The k-functor in the theory of extensions of C*-algebras

Kasparov, Gennadi

doi:10.1007/bf01078376

Cited by 282 publications

(556 citation statements)

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“…Therefore, since this later algebra is non commutative, all the construction could be used for any C * -algebra. Then Kasparov [39,40] defined the notion of KK-theory generalizing even more the K groups to correspondences between two C * -algebras. The problem investigated in the present paper is the latest development of a program that was initiated in the early eighties [38,5] when the first version of the gap labeling theorem was proved (see [6,9,13] for later developments).…”

Section: Historic Backgroundmentioning

confidence: 99%

A spectral sequence for theK-theory of tiling spaces

Savinien

Bellissard

2009

Ergod. Th. Dynam. Sys.

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Let T be an aperiodic and repetitive tiling of R d with finite local complexity. We present a spectral sequence that converges to the K-theory of T with page-2 given by a new cohomology that will be called PV in reference to the Pimsner-Voiculescu exact sequence. It is a generalization of the Serre spectral sequence. The PV cohomology of T generalizes the cohomology of the base space of a fibration with local coefficients in the K-theory of its fiber. We prove that it is isomorphic to thě Cech cohomology of the hull of T (a compactification of the family of its translates). Main ResultsLet T be an aperiodic and repetitive tiling of R d with finite local complexity (definition 3). The hull Ω is a compactification, with respect to an appropriate topology, of the family of translates of T by vectors of R d (definition 4). The tiles of T are given compatible ∆-complex decompositions (section 4.2), with each simplex punctured, and the ∆-transversal Ξ ∆ is the subset of Ω corresponding to translates of T having the puncture of one of those simplices at the origin 0 R d . The prototile space B 0 (definition 9) is built out of the prototiles of T (translational equivalence classes of tiles) by gluing them together according to the local configurations of their representatives in the tiling. The hull is given a dynamical system structure via the natural action of the group R d on itself by translation [13]. The C * -algebra of the hull is isomorphic to the crossed-productThere is a map p o from the hull onto the prototile space (proposition 1)which, thanks to a lamination structure on Ω (remark 2), resembles (although is not) a fibration with base space B 0 and fiber Ξ ∆ (remark 4). The Pimsner-Voiculescu (PV) *

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Section: Historic Backgroundmentioning

confidence: 99%

A spectral sequence for theK-theory of tiling spaces

Savinien

Bellissard

2009

Ergod. Th. Dynam. Sys.

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“…(Restriction to a finite subcomplex makes X compact, and thus makes C(X) a commutative C * -algebra with unit.) The "slant product" to be used here is the Kasparov intersection product of [Kas1] …”

Section: Theorem (Essentially Due To Karoubimentioning

confidence: 99%

Analytic Novikov for topologists

Rosenberg¹

1995

Novikov Conjectures, Index Theorems, and Rigidity

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Abstract. We explain for topologists the "dictionary" for understanding the analytic proofs of the Novikov conjecture, and how they relate to the surgery-theoretic proofs. In particular, we try to explain the following points:(1) Why do the analytic proofs of the Novikov conjecture require the introduction of C * -algebras? (2) Why do the analytic proofs of the Novikov conjecture all use Ktheory instead of L-theory? Aren't they computing the wrong thing? (3) How can one show that the index map µ or β studied by operator theorists matches up with the assembly map in surgery theory? (4) Where does "bounded surgery theory" appear in the analytic proofs?Can one find a correspondence between the sorts of arguments used by analysts and the controlled surgery arguments used by topologists?The literature on the Novikov conjecture (see [FRR]) consists of several different kinds of papers. Most of these fall into two classes: those based on topological arguments, usually involving surgery theory, and those based on analytic arguments, usually involving index theory. The purpose of this note is to "explain" the second class of papers to those familiar with the first class. I do not intend here to give a detailed sketch of the Kasparov KK-approach to the Novikov conjecture (for which the key details appear in [HsR]. Rather, I intend to concentrate on explaining the "dictionary" for relating the two main classes of papers on the Novikov conjecture, the topological and the analytic, and on trying

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“…This was pointed out in [12], but is really something which follows from [11], [7] and [9]. Let us assume that A is in the Bootstrap-category for which the UCT holds, cf.…”

Section: The Six-term Exact Sequencesmentioning

confidence: 99%

“…In the notation from the proof of Proposition 2.4 this shows that every element (11) (8)- (10) of Definition 3.2 ensure that is a * -homomorphism, and is G-equivariant by condition (11). By the Bartle-Graves selection theorem there is a continuous right inverse s for q and we set λ s…”

Section: Products and Pairingsmentioning

confidence: 99%