Principles of operator algebras

TL;DR

This paper introduces operator algebras, especially von Neumann algebras, exploring their structure, properties, and applications in quantum mechanics, algebra, geometry, analysis, and probability on quantum measured spaces.

Contribution

It provides an overview of the foundational principles and properties of operator algebras, emphasizing the role of von Neumann algebras in quantum theory and mathematics.

Findings

Von Neumann algebras are stable under adjoints and weak closure.

Trace functions relate algebras to quantum measured spaces.

The free case has a trivial center, highlighting non-commutative structures.

Abstract

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 $X$.