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