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K-Theory for Operator Algebras
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TLDR
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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A lefschetz fixed-point formula for certain orbifold c*-algebras
TL;DR: Using Poincare duality in K-theory, the authors proved a Lefschetz fixed point formula for endomorphisms of crossed product C � -algebras C0(X)⋊ G coming from covariant pairs.
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Full coactions on Hilbert C*-modules
TL;DR: In this paper, the authors introduce a natural notion of full coactions of a locally compact group on a Hilbert C*-module, and associate each full coaction in a natural way to an ordinary coaction.
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On the topology of the group of invertible elements
TL;DR: A survey on the homotopy theory of the regular group of Banach algebras with emphasis on the unstable K-Theory of real and complex C*-algebra is given in this paper.
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On the Homotopy Classification of Elliptic Operators on Manifolds with Edges
TL;DR: In this paper, a classification of elliptic operators modulo stable homotopy on manifolds with edges is presented, and the main part of the proof is the computation of the boundary map using semiclassical quantization.
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Computing Ext for graph algebras
TL;DR: For a row-nite graph G with no sinks and in which every loop has an exit, this paper constructed an isomorphism between Ext(C (G)) and coker(A I), where A is the vertex matrix of G and C is the class associated to a graph obtained by attaching a sink to G, and this isomor-phism maps c to the class of a vector that describes how the sink was added.