Book•
K-Theory for Operator Algebras
10 Sep 1986-
TL;DR: A survey of topological K-theory can be found in this paper, where the authors present a survey of applications to geometry and topology, including the Pimsner-Voiculescu exact sequence and Connes' Thorn isomorphism.
Abstract: I. Introduction To K-Theory.- 1. Survey of topological K-theory.- 2. Overview of operator K-theory.- II. Preliminaries.- 3. Local Banach algebras and inductive limits.- 4. Idempotents and equivalence.- III. K0-Theory and Order.- 5. Basi K0-theory.- 6. Order structure on K0.- 7. Theory of AF algebras.- IV. K1-Theory and Bott Periodicity.- 8. Higher K-groups.- 9. Bott Periodicity.- V. K-Theory of Crossed Products.- 10. The Pimsner-Voiculescu exact sequence and Connes' Thorn isomorphism.- 11. Equivariant K-theory.- VI. More Preliminaries.- 12. Multiplier algebras.- 13. Hilbert modules.- 14. Graded C*-algebras.- VII. Theory of Extensions.- 15. Basic theory of extensions.- 16. Brown-Douglas-Fillmore theory and other applications.- VIII. Kasparov's KK-Theory.- 17. Basic theory.- 18. Intersection product.- 19. Further structure in KK-theory.- 20. Equivariant KK-theory.- IX. Further Topics.- 21. Homology and cohomology theories on C*-algebras.- 22. Axiomatic K-theory.- 23. Universal coefficient theorems and Kunneth theorems.- 24. Survey of applications to geometry and topology.
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1,025 citations
TL;DR: An overview of the integer quantum Hall effect is given in this paper, where a mathematical framework using non-ommutative geometry as defined by Connes is prepared. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localized states.
Abstract: An overview of the integer quantum Hall effect is given. A mathematical framework using nonommutative geometry as defined by Connes is prepared. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localized states.
626 citations
TL;DR: The concept of real rank of a C ∗ -algebra is introduced as a non-commutative analogue of dimension in this paper, and it is shown that real rank zero is equivalent to the previously defined conditions FS and HP, and is invariant under strong Morita equivalence, in particular under stable isomorphism.
604 citations
Book•
20 Feb 2007TL;DR: This book is intended primarily for graduate students who wish to begin research using crossed product C ∗ -algebras and is now essentially a final draft, and the final version will appear in the Surveys and Monograph series of the American Mathematical Society.
Abstract: Locally compact groups Dynamical systems and crossed products Special cases and basic constructions Imprimitivity theorems Induced representations and induced ideals Orbits and quasi-orbits Properties of crossed products Ideal structure The proof of the Gootman-Rosenberg-Sauvageot theorem Amenable groups The Banach *-algebra $L^1(G,A)$ Bundles of $C*$-algebras Groups Representations of $C*$-algebras Direct integrals Effros's ideal center decomposition The Fell topology Miscellany Notation and Symbol Index Index Bibliography.
589 citations
Cites background from "K-Theory for Operator Algebras"
...Summaries and references for these results can be found in Blackadar’s treatise [8]....
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01 Jan 2004
TL;DR: In this paper, a reformulation of the conjecture is presented, which is simpler and applies more generally than the earlier statement. But the universal example for proper actions is not considered.
Abstract: We announce a reformulation of the conjecture in [8,9,10]. The advantage of the new version is that it is simpler and applies more generally than the earlier statement. A key point is to use the universal example for proper actions introduced in [10]. There, the universal example seemed somewhat peripheral to the main issue. Here, however, it will play a central role.
585 citations
Cites background from "K-Theory for Operator Algebras"
...An operator is Fredholm if it is invertible modulo compact operators. See [ 14 ] for details....
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