\(K\)-homology relative to semisplit ideals (Q1382119)

Let \(A\) be a separable \(C^*\)-algebra, \(Q(A)= A*A\) be the free product of \(A\) by itself. Identifying the two copies of \(A\) in \(Q(A)\) gives rise to a canonical multiplication surjective map \(\mu: Q(A)\to A\). The kernel of \(\mu\) is an ideal denoted by \(q(A)\). The construction \(A\mapsto q(A)\) is functorial in the sense that any \(C^*\)-homomorphism \(\alpha: A\to B\) gives rise to a homomorphism \(\alpha^q: q(A)\to q(B)\). But this functor does not preserve exact sequences. Given an ideal \(J\) of \(A\), the natural object to consider is the relative algebra \(q(A,J)= \ker (\pi^q)\), where \(\pi:A \to A/J\) is the natural epimorphism. It was shown by Cuntz that the Kasparov group \(KK^0 (A,B)\) can be described as the group of stable homotopy classes of homomorphisms from \(q(A)\) to \(B\). The author's method to study the pair \((A,J)\) is inspired and closely connected with this approach of Cuntz. Namely, an abelian group \(\widehat K^0 (A,J)\) is defined as a group of operator homotopy classes of homomorphisms from \(q(A,J)\) to the algebra of compact operators on a separable Hilbert space. It is shown that when \(J\) is a semisplit ideal of \(A\), \(\widehat K^0 (A,J)\) is isomorphic to the relative \(K\)-homology \(K^0(A,J)\), as defined by Baum-Douglas-Taylor. One of main results of the Baum-Douglas-Taylor theory is strengthened by showing that the isomorphism \(\widehat K^0 (A,J) \cong K^0 (J)\) holds without any assumption that \(A\) be separable.