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Enveloping von Neumann algebra

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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.

Definition

Suppose that A is a C*-algebra and πU its universal representation, acting on the Hilbert space HU. The image of πU, denoted πU(A), is a C*-subalgebra of bounded operators on HU. The enveloping von Neumann algebra of A is defined to be the closure of πU(A) in the weak operator topology. It is sometimes denoted by A′′.

Properties

The universal representation πU 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 πU(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.

See also