Relative K-homology and normal operators

Vladimir Manuilov, Klaus Thomsen

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    Abstract

    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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