Spectral flow and the unbounded Kasparov product

Kaad, Jens; Lesch, Matthias

doi:10.1016/j.aim.2013.08.015

Cited by 56 publications

(113 citation statements)

References 17 publications

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“…The following definition of an ‘operator algebra with involution’ can be found in . The terminology is taken from . Definition An operator algebra

scriptA

is an operator

*

‐algebra when it comes equipped with an involution

† : A \to A

such that (1)

scriptA

becomes a

*

‐algebra; (2)the involution is a complete isometry, thus

false∥ x^{†} ∥_{scriptA} = {∥ x ∥}_{scriptA} for all n \in N and x \in M_{n} false(scriptA false),

where

{(x^{†})}_{i j} = {(x_{j i})}^{†}

for all

i, j \in {1, \dots, n}

. We say that two operator

*

‐algebras

scriptA

and

scriptB

are cb‐isomorphic when there exists a cb‐isomorphism

ϕ : A \to B

of the underlying operator algebras such that

ϕ false(a^{†} false) = ϕ {(a)}^{†}

for all

a \in A

.…”

Section: Operator ∗‐Algebrasmentioning

confidence: 99%

“…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%

“…In , the rigged modules of were considered in the context of spectral triples, and inner products with values in a relevant operator

*

‐algebra were shown to arise naturally. Various analogues and intrinsic definitions in the general operator

*

‐algebra setting have been proposed starting in , and continuing in several papers of the second two authors and their coauthors.…”

Section: Introductionmentioning

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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“…The following definition of an ‘operator algebra with involution’ can be found in . The terminology is taken from . Definition An operator algebra

scriptA

is an operator

*

‐algebra when it comes equipped with an involution

† : A \to A

such that (1)

scriptA

becomes a

*

‐algebra; (2)the involution is a complete isometry, thus

false∥ x^{†} ∥_{scriptA} = {∥ x ∥}_{scriptA} for all n \in N and x \in M_{n} false(scriptA false),

where

{(x^{†})}_{i j} = {(x_{j i})}^{†}

for all

i, j \in {1, \dots, n}

. We say that two operator

*

‐algebras

scriptA

and

scriptB

are cb‐isomorphic when there exists a cb‐isomorphism

ϕ : A \to B

of the underlying operator algebras such that

ϕ false(a^{†} false) = ϕ {(a)}^{†}

for all

a \in A

.…”

Section: Operator ∗‐Algebrasmentioning

confidence: 99%

“…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%

“…In , the rigged modules of were considered in the context of spectral triples, and inner products with values in a relevant operator

*

‐algebra were shown to arise naturally. Various analogues and intrinsic definitions in the general operator

*

‐algebra setting have been proposed starting in , and continuing in several papers of the second two authors and their coauthors.…”

Section: Introductionmentioning

confidence: 99%

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Operator ∗‐correspondences in analysis and geometry

Blecher

Kaad

Mesland

2018

Proc. London Math. Soc.

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“…The statement in bounded KK-theory will have an immediate consequence in the KK-bordism groups only for algebras A such that the bounded transform is injective. We note that since any element of I has bounded adjointable commutators with D, the continuity of I → Lip(D) ensures the existence of a C > 0 such that (21) [D, a] End * B (Ẽ) ≤ C a I . We take matrix units (e jk ) ∞ j,k=0 .…”

Section: 2mentioning

confidence: 99%

The bordism group of unbounded KK-cycles

2018

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Abstract. We consider Hilsum's notion of bordism as an equivalence relation on unbounded KK-cycles and study the equivalence classes. Upon fixing two C * -algebras, and a * -subalgebra dense in the first C * -algebra, a Z/2Z-graded abelian group is obtained; it maps to the Kasparov KK-group of the two C * -algebras via the bounded transform. We study properties of this map both in general and in specific examples. In particular, it is an isomorphism if the first C * -algebra is the complex numbers (i.e., for K-theory) and is a split surjection if the first C * -algebra is the continuous functions on a compact manifold with boundary when one uses the Lipschitz functions as the dense * -subalgebra.

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“…(We note, by the way, that in the general Hilbert C * -module context, Kaad and Lesch ( [14,15]) give general conditions that ensure self-adjointness and regularity for a class of two-bytwo matrix operators that include D below.) Proof.…”

Section: Appendix a Unbounded Operators With Compact Resolventmentioning

confidence: 99%

Contractive Spectral Triples for Crossed Products

Paterson¹

2014

MATH. SCAND.

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Abstract. Connes showed that spectral triples encode (noncommutative) metric information. Further, Connes and Moscovici in their metric bundle construction showed that, as with the Takesaki duality theorem, forming a crossed product spectral triple can substantially simplify the structure. In a recent paper, Bellissard, Marcolli and Reihani (among other things) studied in depth metric notions for spectral triples and crossed product spectral triples for Z-actions, with applications in number theory and coding theory. In the work of Connes and Moscovici, crossed products involving groups of diffeomorphisms and even ofétale groupoids are required. With this motivation, the present paper develops part of the Bellissard-Marcolli-Reihani theory for a general discrete group action, and in particular, introduces coaction spectral triples and their associated metric notions. The isometric condition is replaced by the contractive condition.

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Spectral flow and the unbounded Kasparov product

Cited by 56 publications

References 17 publications

Operator ∗‐correspondences in analysis and geometry

Operator ∗‐correspondences in analysis and geometry

The bordism group of unbounded KK-cycles

Contractive Spectral Triples for Crossed Products

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