When are Fredholm triples operator homotopic?

Kucerovsky, Dan

doi:10.1090/S0002-9939-06-08481-4

When are Fredholm triples operator homotopic?
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Proc. Amer. Math. Soc. 135 (2007), 405-415 Request permission

Abstract:

Fredholm triples are used in the study of Kasparov’s $KK$-groups, and in Connes’s noncommutative geometry. We define an absorption property for Fredholm triples, and give an if and only if condition for a Fredholm triple to be absorbing. We study the interaction of the absorption property with several of the more common equivalence relations for Fredholm triples. In general these relations are coarser than homotopy in the norm topology. We give simple conditions for an equivalence of triples to be implemented by an operator homotopy (i.e. a homotopy with respect to the norm topology). This can be expected to have applications in index theory, as we illustrate by proving two theorems of Pimsner-Popa-Voiculescu type. We show that there is some relationship with the interesting Toms–Winter characterization of ${\mathcal D}$-absorbing algebras, recently obtained as part of Elliott’s classification program.

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