From Rational Homotopy to K-Theory for Continuous Trace Algebras

TL;DR

This paper explores the rational homotopy groups of the unitary group of certain continuous trace $C^*$-algebras, connecting topological and algebraic invariants through explicit calculations and examples.

Contribution

It computes the rational homotopy groups of the unitary group for specific continuous trace $C^*$-algebras and analyzes the map to their $K$-theory.

Findings

Calculated rational homotopy groups for $A = C(X) imes M_n(b C)$ and continuous trace algebras.

Analyzed the $b Z$-graded map from rational homotopy to $K$-theory.

Provided concrete examples illustrating the relationship between topological and algebraic invariants.

Abstract

Let $A$ be a unital $C^*$-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 $\pi_*(U(A\otimes \KK ))$ which is isomorphic (with a degree shift) to $K_*(A)$. This approach has led to spectacular success in many arenas, as is well-known. A different approach is to consider $\pi_*(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 $C^*$-algebra. In this note we look at some concrete examples of this calculation and, in particular, at the $\ZZ$-graded map \[ \pi _*(UA)\otimes\QQ \longrightarrow K_{*+1}(A)\otimes\QQ . \]