Stokes drift: Difference between revisions

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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 ==
 
Let ''A'' be a [[C*-algebra]] and ''&pi;''<sub>''U''</sub> be its [[GNS construction|universal representation]], acting on Hilbert space ''H''<sub>''U''</sub>. The image of ''&pi;''<sub>''U''</sub>, ''&pi;''<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 ''&pi;''<sub>''U''</sub>(''A'') in the weak operator topology. It is sometimes denoted by ''A''&prime;&prime;.
 
== Properties ==
 
The universal representation ''&pi;''<sub>''U''</sub> and ''A''&prime;&prime; satisfies the following [[universal property]]: for any representation ''&pi;'', there is a unique *-homomorphism
 
:<math> \Phi: \pi_U(A)'' \rightarrow \pi(A)'' </math>
that is continuous in the weak operator topology and the restriction of &Phi; to ''&pi;''<sub>''U''</sub>(''A'') is ''&pi;''.
 
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''&prime;&prime;, as [[Banach space]]s.
 
Every representation of ''A'' uniquely determines a [[central projection]] in ''A''&prime;&prime;; it is called the '''central cover''' of that projection.
 
[[Category:C*-algebras]]
 
{{mathanalysis-stub}}

Revision as of 23:10, 21 February 2014

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