An Introduction to Ktheory and Cyclic Cohomology
Abstract
These lecture notes contain an exposition of basic ideas of Ktheory and cyclic cohomology. I begin with a list of examples of various situations in which the Kfunctor of Grothendieck appears naturally, including the rudiments of the topological and algebraic Ktheory, Ktheory of C^*algebras, and Khomology. I then discuss elementary properties of cyclic cohomology using the CuntzQuillen 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 indextheorem 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 semidirect 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.
 Publication:

arXiv eprints
 Pub Date:
 June 1996
 arXiv:
 arXiv:functan/9606001
 Bibcode:
 1996funct.an..6001B
 Keywords:

 Mathematics  Functional Analysis;
 High Energy Physics  Theory;
 Mathematics  Operator Algebras
 EPrint:
 113 pages, LaTeX