Enveloping von Neumann algebra

Enveloping von Neumann algebra

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.

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′′".

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)" ightarrow 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 extension 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.


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