Dissertation/Thesis Abstract

A geometric construction of cyclic cocycles on twisted convolution algebras
by Angel, Eitan, Ph.D., University of Colorado at Boulder, 2010, 86; 3404042
Abstract (Summary)

In this thesis we give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. In his seminal book, Connes constructs a map from the equivariant cohomology of a manifold carrying the action of a discrete group into the periodic cyclic cohomology of the associated convolution algebra. Furthermore, for proper ├ętale groupoids, J.-L. Tu and P. Xu provide a map between the periodic cyclic cohomology of a gerbe twisted convolution algebra and twisted cohomology groups.

Our focus will be the convolution algebra with a product defined by a gerbe over a discrete translation groupoid. When the action is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial notions related to ideas of J. Dupont to construct a simplicial form representing the Dixmier-Douady class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial Dixmier-Douady form to the mixed bicomplex of certain matrix algebras. Finally, we define a morphism from this complex to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras.

Indexing (document details)
Advisor: Gorokhovsky, Alexander
Commitee: Farsi, Carla, Packer, Judith, Ramsay, Arlan, de Alwis, Senarath
School: University of Colorado at Boulder
Department: Mathematics
School Location: United States -- Colorado
Source: DAI-B 71/06, Dissertation Abstracts International
Subjects: Mathematics
Keywords: Convolution algebras, Cyclic cocycles, Translation groupoids
Publication Number: 3404042
ISBN: 978-1-109-78219-6
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