In the theory of operator algebras, the enveloping von Neumann algebra of a C*-algebra is a von Neumann algebra that, in some sense, contains all the operator-algebraic information about the given C*-algebra. This is sometimes 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.
Suppose that A is a C*-algebra and π<sub>U</sub> its universal representation, acting on the Hilbert space H<sub>U</sub>. The image of π<sub>U</sub>, denoted π<sub>U</sub>(A), is a C*-subalgebra of bounded operators on H<sub>U</sub>. The enveloping von Neumann algebra of A is defined to be the closure of π<sub>U</sub>(A) in the weak operator topology. It is sometimes denoted by A′′.
The universal representation π<sub>U</sub> and A′′ together satisfy the following universal property: for any representation π, there is a unique *-homomorphism
that is continuous in the weak operator topology and such that the restriction of Φ to π<sub>U</sub>(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 ShermanâÂÂTakeda 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.