Hopf–von Neumann Algebras

In this section, we consider a category of objects called Hopf–von Neumann algebras, which in a certain sense generalize locally compact groups, and also their actions on W*-algebras. The main interest of these objects consists in their giving a natural framework for the duality theory of locally compact groups.

We first introduce certain notation and conventions that will be used frequently in what follows.

Let be Hilbert spaces and von Neumann algebras. There exists a unique unitary operator

such that for all. The mapping

is then a *-isomorphism, uniquely determined, such that

The value of a linear form on a vector will also be denoted by

Let G be a locally compact group. The elements of the Hilbert space will be identified with vector-valued functions on G. Also, the linear operators on will usually be defined by specifying the elements. The formal versions of these procedures are standard and well known.

The identity mapping on will be denoted by, or by, where indicates a position in tensor products. The same convention will be used for the unit element

A Hopf–von Neumann algebra is a pair, where is a W *-algebra and is an injective unital normal *-homomorphism, called comultiplication, which is coassociative, that is

The Hopf–von Neumann algebra is said to be commutative if the algebra is commutative and is called cocommutative if

A coinvolutive Hopf–von Neumann algebra is a triple, where is a Hopf–von Neumann algebra and is an involutive *-antiautomorphism, called coinvolution, such that

An action of the Hopf–von Neumann algebra on the W *-algebra is an injective unital normal *-homomorphism such that

In this case, we also say that is an -comodule via.

Note that is an action of on.

The centralizer of the action

Clearly, is a unital W *-subalgebra of.

Let be -comodules via the actions respectively. We shall say that a normal completely positive linear mapping intertwines the actions or that is an -comodule mapping if

In particular, if there exists a *-isomorphism intertwining then we say that the actions are isomorphic.