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.
Mots clés
- noncommutative geometry
- 2020 Mathematics subject classification: 58B34 47D03 46L80 47B90 K-homology K-theory spectral triples Riesz transforms Hilbert transforms L p -spaces Fredholm modules noncommutative geometry
- Fredholm modules
- L p -spaces
- Hilbert transforms
- Riesz transforms
- spectral triples
- K-theory
- 47B90 K-homology
- 46L80
- 47D03
- 2020 Mathematics subject classification: 58B34
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