

‐algebra has been first used in and later in , in the context of the unbounded Kasparov product.…”

Section: Operator ∗‐Algebrasmentioning

confidence: 99%

“…More generally, this fact presents an obstruction to a straightforward definition of the analogue of adjointable operators on operator

*

‐modules. When the operator

*

‐algebra

K (X)

has a bounded approximate unit, the use of multiplier algebras provides a solution to this problem, as has been employed in . In many situations, notably in case the Riemannian manifold

M

in Example above is not complete, such an approximate unit does not exist.…”

Section: Operator ∗‐Correspondencesmentioning

confidence: 99%

See 1 more Smart Citation

Operator ∗‐correspondences in analysis and geometry

Blecher

Kaad

Mesland

2018

Proc. London Math. Soc.

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“…Coupling the unbounded KK-cycle for (C * r (G), C(Ω 0 )) to such a spectral triple for C(Ω 0 ) using the unbounded Kasparov product, gives us K-homology representatives for C * r (G). The construction of the product operator employs the techniques developed in [66], but the commutators with C * r (G) turn out to be unbounded. Nonetheless, using arguments similar to [36] and recent results in [63], we are able to show that the operator represents the Kasparov product of the given classes via the bounded transform.…”

Section: Introductionmentioning

confidence: 99%

Index Theory and Topological Phases of Aperiodic Lattices

Bourne

Mesland

2019

Ann. Henri Poincaré

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We examine the noncommutative index theory associated to the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological phases of aperiodic lattices and materials, and applies to invariants from tilings as well. Our discussion concerns semifinite index pairings, factorisation properties of Kasparov modules and the construction of unbounded Fredholm modules for lattices with finite local complexity.

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“…Instead we closely follow the "classical" theory as detailed in [18,35,14,19], inspired by the very modern approach of [1]. The final two sections are concerned with crossed products, both full and reduced, and the associated descent map on equivariant KK-theory, for which we draw greatly on [21,31].Throughout the exposition, we include a description of the unbounded picture [2] for future applications where a need to compute explicit representatives of Kasparov products [22,26,29,17,3,30] and index formulae [11,15,6,7,8,9,4,5] may arise. The results regarding the unbounded picture have already appeared, in the context of Lie groupoids, in [27], which itself draws on the theory outlined in [33] in the Hausdorff setting.…”

mentioning

confidence: 99%

Equivariant KK-theory for non-Hausdorff groupoids

MacDonald

2020

Journal of Geometry and Physics

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We give a detailed and unified survey of equivariant KK-theory over locally compact, second countable, locally Hausdorff groupoids. We indicate precisely how the "classical" proofs relating to the Kasparov product can be used almost wordfor-word in this setting, and give proofs for several results which do not currently appear in the literature. This article is intended as a detailed survey of the theory of groupoid-equivariant KKtheory, in the setting of locally compact, second countable, locally Hausdorff groupoids.Following a substantial period of innovation and development [18,12,35,14,19] in the non-equivariant and group-equivariant cases, groupoid-equivariant KK-theory was first treated, for Hausdorff groupoids, by Le Gall in his PhD thesis [24]. Since many examples of groupoids are only locally Hausdorff (for instance, the holonomy groupoids of foliated manifolds [10]), it is desirable to have a groupoid-equivariant KK-theory for non-Hausdorff groupoids as well.Despite the demand arising from applications to foliation theory, the literature on equivariant KK-theory over non-Hausdorff groupoids has, up until this point, been rather sparse. The non-Hausdorff theory makes its first appearance in [36], in which, for details on the inner workings of the theory, the reader is referred to the expositions given forHausdorff groupoids in [25,37]. Since then the theory has been very rapidly expounded upon, in an extremely general context that covers not only groupoids but Hopf algebras, in [1] for applications to singular foliations. Finally, the theory has found use in [27] in the study of the Godbillon-Vey invariant of regular foliated manifolds.The common denominator in all of the treatments [36, 1, 27] of equivariant KK-theory over non-Hausdorff groupoids is that each has been focussed on a particular application of equivariant KK-theory, rather than on a detailed development of the theory itself.As a consequence, most details (some of which are somewhat nontrivial) are skipped over, and it is difficult to glean a precise understanding of how the non-Hausdorffness of the underlying groupoid affects the "classical" definitions, statements of results, and their proofs. The goal of the present paper is to rectify this state of the literature, by providing 1 a unified and detailed description of the theory. In particular, we include all the necessary definitions (of which there are many), and include detailed proofs for several results which do not yet appear in the literature. We will see, as remarked in [1], that much of the theory can be proved in exactly the manner of the "classical" theory, provided one makes some conceptual substitutions.Let us briefly describe the content of the paper. The first three sections consist of the relevant definitions for groupoids, upper-semicontinuous bundles and actions of groupoids on algebras and Hilbert modules. Note that we do not go to the generality of Fell bundles, and we refer the reader to [37] and [36] for an indication of how the theory works at this level of general...

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Nonunital spectral triples and metric completeness in unbounded KK-theory

Cited by 37 publications

References 42 publications

Operator ∗‐correspondences in analysis and geometry

Operator ∗‐correspondences in analysis and geometry

Index Theory and Topological Phases of Aperiodic Lattices

Equivariant KK-theory for non-Hausdorff groupoids

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