Unbounded bivariant <i>K</i>-theory and correspondences in noncommutative geometry

Mesland, Bram

doi:10.1515/crelle-2012-0076

Cited by 72 publications

(132 citation statements)

References 21 publications

(37 reference statements)

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“…Although this follows from the (more general) framework of Mesland [Mes14], we will prove it directly using the following result.…”

Section: Definition 224 ([Bj83]mentioning

confidence: 99%

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On globally non-trivial almost-commutative manifolds

Boeijink

Dungen

2014

Journal of Mathematical Physics

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Within the framework of Connes' noncommutative geometry, we define and study globally non-trivial (or topologically non-trivial) almost-commutative manifolds. In particular, we focus on those almostcommutative manifolds that lead to a description of a (classical) gauge theory on the underlying base manifold. Such an almost-commutative manifold is described in terms of a 'principal module', which we build from a principal fibre bundle and a finite spectral triple. We also define the purely algebraic notion of 'gauge modules', and show that this yields a proper subclass of the principal modules. We describe how a principal module leads to the description of a gauge theory, and we provide two basic yet illustrative examples.

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“…Although this follows from the (more general) framework of Mesland [Mes14], we will prove it directly using the following result.…”

Section: Definition 224 ([Bj83]mentioning

confidence: 99%

“…The construction of I × ∇ M via Kasparov products fits naturally in the framework of Mesland's category of spectral triples [Mes14], where the internal space I ∞ with the connection ∇ can be seen as (a representative of) a morphism from the canonical triple for M to the almost-commutative manifold I ∞ × ∇ M .…”

Section: The Idea Is To Prove That (mentioning

confidence: 99%

On globally non-trivial almost-commutative manifolds

Boeijink

Dungen

2014

Journal of Mathematical Physics

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“…[22]): we discuss here only the case of our interest. A connection on a right Hilbert A -module E (with A the C * -completion of A ) will be then a map with dense domain E 1 ⊂ E and image in…”

Section: On the Relation Between The Moyal Plane And The 1-point Spacementioning

confidence: 99%

“…Following [26] we work with connections associated to Connes' differential calculus, rather than universal connections as in [22]. Objects in our category are spectral triples, rather than unitary equivalence classes.…”

Section: Introductionmentioning

confidence: 99%

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Matrix geometries emergent from a point

2014

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We describe a categorical approach to finite noncommutative geometries. Objects in the category are spectral triples, rather than unitary equivalence classes as in other approaches. This enables to treat fluctuations of the metric and unitary equivalences on the same footing, as representatives of particular morphisms in this category. We then show how a matrix geometry (Moyal plane) emerges as a fluctuation from one point, and discuss some geometric aspects of this space.

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Homotopy Theory of C*-Algebras

Østvær¹

2010

Frontiers in Mathematics

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In this work we construct from ground up a homotopy theory of C * -algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model structure and second a stable model structure. The theory makes use of a full fledged import of homotopy theoretic techniques into the subject of C * -algebras.The spaces in C * -homotopy theory are certain hybrids of functors represented by C * -algebras and spaces studied in classical homotopy theory. In particular, we employ both the topological circle and the C * -algebra circle of complex-valued continuous functions on the real numbers which vanish at infinity. By using the inner workings of the theory, we may stabilize the spaces by forming spectra and bispectra with respect to either one of these circles or their tensor product. These stabilized spaces or spectra are the objects of study in stable C * -homotopy theory.The stable homotopy category of C * -algebras gives rise to invariants such as stable homotopy groups and bigraded cohomology and homology theories. We work out examples related to the emerging subject of noncommutative motives and zeta functions of C * -algebras. In addition, we employ homotopy theory to define a new type of K-theory of C * -algebras. for inspiring correspondence and discussions. We extend our gratitude to Michael Joachim for explaining his joint work with Mark Johnson on a model category structure for sequentially complete locally multiplicatively convex C * -algebras with respect to some infinite ordinal number [42]. The two viewpoints turned out to be wildly different. Aside from the fact that we are not working with the same underlying categories, one of the main differences is that the model structure in [42] is right proper, since every object is fibrant, but it is not known to be left proper. Hence it is not suitable fodder for stabilization in terms of today's (left) Bousfield localization machinery. One of the main points in our work is that fibrancy is a special property; in fact, it governs the whole theory, while left properness is required for defining the stable C * -homotopy category.

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Unbounded bivariant K-theory and correspondences in noncommutative geometry

Cited by 72 publications

References 21 publications

On globally non-trivial almost-commutative manifolds

On globally non-trivial almost-commutative manifolds

Matrix geometries emergent from a point

Homotopy Theory of C*-Algebras

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