Enveloping von Neumann algebra

Type of algebra

This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.
Find sources: "Enveloping von Neumann algebra" – news · newspapers · books · scholar · JSTOR (January 2022) (Learn how and when to remove this message)

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

Properties

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

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

that is continuous in the weak operator topology and the restriction of Φ to π_U(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.

Misplaced Pages