Uma descrição das aplicações de conexão em K-teoria de C*-álgebras usando cones

Abstract

MAEKAWA, R. A. A description of the connecting maps in K-theory for C∗-algebras using cones. 2013. 73 f. Dissertacao (Mestrado) Instituto de Matematica e Estatistica, Universidade de Sao Paulo, Sao Paulo, 2013. If f : B −→ A is a map between the C∗-algebras A and B, the mapping cone is the set of pairs (b, φ) in B ⊕ CA such that f(b) = φ(0), where CA is the cone of A. In this work, we study the functor determined by the assignment of the exact sequence 0 −→ SA −→ Cf −→ B −→ 0 to each *-homomorphism f : B −→ A, and we show that this functor is exact. We characterize the connecting maps associated with the short exact sequence 0 −→ SA −→ Cf −→ B −→ 0 and we prove that its index map is θA ◦K1(f) and that its exponential map is βA ◦K0(f), where θA is the isomorphism between K1(A) and K0(SA), and βA is the Bott map. Finally, using that every short exact sequence of C∗-algebras can be seen as 0 −→ Kerf ↪→ B f −→ A −→ 0, we prove that the connecting maps, δ1 and δ0, associated with a short exact sequence are given by δn = Kn+1(j) −1 ◦Kn+1(i) ◦αn, where j is the inclusion of f 's kernel in Cf , i is the inclusion of the suspension SA in Cf , α0 = βA and α1 = θA.