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An Introduction to K-theory and cyclic cohomology

An Introduction to K-theory and cyclic cohomology
An Introduction to K-theory and cyclic cohomology
These lecture notes contain an exposition of basic ideas of K-theory and cyclic cohomology. I begin with a list of examples of various situations in which the K-functor of Grothendieck appears naturally, including the rudiments of the topological and algebraic K-theory, K-theory of C^*-algebras, and K-homology. I then discuss elementary properties of cyclic cohomology using the Cuntz-Quillen version of the calculus of noncommutative differential forms on an algebra. As an example of the relation between the two theories we describe the Chern homomorphism and various index-theorem type statements. The remainder of the notes contains some more detailed calculations in cyclic and reduced cyclic cohomology. A key tool in this part is Goodwillie's theorem on the cyclic complex of a semi-direct product algebra. The final chapter gives an exposition of the entire cyclic cohomology of Banach algebras from the point of view of supertraces on the Cuntz algebra. The results discussed here include the simplicial normalization of the entire cyclic cohomology, homotopy invariance and the action of derivations.
operator algebras
Polish Scientific Publishers
Brodzki, J.
b1fe25fd-5451-4fd0-b24b-c59b75710543
Brodzki, J.
b1fe25fd-5451-4fd0-b24b-c59b75710543

Brodzki, J. (1998) An Introduction to K-theory and cyclic cohomology , Warsaw, Poland. Polish Scientific Publishers, 132pp.

Record type: Book

Abstract

These lecture notes contain an exposition of basic ideas of K-theory and cyclic cohomology. I begin with a list of examples of various situations in which the K-functor of Grothendieck appears naturally, including the rudiments of the topological and algebraic K-theory, K-theory of C^*-algebras, and K-homology. I then discuss elementary properties of cyclic cohomology using the Cuntz-Quillen version of the calculus of noncommutative differential forms on an algebra. As an example of the relation between the two theories we describe the Chern homomorphism and various index-theorem type statements. The remainder of the notes contains some more detailed calculations in cyclic and reduced cyclic cohomology. A key tool in this part is Goodwillie's theorem on the cyclic complex of a semi-direct product algebra. The final chapter gives an exposition of the entire cyclic cohomology of Banach algebras from the point of view of supertraces on the Cuntz algebra. The results discussed here include the simplicial normalization of the entire cyclic cohomology, homotopy invariance and the action of derivations.

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Published date: 1998
Keywords: operator algebras

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Local EPrints ID: 29843
URI: http://eprints.soton.ac.uk/id/eprint/29843
PURE UUID: 4afd9442-f778-4c1e-99bc-4794b57de663
ORCID for J. Brodzki: ORCID iD orcid.org/0000-0002-4524-1081

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Date deposited: 03 May 2007
Last modified: 14 Jan 2023 02:38

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