Relative K-homology and Normal Operators

Let X be a compact metric space. By results of Brown, Douglas and Fillmore, [BDF2], the K-homology of X is realized by Ext(X), the equivalence classes of unital and essential extensions of C(X) by the compact operators K on a separable infinite dimensional Hilbert space H , or equivalently, the equivalence classes of unital and injective ∗-homomorphisms C(X) → Q, where Q = L(H)/K is the Calkin algebra. This discovery came out of questions and problems related to essential normal operators, and it led quickly to the development of a vast new area of mathematics which combines operator theory with algebraic topology. In particular, the BDF-theory was generalized by Kasparov in form of KK-theory, which has proven to be a powerful tool in the theory of operator algebras as well as in algebraic topology. It is the purpose of the present paper to develop a relative theory in this context. The point of departure here is a generalization of the six-term exact sequence of extension theory which relates the group of extensions of a unital C∗-algebra to the group of unital extensions. This sequence was discovered by Skandalis, cf. [S], and a construction of it was presented in [MT]. It is the latter construction which we here generalize to get a relative extension theory. Subsequently we investigate the relative K-homology which arises from it by specializing to abelian C∗-algebras. It turns out that relative K-homology carries substantial information also in the operator theoretic setting from which the BDFtheory was developed, cf. [BDF1], and we conclude the paper by extracting some of this information. In the remaining part of this introduction we give a more detailed account of the content of the paper. Let A be a C∗-algebra, J ⊆ A a C∗-subalgebra, and let B be a stable C∗algebra. Under modest assumptions we organize the C∗-extensions of A by B that are trivial when restricted onto J to become a semi-group ExtJ(A,B) which is the semigroup Ext(A,B) of Kasparov, [K1], when J = {0}. The group Ext−1 J (A,B) of invertible elements in ExtJ(A,B) can be effectively computed by a six-term exact sequence which generalizes the excision six-term exact sequence in the first variable of KK-theory, and it turns out that there is a natural identification Ext−1 J (A,B) = KK (Ci, B), where Ci is the mapping cone of the inclusion i : J → A. Thus, as an abstract group, the relative extension group is a familiar object, and the six-term exact sequence which calculates it, is a version of the Puppe exact sequence of Cuntz and Skandalis; [CS]. But the realization of KK(Ci, B) as a relative extension group has non-trivial consequences already in the setup from which KK-theory developed, namely the setting of (essential) normal operators, and the second half of the paper is devoted to the extraction of the information which the relative extension group contains about normal operators when specialized to the case where B = K and X and Y are compact metric spaces, and f : X → Y is a continuous surjection giving rise to an embedding of J = C(Y ) into A = C(X). In this setting ExtJ(A,K) is a group, and we denote it by ExtY,f(X). As an abstract group this is the