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K-Theory for Operator Algebras
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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.read more
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Journal ArticleDOI
Generalized Connes-Chern characters in KK-theory with an application to weak invariants of topological insulators
TL;DR: In this article, constructive bounded Kasparov K-theory was used to investigate numerical invariants stemming from the internal Kas-parov products of Dirac cycles, where the last morphism is provided by a tracial state.
Journal ArticleDOI
The Heat Asymptotics on Filtered Manifolds.
TL;DR: In this article, a universal heat kernel expansion for formally self-adjoint non-negative Rockland differential operators on general closed filtered manifolds is established, which implies that the new calculus, a more general version of the Heisenberg calculus, also has a noncommutative residue.
Book ChapterDOI
Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds
Anton Savin,Boris Ju. Sternin +1 more
TL;DR: A survey of elliptic theory for operators associated with diffeomorphisms of smooth manifolds is given in this article, with a general overview of the area of research and a detailed formulation of the most important results mentioned in the introduction.
Journal ArticleDOI
Real C*-algebras, United KK-theory, and the Universal Coefficient Theorem
TL;DR: For real C*-algebras A and B such that A is separable and B is sigma-unital, this article defined United KK-theory for real C *-algeses and proved that KK\crt(A,B) can be written as the middle term of a short exact sequence whose outer terms involve the united Ktheory.
Journal ArticleDOI
Equivariant Euler characteristics and K-homology Euler classes for proper cocompact G-manifolds
TL;DR: In this article, the universal equivariant Euler characteristic is shown to carry the same information as the collection of the orbifold Euler characteristics of the components of the L-fixed point sets M^L, where L runs through the finite cyclic subgroups of G.