Extremal \(K\)-theory and index for \(C^*\)-algebras (Q1591575)

Let \({\mathcal E}(A)\) be the set of extreme points in the unit ball of a unital \(C^*\)-algebra \(A,\) that is the partial isometries \(v\) such that\((I-v^*v)(I-vv^*)=0.\) The two orthogonal, closed ideals \(J_+\) and \(J_-\) generated by projections \(p_+=I-v^*v\) and \(p_-=I-vv^*\) will be known as the defect ideals of \(v.\) For each \(n\) let us consider the set\([{\mathcal E}({\mathbf M}_n(A))]\) of homotopy classes of extreme partial isometries in the \(C^*\)-algebra \({\mathbf M}_n(A)\) of \((n\times n)\)-matrices over \(A.\) For any two elements \(a\) and \(b\) in \([{\mathcal E}_{\infty}(A)]=\lim_{n\to\infty }[{\mathcal E}({\mathbf M}_n(A))]\) define \(a \sim b\) if \(a+c=b+c\) for some \(c\) in \([{\mathcal E}_{\infty}(A)]\) with smaller defects than \(a\) and \(b.\) Evidently this equivalence implies that \(a\) and \(b\) had the same defect ideals. The extremal \(K\)-set of the \(C^*\)-algebra \(A\) is defined as \(K_e(A)=[{\mathcal E}_{\infty}(A)]/ \sim .\) If \(A\) is a \(C^*\)-algebra without unit then \(K_e(A)=K_e(A^+)\) where \(A^+\) denotes the unitalisation of the \(C^*\)-algebra \(A.\) The paper is devoted to the development of the general theory for a new functor \(K_e\) on the category of \(C^*\)-algebras. \(K_e(A)\) contains \(K_1(A)\) and admits a partially defined addition extending the addition in \(K_1(A),\) so there is an action of \(K_1(A)\) on \(K_e(A).\) The extremal \(K\)-set is used to extend the classical index theory for Fredholm and semi-Fredholm operators.