On the Unbounded Picture of 𝛫𝛫-Theory

Abstract

In the founding paper on unbounded 𝛫𝛫-theory, it was established by Baaj and Julg that the bounded transform, which associates a class in 𝛫𝛫-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 𝛫𝛫-theory, which becomes isomorphic to 𝛫𝛫-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.