RELATIVE K-THEORY FOR C∗-ALGEBRAS

Abstract

Given ${\mathrm{C}}^{\ast }$-algebras $A$ and $B$ and a ${}^{\ast }$-homomorphism $\varphi :A\to B$, we adopt the portrait of the relative $K$-theory $K_{\ast }(\varphi )$ due to Karoubi using Banach categories and Banach functors. We show that the elements of the relative groups may be represented in a simple form. We prove the existence of two six-term exact sequences, and we use these sequences to deduce the fact that the relative theory is isomorphic, in a natural way, to the $K$-theory of the mapping cone.