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Relative K-homology and normal operators

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  • Institut for Matematiske Fag
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.
OriginalsprogEngelsk
TidsskriftJournal of Operator Theory
Vol/bind62
Nummer2
Sider (fra-til)249-279
Antal sider31
ISSN0379-4024
StatusUdgivet - 2009

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