Classical harmonic analysis viewed through the prism of noncommutative geometry
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915691856134144 |
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| author | Arhancet, Cédric |
| author_facet | Arhancet, Cédric |
| contents | The aim of this paper is to bridge noncommutative geometry with classical harmonic analysis on Banach spaces, focusing primarily on both classical and noncommutative $\mathrm{L}^p$ spaces. Introducing a notion of Banach Fredholm module, we define new abelian groups, $\mathrm{K}^{0}(\mathcal{A},\mathscr{B})$ and $\mathrm{K}^{1}(\mathcal{A},\mathscr{B})$, of $\mathrm{K}$-homology associated with an algebra $\mathcal{A}$ and a suitable class $\mathscr{B}$ of Banach spaces, such as the class of $\mathrm{L}^p$-spaces. We establish index pairings of these groups with the $\mathrm{K}$-theory groups of the algebra $\mathcal{A}$. Subsequently, by considering (noncommutative) Hardy spaces, we uncover the natural emergence of Hilbert transforms, leading to Banach Fredholm modules and culminating in new 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 $\mathrm{L}^p$-spaces into the expansive framework of noncommutative geometry, offering new perspectives. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07750 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classical harmonic analysis viewed through the prism of noncommutative geometry Arhancet, Cédric Functional Analysis Classical Analysis and ODEs K-Theory and Homology Operator Algebras The aim of this paper is to bridge noncommutative geometry with classical harmonic analysis on Banach spaces, focusing primarily on both classical and noncommutative $\mathrm{L}^p$ spaces. Introducing a notion of Banach Fredholm module, we define new abelian groups, $\mathrm{K}^{0}(\mathcal{A},\mathscr{B})$ and $\mathrm{K}^{1}(\mathcal{A},\mathscr{B})$, of $\mathrm{K}$-homology associated with an algebra $\mathcal{A}$ and a suitable class $\mathscr{B}$ of Banach spaces, such as the class of $\mathrm{L}^p$-spaces. We establish index pairings of these groups with the $\mathrm{K}$-theory groups of the algebra $\mathcal{A}$. Subsequently, by considering (noncommutative) Hardy spaces, we uncover the natural emergence of Hilbert transforms, leading to Banach Fredholm modules and culminating in new 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 $\mathrm{L}^p$-spaces into the expansive framework of noncommutative geometry, offering new perspectives. |
| title | Classical harmonic analysis viewed through the prism of noncommutative geometry |
| topic | Functional Analysis Classical Analysis and ODEs K-Theory and Homology Operator Algebras |
| url | https://arxiv.org/abs/2409.07750 |