Classical harmonic analysis viewed through the prism of noncommutative geometry

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 index theorems. Moreover, by associating each reasonable sub-Markovian semigroup with a <>, 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.