From Rational Homotopy to K-Theory for Continuous Trace Algebras

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 $A=M_2(\text{CC})$. There are two simplifications which have been considered. The first, well-traveled road, is to pass to $\pi_(U(A\otimes \KK ))$whichisisomorphic(withadegreeshift)to$K_(A)$. This approach has led to spectacular success in many arenas, as is well-known. A different approach is to consider ${\pi }_{\ast }(UA)\otimes \text{QQ}$, the rational homotopy of $UA$. In joint work with G. Lupton and N. C. Phillips we have calculated this functor for the cases $A=C(X)\otimes M_n(\text{CC})$ 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 $\text{ZZ}$-graded map $${\pi }_{\ast }(UA)\otimes \text{QQ}⟶K_{\ast +1}(A)\otimes \text{QQ}\mathrm{.}$$