Principles of operator algebras

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 $X$.