K-Theory for Operator Algebras

Blackadar, Bruce

doi:10.1007/978-1-4613-9572-0

Cited by 839 publications

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“…We refer the reader to [9] and [26] for background and notation for C*-algebras. In particular, we use'"" and ;S to denote Murray-von Neumann equivalence and subequivalence of projections, and we write M 00 (A) for the (non-unital) algebra consisting of w xw matrices over an algebra A with only finitely many nonzero entries.…”

Section: Applications To Operator Algebrasmentioning

confidence: 99%

Separative cancellation for projective modules over exchange rings

et al. 1998

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-A separative ring is one whose finitely generated projective modules satisfy the property A EB A � A EB B � B EB B ==} A ,...., B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separate exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ring R has an ideal I with I and Rf I both separative, then R is separative. SEPARATIVE CANCELLATION FOR PROJECTIVE MODULES OVER EXCHANGE RINGSP. ARA, K.R. GOODEARL, K.C. O'MEARA AND E. PARDO ABSTRACT. A separative ring is one whose finitely generated projective modules satisfy the property A EB A ~ A EB B ~ B EB B ~ A ~ B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separative exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ring R has an ideal I with I and R/ I both separative, then R is separative.

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Section: Applications To Operator Algebrasmentioning

confidence: 99%

Separative cancellation for projective modules over exchange rings

et al. 1998

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“…This example serves as the intuition for Kasparov's definition of KK-groups, which seem to be the most natural framework for our purposes. The precise definitions are a little technical [12][13][14][15] so all mathematical precision has been eliminated from the following discussion. The tensor products are all Z 2 graded tensor products.…”

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confidence: 99%

“…In general these algebras are not KK-equivalent. An example of this duality can be constructed as follows [11,14,13]: Suppose that E is the total space of a real vector bundle with a fibre metric over a differentiable manifold X. Consider the algebra of the sections of the complex Clifford bundle A ≡ Γ(Cliff(E)) and the algebra B ≡ C(X).…”

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confidence: 99%

“…for p a point in X, v p a vector in E p , with the norm computed with the fibre metric at p. If E has a spin c structure, Γ(Cliff(E)) is KK-equivalent to C(X) when E has evendimensional fibres, and KK-equivalent to C(X)⊗C 1 when E has odd-dimensional fibres [13], where C 1 is the Clifford algebra for C, the superalgebra with one generator of odd degree σ : σ 2 = 1. Since KK-groups have the appropriate Clifford periodicity properties, these algebras are KK-equivalent up to the shift for the odd-dimensional case.…”

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confidence: 99%

“…Of course, a U(1) action is not general enough to encompass all manifestations of IIA-IIB duality. The general setting is probably that of Takai duality [13] for Z 2 but with specific constraints so that the Takai dual is also the KK-dual.…”

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D-brane charges and K-homology

Periwal

2000

J. High Energy Phys.

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It is argued that D-brane charge takes values in K-homology. For smooth manifolds with spin structure, this could explain why the phase factor Ω(x) calculated with a D-brane state x in IIB theory appears in Diaconescu, Moore and Witten's computation of the partition function of IIA string theory.Diaconescu, Moore and Witten [1] recently compared the partition functions of IIA string theory and M-theory. They found agreement, amusingly interpreted as a derivation of K-theory from M-theory. Many subtle mathematical effects have to conspire for this agreement, so it seems important to explore further the building blocks of their results.K-theory will enter into the discussion in a variety of ways so here is a summary of K-theory in string theory. Witten [2] argued that D-brane charge should take values in K-theory, following earlier work by Cheung and Yin [5] and Moore and Minasian [4], and providing a mathematical setting for the results of Sen [6]. To be precise, the groups K 0(1) (X) are associated with D-branes in IIB(A) string theory on the spacetime X, respectively. In later work, Moore and Witten [3] argued that Ramond-Ramond fields should take values in K-theory as well, with K 1(0) (X) classifying these fields in IIB(A) string theory.An important point in [1] is the computation of a phase factor Ω(x) associated with an element x which is a Ramond-Ramond field in IIA string theory on a manifold X. Somewhat surprisingly, this phase factor is computed from a D-brane state in IIB string theory, also associated with x by means of the above arguments. The perplexing appearance of IIB string theory in a computation that a priori involves only IIA theory is pointed out in footnote 14 in [1].Regressing a little bit to Polchinski's basic operational definition of D-branes [7], recall that string theory in the presence of a D-brane is naïvely defined by specifying a submanifold M of X and including open strings which have both ends on M. Suppose we act with a diffeomorphismthen the submanifold1

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Decay of scalar curvature on uniformly contractible manifolds with finite asymptotic dimension

Wang,

Xie,

2023

Comm Pure Appl Math

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Gromov proved a quadratic decay inequality of scalar curvature for a class of complete manifolds. In this paper, we prove that for any uniformly contractible manifold with finite asymptotic dimension, its scalar curvature decays to zero at a rate depending only on the contractibility radius of the manifold and the diameter control of the asymptotic dimension. We construct examples of uniformly contractible manifolds with finite asymptotic dimension whose scalar curvature functions decay arbitrarily slowly. This shows that our result is the best possible. We prove our result by studying the index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control. A key technical ingredient for the proof of our main result is a Lipschitz control for the topological K‐theory of finite dimensional simplicial complexes.

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K-Theory for Operator Algebras

Cited by 839 publications

References 0 publications

Separative cancellation for projective modules over exchange rings

Separative cancellation for projective modules over exchange rings

D-brane charges and K-homology

Decay of scalar curvature on uniformly contractible manifolds with finite asymptotic dimension

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