K-Homology Relative to Semisplit Ideals

Given a C^*-algebra A, and an ideal J of A, we define a relative group K^⁰(A,J) in terms of a relative universal C^*-algebra for the pair (A,J). We show that the natural restriction map K^⁰(A,J) K⁰(J) is an isomorphism, and that, if J is a semisplit ideal of A, the Baum–Douglas–Taylor relative K-homology is recovered. This provides a generalization of one of the main results of the Baum–Douglas–Taylor theory.