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Pré-Publication, Document De Travail Année : 2022

Principles of operator algebras

Teo Banica
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Résumé

This is an introduction to the algebras $A\subset B(H)$ that the bounded linear operators $T:H\to H$ can form, once a complex Hilbert space $H$ is given. Motivated by quantum mechanics, we are mostly interested in the von Neumann algebras, which are stable under taking adjoints, $T\to T^*$, and weakly closed. When the algebra has a trace $tr:A\to\mathbb C$, we can think of it as being of the form $A=L^\infty(X)$, with $X$ being a quantum measured space, and of particular interest is the free case, where the center of the algebra is $Z(A)=\mathbb C$. Following Murray, von Neumann, Connes, Jones, Voiculescu, Woronowicz, we discuss here the basic properties of such algebras $A$, and how to do algebra, geometry, analysis and probability on the underlying quantum spaces $X$.
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Dates et versions

hal-03747024 , version 1 (07-08-2022)
hal-03747024 , version 2 (11-11-2022)
hal-03747024 , version 3 (21-12-2022)
hal-03747024 , version 4 (14-06-2023)

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  • HAL Id : hal-03747024 , version 3

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Teo Banica. Principles of operator algebras. 2022. ⟨hal-03747024v3⟩
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