Lecture II: Dixmier-Douady theory and twisted K-theory

From CRCG-Wiki

A classical theorem of Dixmier-Douady identifies the integral degree 3 cohomology group $H^3(X,\Z)$ with Morita isomorphism classes of *-algebra bundles $A\to X$, with typical fiber the compact operators on a Hilbert space. Such bundles (sometimes called `gerbes') are thus a higher degree analogs of line bundles. We will consider the definition of the twisted K-theory of X in terms of such bundles, and discuss some basic examples.