Enveloping von Neumann algebra
Type of algebra
From Wikipedia, the free encyclopedia
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 (equivalently, by the von Neumann bicommutant theorem, the enveloping von Neumann algebra is given by πU(A)′′).[1] 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.[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]
