Extremal K -Theory and Index for C*-Algebras

We develop the general theory for a new functor K_e on the category of C^*-algebras. The extremal K-set, K_e(A), of a C^*-algebra A is defined by means of homotopy classes of extreme partial isometries. It contains K₁(A) and admits a partially defined addition extending the addition in K₁(A), so that we have an action of K₁(A) on K_e(A). We show how this functor relates to K₀ and K₁, and how it can be used as a carrier of information relating the various K-groups of ideals and quotients of A. The extremal K-set is then used to extend the classical theory of index for Fredholm and semi-Fredholm operators.