From Rational Homotopy to K-theory for Continuous Trace Algebras

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 = M2(C). There are two simplifications which have been considered. The first, well-traveled road, is to pass to π∗(U(A⊗K)) 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 π∗(UA)⊗ Q, 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) ⊗ Mn(C) 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 Z-graded map π∗(UA)⊗ Q −→ K∗+1(A)⊗ Q. 1. Statement of the Main Theorem Let Mn = Mn(C) be the complex matrices, Un its group of unitaries, and let PUn be the quotient of Un by its center. Let ζ :T → X be a principal PUn-bundle over a compact space X , let PUn act on Mn by conjugation and let T ×PUn Mn → X be the associated n-dimensional complex matrix bundle. Define Aζ to be the set of continuous sections of the latter. These sections have natural pointwise addition, multiplication, and ∗-operations and give Aζ the structure of a unital C -algebra. The algebra Aζ is the most general unital continuous trace C -algebra. Let UAζ denote the topological group of unitaries of Aζ . We have determined the rational homotopy type of UAζ . To state our calculation of the rational homotopy groups of UAζ , we introduce some notation. Given graded Z-graded vector spaces V and W , we grade the tensor product V ⊗ W by declaring that v ⊗ w has grading |v| + |w|. Let V ⊗̃W be the effect of considering only tensors with non-negative grading. Given elements x1, . . . , xn, each of homogeneous degree, write 〈x1, . . . , xn〉 for the graded vector space with basis x1, . . . , xn. Given a topological group G, write G◦ for the path component of the identity in G. The following is the principal result of our recent paper [2]. Theorem A. [2] Let ζ be a principal PUn bundle over a compact metric space X. Let Aζ be the associated continuous trace C -algebra, and let UAζ its group of unitaries. Then the rationalization of (UAζ)◦ is rationally H-equivalent to a 2000 Mathematics Subject Classification. 46J05, 46L85, 55P62, 54C35, 55P15, 55P45.