nLab
W*-representation

Definition

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

Warning

If the map AB(H)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 AA is equivalent to the category of Hilbert W*-modules over AA.

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

Last revised on July 3, 2021 at 13:39:22. See the history of this page for a list of all contributions to it.