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 &pi;U be its universal representation, acting on Hilbert space HU. The image of &pi;U, &pi;U(A), is a C*-subalgebra of bounded operators on HU. The enveloping von Neumann algebra of A is the closure of &pi;U(A) in the weak operator topology. It is sometimes denoted by A&prime;&prime;.

Properties
The universal representation &pi;U and A&prime;&prime; satisfies the following universal property: for any representation &pi;, there is a unique *-homomorphism


 * $$ \Phi: \pi_U(A) \rightarrow \pi(A) $$

that is continuous in the weak operator topology and the restriction of &Phi; to &pi;U(A) is &pi;.

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&prime;&prime;, as Banach spaces.

Every representation of A uniquely determines a central projection (i.e. a projection in the center of the algebra) in A&prime;&prime;; it is called the central cover of that projection.