Enveloping von Neumann algebra

From Wikipedia, the free encyclopedia

In operator algebras, the enveloping von Neumann algebra of a C*-algebra is an object that contains all the operator-algebraic information about the given C*-algebra.

[edit] Definition

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

[edit] Properties

The universal representation πU 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 πU(A) is π.

As a particular case, one can consider the continuous functional calculus, whose unique extention gives a canonical Borel functional calculus.

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 in A′′; it is called the central cover of that projection.

This mathematical analysis-related article is a stub. You can help Wikipedia by expanding it.