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Pré-Publication, Document De Travail Année : 2023

Affine noncommutative geometry

Teo Banica
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Résumé

This is an introduction to noncommutative geometry, from an algebraic and probabilistic viewpoint. The spaces $\mathbb R^N,\mathbb C^N$ have no free analogues in the operator algebra sense, but the corresponding unit spheres $S^{N-1}_\mathbb R,S^{N-1}_\mathbb C$ do have free analogues $S^{N-1}_{\mathbb R,+},S^{N-1}_{\mathbb C,+}$. There are many examples of real algebraic submanifolds $X\subset S^{N-1}_{\mathbb R,+},S^{N-1}_{\mathbb C,+}$, some of which are of Riemannian flavor, coming with a Haar integration functional $\int:C(X)\to\mathbb C$, that we will study here. We will mostly focus on the free geometry, but we will discuss as well some related geometries, called easy, completing the picture formed by the 4 main geometries, namely classical/free, real/complex.
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Dates et versions

hal-03312800 , version 1 (02-08-2021)
hal-03312800 , version 2 (16-11-2021)
hal-03312800 , version 3 (14-04-2022)
hal-03312800 , version 4 (31-10-2022)
hal-03312800 , version 5 (02-05-2023)

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  • HAL Id : hal-03312800 , version 5

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Teo Banica. Affine noncommutative geometry. 2023. ⟨hal-03312800v5⟩
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