On the unbounded picture of kk-theory

Jens Kaad*

*Corresponding author for this work

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In the founding paper on unbounded KK-theory it was established by Baaj and Julg that the bounded transform, which associates a class in KK-theory to any unbounded Kasparov module, is a surjective homomorphism (under a separability assumption). In this paper, we provide an equivalence relation on unbounded Kasparov modules and we thereby describe the kernel of the bounded transform. This allows us to introduce a notion of topological unbounded KK-theory, which becomes isomorphic to KK-theory via the bounded transform. The equivalence relation is formulated entirely at the level of unbounded Kasparov modules and consists of homotopies together with an extra degeneracy condition. Our degenerate unbounded Kasparov modules are called spectrally decomposable since they admit a decomposition into a part with positive spectrum and a part with negative spectrum.

Original languageEnglish
Article number082
JournalSymmetry, Integrability and Geometry: Methods and Applications
Number of pages21
Publication statusPublished - 2020

Bibliographical note

In: Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi for his sixtieth birthday


  • Bounded transform
  • Equivalence relations
  • KK-theory
  • Unbounded KK-theory


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