Enveloping von Neumann algebra

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_U its universal representation, acting on the Hilbert space H_U. The image of π_U, denoted π_U(A), is a C*-subalgebra of bounded operators on H_U. The enveloping von Neumann algebra of A is defined to be the closure of π_U(A) in the weak operator topology (equivalently, by the von Neumann bicommutant theorem, the enveloping von Neumann algebra is given by π_U(A)′′).^[1] It is sometimes denoted by A′′.

The universal representation π_U and A′′ together satisfy the following universal property: for any representation π, there is a unique *-homomorphism

${\displaystyle \Phi$ :\pi _{U}(A)''\rightarrow \pi (A)''}

that is continuous in the weak operator topology and such that the restriction of Φ to π_U(A) is π.^[1]

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.^[1]

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 or support projection of that representation.^[1]

• Universal enveloping algebra