Functions and Analysis

    

Affine Noncommutative Geometry

Authors: Teo Banica

This is an introduction to noncommutative geometry, from an affine viewpoint, that is, by using coordinates. 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 $Xsubset S^{N-1}_{mathbb R,+},S^{N-1}_{mathbb C,+}$, some of which are of Riemannian flavor, coming with a Haar integration functional $int:C(X)tomathbb C$, that we will study here. We will mostly focus on free geometry, but we will discuss as well some related geometries, called easy, completing the picture formed by the 4 main geometries, namely real/complex, classical/free.

Comments: 400 Pages.

Download: PDF

Submission history

[v1] 2020-05-13 22:52:53
[v2] 2020-06-02 19:06:21
[v3] 2020-09-03 19:01:37
[v4] 2021-08-02 12:20:28
[v5] 2024-08-03 18:56:15

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