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.