Enveloping von Neumann algebra
# Enveloping von Neumann algebra

In operator algebras, the enveloping von Neumann algebra of a C*-algebra is a von Neumann algebra that contains all the operator-algebraic information about the given C*-algebra. This may also be called the universal enveloping von Neumann algebra, since it is given by a universal property; and (as always with von Neumann algebras) the term W*-algebra may be used in place of von Neumann algebra.

## Definition

Let A be a C*-algebra and $π_U$ be its universal representation, acting on Hilbert space $H_U.$ The image of $π_U,$ $π_U(A),$ is a C*-subalgebra of bounded operators on $H_U.$ The enveloping von Neumann algebra of A is the closure of $π_U(A)$ in the weak operator topology. It is sometimes denoted by A′′.

## Properties

The universal representation $π_U$ and A′′ satisfies the following universal property: for any representation π, there is a unique *-homomorphism

    $\Phi :\pi _{U}(A)''\rightarrow \pi (A)''$

that is continuous in the weak operator topology and the restriction of Φ to $π_U(A)$ is π.

As a particular case, one can consider the continuous functional calculus, whose unique extension gives a canonical Borel functional calculus.

By the Sherman–Takeda theorem, the double dual of a C*-algebra A, A**, can be identified with A′′, as Banach spaces.

Every representation of A uniquely determines a central projection (i.e. a projection in the center of the algebra) in A′′; it is called the central cover of that projection.

