Large Fredholm triples (Q2497926)

The paper deals with simplifications of the standard equivalence relation on \(KK\)-theory. Let \(A\) be a separable \(C^*\)-algebra and \(B\) a \(\sigma\)-unital stable \(C^*\)-algebra. Several equivalent conditions for Busby extensions of a given \(B\) to be absorbing are given. In particular, in the unital case (i.e., when \(A\) is unital and \(B\) is the stabilization of a unital \(C^*\)-algebra) it is shown that a full extension \(\tau:A\to M(B)/B\) is absorbing if, for each positive \(a\in A\), the algebra \(\tau(C^*(a))\) is contained in the image of some injective and purely large Busby map \(l^\infty/c_0\to M(B)/B\). Then the notion of absorption is transferred from extensions to Fredholm triples. Let a triple \(\mathcal F=(M(B),\phi,F)\), where \(\phi\) is a \(*\)-homomorphism from \(A\) to \(M(B)\) and \(F\in M(B)\), be a (stabilized ungraded) Fredholm cycle in \(KK^1(A,B)\). A Fredholm triple \(\mathcal F\) is called absorbing if \(\mathcal F\oplus \mathcal T\) is unitarily equivalent to \(\mathcal F\) for any trivial cycle \(\mathcal T\). It is shown that \(\mathcal F\in KK^1(A,B)\) is absorbing if \(F\phi(a)F^*\) is a quasi-Fredholm operator whenever it is positive. As an application, a multivariable BDF type theorem is proven.