Smooth K-theory and locally convex algebras

In this thesis, we improve the loop linearization process from the classical article of Atiyah and Bott on Bott periodicity. The linearization process is made explicit in terms of formulae for smooth loops. Using this improvement allows us to extend K-theory (including periodicity) to a class of locally convex algebras vastly larger then the one of Banach algebras. We find various ways to represent periodicity by explicit formulae. For finite Laurent loops formulae yielding finite matrices to represent the associated Ko classes are obtained. The methods used also allow us to reinterpret some recent results of Melrose on smooth classifying spaces for K-theory. The relationship between the universal even and odd Chern characters and periodicity is investigated, giving correspondences between the various representatives in the form of family index theorems for loop groups. In the discussion Ko and the even Chern character are primarily formulated in the language of involutions. The paper also demonstrates the universality of the involution terminology with respect to vector bundles.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2003

Author dc:creator

• Lakos, Gyula, 1973-

Advisor dc:contributor.advisor

• Richard B. Melrose.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/29357

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/29357