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Добірка наукової літератури з теми "Kasparov's KK-theory"

Автор: Grafiati
Опубліковано: 10 грудня 2022
Оновлено: 28 січня 2023

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Статті в журналах з теми "Kasparov's KK-theory"

1

Grensing, Martin. "A note on Kasparov products." Journal of K-Theory 10, no. 2 (January 24, 2012): 233–40. http://dx.doi.org/10.1017/is012001012jkt180.

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Анотація:
AbstractCombining Kasparov's generalization of a theorem of Voiculescu and Cuntz's description of KK-theory in terms of quasihomomorphisms (sections one and two), we give a simple construction of the Kasparov product (section three). This construction will be generalized in [Gre] to give a version of the product for so-called locally convex Kasparov modules over locally convex algebras in order to treat products of certain universal cycles.
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2

Leung, Ho-Hon. "Divided difference operators in equivariant KK-theory." Journal of Topology and Analysis 06, no. 02 (April 9, 2014): 237–61. http://dx.doi.org/10.1142/s1793525314500071.

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Let G be a compact connected Lie group with a maximal torus T. Let A, B be G-C*-algebras. We define certain divided difference operators on Kasparov's T-equivariant KK-group KKT(A, B) and show that KKG(A, B) is a direct summand of KKT(A, B). More precisely, a T-equivariant KK-class is G-equivariant if and only if it is annihilated by an ideal of divided difference operators. This result is a generalization of work done by Atiyah, Harada, Landweber and Sjamaar.
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3

Schick, Thomas. "Real versus complex K–theory using Kasparov’s bivariant KK–theory." Algebraic & Geometric Topology 4, no. 1 (May 29, 2004): 333–46. http://dx.doi.org/10.2140/agt.2004.4.333.

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4

Schochet, Claude. "Equivariant KK-theory for inverse limits of G-C*-algebras." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 56, no. 2 (April 1994): 183–211. http://dx.doi.org/10.1017/s1446788700034832.

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Анотація:
AbstractThe Kasparov groups are extended to the setting of inverse limits of G-C*-algebras, where G is assumed to be a locally compact group. The K K-product and other important features of the theory are generalized to this setting.
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5

Skandalis, George. "Exact Sequences for the Kasparov Groups of Graded Algebras." Canadian Journal of Mathematics 37, no. 2 (April 1, 1985): 193–216. http://dx.doi.org/10.4153/cjm-1985-013-x.

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Анотація:
In [11] G. G. Kasparov defined the “operator K-functor” KK(A, B) associated with the graded C*-algebras A and B. If the algebras A and B are trivially graded and A is nuclear he proves six term exact sequence theorems. He asks whether this extends to the graded case.Here we prove such “six-term exact sequence” results in the graded case. Our proof does not use nuclearity of the algebra A. This condition is replaced by a completely positive lifting condition (Theorem 1.1).Using our result we may extend the results by M. Pimsner and D. Voiculescu on the K groups of crossed products by free groups to KK groups [15]. We give however a different way of computing these groups using the equivariant KK-theory developed by G. G. Kasparov in [12]. This method also allows us to compute the KK groups of crossed products by PSL2(Z).
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6

Dadarlat, Marius, and Wilhelm Winter. "On the $KK$-theory of strongly self-absorbing $C^{*}$-algebras." MATHEMATICA SCANDINAVICA 104, no. 1 (March 1, 2009): 95. http://dx.doi.org/10.7146/math.scand.a-15086.

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Анотація:
Let $\mathcal D$ and $A$ be unital and separable $C^{*}$-algebras; let $\mathcal D$ be strongly self-absorbing. It is known that any two unital ${}^*$-homomorphisms from $\mathcal D$ to $A \otimes \mathcal D$ are approximately unitarily equivalent. We show that, if $\mathcal D$ is also $K_{1}$-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of $\mathcal D$ is asymptotically inner. Moreover, the space of automorphisms of $\mathcal D$ is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space $X$, the set of homotopy classes $[X,(\mathrm{Aut}(\mathcal D)]$ reduces to a point. The respective statement holds for the space of unital endomorphisms of $\mathcal D$. As an application, we give a description of the Kasparov group $KK(\mathcal D, A\otimes \mathcal D)$ in terms of $^*$-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group $KK(\mathcal D, A\otimes \mathcal D)$ is isomorphic to $K_0(A\otimes \mathcal D)$.
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7

Bárcenas, Noé. "Twisted geometric K-homology for proper actions of discrete groups." Journal of Topology and Analysis 12, no. 04 (December 14, 2018): 1019–40. http://dx.doi.org/10.1142/s1793525319500729.

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Анотація:
We define Twisted Equivariant [Formula: see text]-homology groups using geometric cycles. We compare them with analytical approaches using Kasparov KK-theory and (twisted) [Formula: see text]-algebras of groups and groupoids.
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8

Kaad, J., R. Nest, and A. Rennie. "KK-Theory and Spectral Flow in von Neumann Algebras." Journal of K-Theory 10, no. 2 (April 4, 2012): 241–77. http://dx.doi.org/10.1017/is012003003jkt185.

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AbstractWe present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko(J).Given a semifinite spectral triple (A, H, D) relative to (N, τ) with A separable, we construct a class [D] ∈ KK1(A, K(N)). For a unitary u ∈ A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product [u]A[D], and is simply related to the numerical spectral flow, and a refined C*-spectral flow.
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9

García-Compeán, H., W. Herrera-Suárez, B. A. Itzá-Ortiz, and O. Loaiza-Brito. "D-branes in orientifolds and orbifolds and Kasparov KK-theory." Journal of High Energy Physics 2008, no. 12 (December 1, 2008): 007. http://dx.doi.org/10.1088/1126-6708/2008/12/007.

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10

Meyer, Ralf, and Ryszard Nest. "C*-Algebras over Topological Spaces: Filtrated K-Theory." Canadian Journal of Mathematics 64, no. 2 (April 16, 2012): 368–408. http://dx.doi.org/10.4153/cjm-2011-061-x.

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Анотація:
AbstractWe define the filtrated K-theory of a C*-algebra over a finite topological spaceXand explain how to construct a spectral sequence that computes the bivariant Kasparov theory overXin terms of filtrated K-theory.For finite spaces with a totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated K-theory is not yet a complete invariant. We describe two C*-algebras over a spaceXwith four points that have isomorphic filtrated K-theory without being KK(X)-equivalent. For this spaceX, we enrich filtrated K-theory by another K-theory functor to a complete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient Theorem.
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Дисертації з теми "Kasparov's KK-theory"

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Kucerovsky, Daniel. "Kasparov, products in KK-theory, and unbound operators with applications to index theory." Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.259932.

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2

Van, Den Dungen Koen. "Lorentzian geometry and physics in Kasparov's theory." Phd thesis, 2015. http://hdl.handle.net/1885/15240.

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We study two geometric themes, Lorentzian geometry and gauge theory, from the perspective of Connes’ noncommutative geometry and (the unbounded version of) Kasparov’s KK-theory. Lorentzian geometry is the mathematical framework underlying Einstein’s description of gravity. The geometric formulation of a gauge theory (in terms of principal bundles) offers a classical description for the interactions between particles. The underlying motivation is the hope that this noncommutative approach may lead to a unified description of gauge theories coupled with gravity on a Lorentzian manifold. The main objects in noncommutative geometry are spectral triples, which encompass and generalise Riemannian spin manifolds. A spectral triple defines a class in K-homology, via which one can access the topology of the (noncommutative) manifold. In this thesis we present two possible definitions for ‘Lorentian spectral triples’, which offer noncommutative generalisations of Lorentzian manifolds as well. We will prove that both definitions preserve the link with analytic K-homology. We will describe under which conditions Lorentzian (or pseudo- Riemannian) manifolds satisfy these definitions. Another main example is the harmonic oscillator, which in particular shows that our framework allows to deal with more than just metrics of indefinite signature. In the context of noncommutative geometry, the description of a gauge theory can be obtained from so-called almost-commutative manifolds. While the usual approach yields by default a topologically trivial gauge theory (in the sense that the corresponding principal fibre bundle is globally trivial), we show in this thesis that the framework can be adapted, using the internal unbounded Kasparov product, to allow for globally non-trivial gauge theories as well. Finally, we combine the two themes of Lorentzian geometry and gauge theory, and we define Krein spectral triples, which generalise spectral triples from Hilbert spaces to Krein spaces. We use this definition to construct almost-commutative Lorentzian manifolds. Furthermore, we propose a Lorentzian alternative for the fermionic action, which allows to derive (the fermionic part of) the Lagrangian of a gauge theory. We show that our alternative action recovers exactly the correct physical Lagrangian.
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3

Forsyth, Iain Graham. "Boundaries and equivariant products in unbounded Kasparov theory." Phd thesis, 2016. http://hdl.handle.net/1885/101227.

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The thesis explores two distinct areas of noncommutative geometry: factorisation and boundaries. Both of these topics are concerned with cycles in Kasparov’s KK-theory which are defined using unbounded operators, and manipulating these cycles. These unbounded operators generalise the Dirac operators of classical geometry. The first topic of the thesis is factorisation, which is a process by which one attempts to represent the class of an equivariant spectral triple as a product of two unbounded Kasparov cycles, which, if they exist, are defined using the group action. We provide sufficient conditions for factorisation to be achieved for actions by compact abelian Lie groups. We apply our results to examples from Dirac operators on manifolds and their noncommutative theta-deformations. In particular, we show that the equivariant spectral triple associated to a Dirac operator on the total space of a compact torus principal bundle always factorises. The second topic of the thesis is relative spectral triples, which can be used to describe (noncommutative) manifolds with boundary. Whereas spectral triples are defined using self-adjoint unbounded operators, relative spectral triples are defined using symmetric unbounded operators. We show that the bounded transform of a relative spectral triple defines a relative Fredholm module, and hence a class in relative K-homology. We use relative spectral triples to investigate the boundary map in the six-term exact sequence of K-homology. We show that the boundary of a relative spectral triple has a simple description in terms of extension theory. With some additional data modelled on the inward normal of a manifold with boundary, we construct a triple which is a candidate for a spectral triple representing the boundary class of a relative spectral triple.
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4

Duwenig, Anna. "Poincaré self-duality of A_θ". Thesis, 2020. http://hdl.handle.net/1828/11678.

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Анотація:
The irrational rotation algebra A_θ is known to be Poincaré self-dual in the KK-theoretic sense. The spectral triple representing the required K-homology fundamental class was constructed by Connes out of the Dolbeault operator on the 2-torus, but so far, there has not been an explicit description of the dual element. We geometrically construct, for certain elements g of the modular group, a finitely generated projective module L_g over A_θ ⊗ A_θ out of a pair of transverse Kronecker flows on the 2-torus. For upper triangular g, we find an unbounded cycle representing the dual of said module under Kasparov product with Connes' class, and prove that this cycle is invertible in KK(A_θ,A_θ), allowing us to 'untwist' L_g to an unbounded cycle representing the unit for the self-duality of A_θ.
Graduate
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5

Nadareishvili, George. "A classification of localizing subcategories by relative homological algebra." Doctoral thesis, 2015. http://hdl.handle.net/11858/00-1735-0000-0028-867A-A.

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Книги з теми "Kasparov's KK-theory"

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Argentina) Luis Santaló Winter School-CIMPA Research School Topics in Noncommutative Geometry (3rd 2010 Buenos Aires. Topics in noncommutative geometry: Third Luis Santaló Winter School-CIMPA Research School Topics in Noncommutative Geometry, Universidad de Buenos Aires, Buenos Aires, Argentina, July 26-August 6, 2010. Edited by Cortiñas, Guillermo, editor of compilation. Providence, RI: American Mathematical Society, 2012.

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Частини книг з теми "Kasparov's KK-theory"

1

Blackadar, Bruce. "Kasparov’s KK-Theory." In Mathematical Sciences Research Institute Publications, 171–265. New York, NY: Springer New York, 1986. http://dx.doi.org/10.1007/978-1-4613-9572-0_8.

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2

Valette, Alain. "Kasparov’s Equivariant KK-theory." In Introduction to the Baum-Connes Conjecture, 41–46. Basel: Birkhäuser Basel, 2002. http://dx.doi.org/10.1007/978-3-0348-8187-6_5.

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3

Jensen, Kjeld Knudsen, and Klaus Thomsen. "The Kasparov Approach to KK-Theory." In Elements of KK-Theory, 47–92. Boston, MA: Birkhäuser Boston, 1991. http://dx.doi.org/10.1007/978-1-4612-0449-7_2.

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4

Jensen, Kjeld Knudsen, and Klaus Thomsen. "The Kasparov Groups for Ungraded C*-Algebras." In Elements of KK-Theory, 121–61. Boston, MA: Birkhäuser Boston, 1991. http://dx.doi.org/10.1007/978-1-4612-0449-7_4.

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