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.
Let A be a C*-algebra and πU be its universal representation, acting on Hilbert space HU.
The enveloping von Neumann algebra of A is the closure of πU(A) in the weak operator topology.
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(A) is π.
By the Sherman–Takeda theorem, the double dual of a C*-algebra A, A**, can be identified with A′′, as Banach spaces.