Pré-Publication, Document De Travail Année : 2024

Classical harmonic analysis viewed through the prism of noncommutative geometry

Résumé

The aim of this paper is to bridge noncommutative geometry with classical harmonic analysis on Banach spaces, focusing primarily on both classical and noncommutative Lp spaces. Introducing a notion of Banach Fredholm module, we define new abelian groups, K0(A,B) and K1(A,B), of K-homology associated with an algebra A and a suitable class B of Banach spaces, such as the class of Lp-spaces. We establish index pairings of these groups with the K-theory groups of the algebra A. Subsequently, by considering (noncommutative) Hardy spaces, we uncover the natural emergence of Hilbert transforms, leading to Banach Fredholm modules and culminating in index theorems. Moreover, by associating each reasonable sub-Markovian semigroup of operators with a «Banach noncommutative manifold», we explain how this leads to (possibly kernel-degenerate) Banach Fredholm modules, thereby revealing the role of vectorial Riesz transforms in this context. Overall, our approach significantly integrates the analysis of operators on Lp-spaces into the expansive framework of noncommutative geometry, offering new perspectives.

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hal-05089869 , version 1 (29-05-2025)

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  • HAL Id : hal-05089869 , version 1

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Cédric Arhancet. Classical harmonic analysis viewed through the prism of noncommutative geometry. 2025. ⟨hal-05089869⟩
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