E-theory for C[0, 1]-algebras with finitely many singular points

Marius Dadarlat, Prahlad Vaidyanathan
2014 Journal of K-Theory  
We study the E-theory group E [0,1] (A, B) for a class of C*-algebras over the unit interval with finitely many singular points, called elementary C[0, 1]-algebras. We use results on E-theory over non-Hausdorff spaces to describe E [0,1] (A, B) where A is a skyscraper algebra. Then we compute E [0,1] (A, B) for two elementary C[0, 1]-algebras in the case where the fibers A(x) and B(y) of A and B are such that E 1 (A(x), B(y)) = 0 for all x, y ∈ [0, 1]. This result applies whenever the fibers
more » ... never the fibers satisfy the UCT, their K0-groups are free of finite rank and their K1-groups are zero. In that case we show that E [0,1] (A, B) is isomorphic to Hom(K0(A), K0(B)), the group of morphisms of the K-theory sheaves of A and B. As an application of this, we provide a streamlined partially new proof of a classification result due to the first author and Elliott. M.D. was partially supported by NSF grant #DMS-1101305. we give a streamlined proof of the main result of [2], see Theorem 5.8. One reason which makes the computation of the KK X -groups difficult is the prevalence of non-semisplit extensions over X. To illustrate this point, let us mention that the exact sequence of C[0, 1]-algebras 0 → C 0 [0, 1) → C[0, 1] → C → 0 is not semisplit over X = [0, 1]. This is more than a technical nuisance, since a six-term sequence of the form cannot be exact for D = C 0 [0, 1). Indeed, after computing each term one gets: Definition 5.1. Let X denote the unit interval and let A be a C[0, 1]-algebra. Let I denote the set of all closed subintervals of X with positive length. To each I ∈ I, associate the group K 0 (A(I)), and to each pair I, J ∈ I such that J ⊂ I, associate the map This data gives a pre-sheaf on I which is denoted by K 0 (A). A morphism of pre-sheaves ϕ : K 0 (A) → K 0 (B) consists of a family of maps ϕ I : K 0 (A(I)) → K 0 (B(I))) such that the following diagram commutes K 0 (A(I)) ϕ I − −−− → K 0 (B(I)) r I J     r I J K 0 (A(J)) ϕ J
doi:10.1017/is013012029jkt252 fatcat:txit5f4hbzgmddm246qrzgckgy