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

摘要

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 sky-scraper 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 satisfy the UCT, their K_0-groups are free of finite rank and their K_1-groups are zero. In that case we show that E_([0,1])(A,B) is'isomorphic to Hom(K_0(A)), K_0(B)), the group of morphisms of the K-theory sheaves of A and B. As an application, we give a streamlined partially new proof of a classification result due to the first author and Elliott.