Relative K-homology and normal operators

TY - JOUR

T1 - Relative K-homology and normal operators

AU - Manuilov, Vladimir

AU - Thomsen, Klaus

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

M3 - Journal article

SN - 0379-4024

VL - 62

SP - 249

EP - 279

JO - Journal of Operator Theory

JF - Journal of Operator Theory

IS - 2

ER -