operator algebras In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study ...

, the enveloping von Neumann algebra of a

C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continu ...

is a

von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann a ...

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 In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently ...

; 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

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 In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is ...

. It is sometimes denoted by ''A''′′.

Properties

The universal representation ''π''_''U'' and ''A''′′ satisfies the following

: 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 In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. Theorem Theorem. Let ' ...

, whose unique extension gives a canonical

Borel functional calculus In functional analysis, a branch of mathematics, the Borel functional calculus is a ''functional calculus'' (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. ...

. By the

Sherman–Takeda theorem In mathematics, the Sherman–Takeda theorem states that if ''A'' is a C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties o ...

, the double dual of a C*-algebra ''A'', ''A''**, can be identified with ''A''′′, as

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...

s. 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.

Definition

Properties

See also