SPECTRAL LOCALIZERS IN KK-THEORY

Speaker(s)

JENS KAAD

Affiliation

Syddansk Universitet, Odense, Denmark

Language of the talk

English

Date

Nov. 12, 2025, 5:15 p.m.

Information about the event

IMPAN - Room 405

Title in Polish

SPECTRAL LOCALIZERS IN KK-THEORY

Seminar

North Atlantic Noncommutative Geometry Seminar

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In this talk, we compute the index homomorphism of even K-groups arising from a class in even KK-theory via the Kasparov product. Due to the seminal work of Baaj and Julg, under mild conditions on the C*-algebras in question, every class in KK-theory can be represented by an unbounded Kasparov module. We then describe the corresponding index homomorphism of even K-groups in terms of spectral localizers. This means that our explicit formula for the index homomorphism does not depend on the full spectrum of the abstract Dirac operator D, but rather on the intersection between this spectrum and a compact interval. The size of this compact interval does, however, reflect the interplay between the K-theoretic input and the abstract Dirac operator. Since the spectral projections for D are not available in the general context of Hilbert C*-modules, we instead rely on certain continuous compactly supported functions applied to D to construct a spectral localizer. In the special case where even KK-theory coincides with even K-homology, our work recovers the pioneering work of Loring and Schulz-Baldes on the index pairing.

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