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# Operator Algebra and Noncommutative Geometry Seminar Series

## July 11 @ 3:30 pm - 4:30 pm

##### Speaker

Asst Prof Chris Bourne (Tohoku University) will be giving a talk.

##### Title

The Cayley transform and K-theory

##### Abstract

The Cayley transform is a way to pass between unitary operators and self-adjoint (generally unbounded) operators on a Hilbert space or C*-module. I will review this construction and its application to an isomorphism between K-theory and Kasparov modules. Time permitting, I will also introduce a Z_2-graded analogue of this transform and its application to van Daele K-theory for graded C*-algebras. Explicit isomorphisms for all 8 real K-theory groups are a special case of this map. This is joint work with Johannes Kellendonk and Adam Rennie.