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 the term W*-algebra may be used in place of von Neumann algebra.

Definition

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

Properties

The universal representation πU and A′′ satisfies the following universal property: for any representation π, there is a unique *-homomorphism
that is continuous in the weak operator topology and the restriction of Φ to πU 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 in A′′; it is called the central cover of that projection.