nLab W*-representation

Definition

A W*-representation, or simply a representation of a von Neumann algebra $A$ on a Hilbert space $H$ is a morphism of von Neumann algebras $A\to B(H)$, where $B(H)$ is the von Neumann algebra of bounded operators on the Hilbert space $H$.

Warning

If the map $A\to B(H)$ is injective, the resulting notion coincides with that of a “concrete von Neumann algebra”, as opposed to an (abstract) von Neumann algebra. An isomorphism of representations is sometimes referred to as a “spatial isomorphism of concrete von Neumann algebras”.

Such terminology is confusing, but is present in some sources, especially older ones.

Properties

The category of W*-representations of $A$ is equivalent to the category of Hilbert W*-modules over $A$.

See the article Hilbert W*-module for more information.

Related concepts

• Hilbert W*-module

• von Neumann algebra

• duality between geometry and algebra