From Rational Homotopy to K-Theory for Continuous Trace Algebras

Abstract

Let $A$ be a unital $C^{\ast }$-algebra. Its unitary group, $UA$, contains a wealth of topological information about $A$. However, the homotopy type of $UA$ is out of reach even for $A = M_2(\CC)$. There are two simplifications which have been considered. The first, well-traveled road, is to pass to $π_*(U(A\otimes \KK ))$ which is isomorphic (with a degree shift) to $K_{\ast }(A)$. This approach has led to spectacular success in many arenas, as is well-known. A different approach is to consider $π_*(UA)\otimes\QQ $, the rational homotopy of $UA$. In joint work with G. Lupton and N. C. Phillips we have calculated this functor for the cases $A = C(X)\otimes M_n(\CC)$ and $A$ a unital continuous trace $C^{\ast }$-algebra. In this note we look at some concrete examples of this calculation and, in particular, at the $\ZZ$-graded map \[ π_*(UA)\otimes\QQ \longrightarrow K_{*+1}(A)\otimes\QQ . \]

6 pages, submitted to proceedings of NSF/CBMS Conference on "Topology, C*-algebras, and String Duality", Principal Lecturer Jonathan Rosenberg, TCU, May 18-22, 2009