# Journal of Operator Theory

Volume 62, Issue 2, Fall 2009 pp. 249-279.

Relative K-homology and normal operators**Authors**: Vladimir Manuilov (1) and Klaus Thomsen (2)

**Author institution:**(1) Dept. of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia

(2) IMF, Department of Mathematics, Ny Munkegade, 8000 Aarhus C, Denmark

**Summary:**Let $A$ be a $C^*$-algebra, $J \subset A$ a $C^*$-subalgebra, and let $B$ be a stable $C^*$-algebra. Under modest assumptions we organize invertible $C^*$-extensions of $A$ by $B$ that are trivial when restricted onto $J$ to become a group $\mathrm{Ext}_J^{-1}(A,B)$, which can be computed by a six-term exact sequence which generalizes the excision six-term exact sequence in the first variable of KK-theory. Subsequently we investigate the relative K-homology which arises from the group of relative extensions by specializing to abelian $C^*$-algebras. It turns out that this relative K-homology carries substantial information also in the operator theoretic setting from which the BDF theory was developed and we conclude the paper by extracting some of this information on approximation of normal operators.

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