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| | 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 alebras) the term ''W*-algebra'' may be used in place of ''von Neumann algebra''. |
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| | == Definition == |
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| | Let ''A'' be a [[C*-algebra]] and ''π''<sub>''U''</sub> be its [[GNS construction|universal representation]], acting on Hilbert space ''H''<sub>''U''</sub>. The image of ''π''<sub>''U''</sub>, ''π''<sub>''U''</sub>(''A''), is a C*-subalgebra of bounded operators on ''H''<sub>''U''</sub>. The '''enveloping von Neumann algebra''' of ''A'' is the closure of ''π''<sub>''U''</sub>(''A'') in the weak operator topology. It is sometimes denoted by ''A''′′. |
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| | == Properties == |
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| | The universal representation ''π''<sub>''U''</sub> and ''A''′′ satisfies the following [[universal property]]: for any representation ''π'', there is a unique *-homomorphism |
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| | :<math> \Phi: \pi_U(A)'' \rightarrow \pi(A)'' </math> |
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| | that is continuous in the weak operator topology and the restriction of Φ to ''π''<sub>''U''</sub>(''A'') is ''π''. |
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| | As a particular case, one can consider the [[continuous functional calculus]], whose unique extension gives a canonical [[Borel functional calculus]]. |
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| | By the [[Sherman–Takeda theorem]], the double dual of a C*-algebra ''A'', ''A''**, can be identified with ''A''′′, as [[Banach space]]s. |
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| | Every representation of ''A'' uniquely determines a [[central projection]] in ''A''′′; it is called the '''central cover''' of that projection. |
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| | [[Category:C*-algebras]] |
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| | {{mathanalysis-stub}} |
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 alebras) the term W*-algebra may be used in place of von Neumann 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
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.
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 in A′′; it is called the central cover of that projection.
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