Principles of operator algebras
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909285333598208 |
|---|---|
| author | Banica, Teo |
| author_facet | Banica, Teo |
| contents | This is an introduction to the algebras $A\subset B(H)$ that the linear operators $T:H\to H$ can form, once a complex Hilbert space $H$ is given. Motivated by quantum mechanics, we are mainly interested in the von Neumann algebras, which are stable under taking adjoints, $T\to T^*$, and are 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. Of particular interest is the free case, where the center of the algebra reduces to the scalars, $Z(A)=\mathbb C$. Following von Neumann, Connes, Jones, Voiculescu and others, we discuss the basic properties of such algebras $A$, and how to do algebra, geometry, analysis and probability on the underlying quantum spaces $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_03600 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Principles of operator algebras Banica, Teo Operator Algebras Mathematical Physics Functional Analysis Quantum Algebra This is an introduction to the algebras $A\subset B(H)$ that the linear operators $T:H\to H$ can form, once a complex Hilbert space $H$ is given. Motivated by quantum mechanics, we are mainly interested in the von Neumann algebras, which are stable under taking adjoints, $T\to T^*$, and are 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. Of particular interest is the free case, where the center of the algebra reduces to the scalars, $Z(A)=\mathbb C$. Following von Neumann, Connes, Jones, Voiculescu and others, we discuss the basic properties of such algebras $A$, and how to do algebra, geometry, analysis and probability on the underlying quantum spaces $X$. |
| title | Principles of operator algebras |
| topic | Operator Algebras Mathematical Physics Functional Analysis Quantum Algebra |
| url | https://arxiv.org/abs/2208.03600 |