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In [[mathematics]], specifically [[functional analysis]], the '''von Neumann bicommutant theorem''' relates the [[closure (mathematics)|closure]] of a set of [[bounded operator]]s on a [[Hilbert space]] in certain [[operator topology|topologies]] to the [[bicommutant]] of that set. In essence, it is a connection between the [[algebra]]ic and topological sides of [[operator theory]].
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The formal statement of the theorem is as follows. Let '''M''' be an [[Operator algebra|algebra]] of bounded operators on a Hilbert space H, containing the identity operator and closed under taking [[Hermitian adjoint|adjoint]]s. Then the [[closure (topology)|closure]]s of '''M''' in the [[weak operator topology]] and the [[strong operator topology]] are equal, and are in turn equal to the [[bicommutant]] '''M&prime;&prime;''' of '''M'''.{{clarify|Algebra of all bounded operators is undefined|date=February 2013}} This algebra is the [[von Neumann algebra]] generated by '''M'''.
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There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If '''M''' is closed in the [[norm topology]] then it is a [[C*-algebra]], but not necessarily a von Neumann algebra. One such example is the C*-algebra of [[compact operator on Hilbert space|compact operator]]s (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are still von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, [[ultraweak topology|ultraweak]], [[ultrastrong topology|ultrastrong]], and *-ultrastrong topologies.
It is related to the [[Jacobson density theorem]].
 
== Proof ==
 
Let ''H'' be a Hilbert space and ''L''(''H'') the bounded operators on ''H''. Consider a self-adjoint subalgebra '''M''' of ''L''(''H''). Suppose also, '''M''' contains the identity operator on ''H''.
 
As stated above, the theorem claims the following are equivalent:
 
:i) '''M''' =  '''M&prime;&prime;'''.
:ii) '''M''' is closed in the [[weak operator topology]].
:iii) '''M''' is closed in the [[strong operator topology]].
 
The adjoint map ''T'' → ''T*'' is continuous in the weak operator topology. So the commutant ''S’'' of any subset ''S'' of ''L''(''H'') is weakly closed. This gives i) ⇒ ii). Since the weak operator topology is weaker than the strong operator topology, it is also immediate that ii) ⇒ iii). What remains to be shown is iii) ⇒ i). It is true in general that ''S'' ⊂ ''S&prime;&prime;'' for any set ''S'', and that any commutant ''S&prime;'' is strongly closed. So the problem reduces to showing '''M&prime;&prime;''' lies in the strong closure of '''M'''.
 
For ''h'' in ''H'', consider the smallest closed subspace '''M'''''h'' that contains {''Mh''| ''M'' &isin; '''M'''}, and the corresponding orthogonal projection ''P''.
 
Since ''M'' is an algebra, one has ''PTP'' = ''TP'' for all ''T'' in '''M'''. Self-adjointness of '''M''' further implies that ''P'' lies in '''M&prime;'''. Therefore for any operator ''X'' in '''M&prime;&prime;''', one has ''XP'' = ''PX''. Since ''M'' is unital, ''h'' &isin; '''M'''''h'', hence ''Xh''∈ '''M'''''h'' and for all ε > 0, there exists ''T'' in '''M''' with ||''Xh - Th''|| < ε.
 
Given a finite collection of vectors ''h<sub>1</sub>'',...''h<sub>n</sub>'', consider the direct sum
 
:<math> {\tilde H} = \oplus_1 ^n H.</math>
 
The algebra '''N''' defined by
 
:<math>{\mathbf N} = \{\oplus_1 ^n M \; | \; M \in {\mathbf M} \} \sub L({\tilde H})</math>
 
is self-adjoint, closed in the strong operator topology, and contains the identity operator. Given a ''X'' in '''M&prime;&prime;''', the operator
 
:<math> \oplus_1 ^n X \in L( {\tilde H} ) </math>
 
lies in '''N&prime;&prime;''', and the argument above shows that,  all ε > 0, there exists ''T'' in '''M''' with ||''Xh''<sub>1</sub> - ''Th''<sub>1</sub>||,...,||''Xh<sub>n</sub> - Th<sub>n</sub>''|| < ε. By definition of the strong operator topology, the theorem holds.
 
=== Non-unital case ===
 
The algebra '''M''' is said to be ''non-degenerate'' if for all ''h'' in ''H'', '''M'''''h'' = {0} implies ''h'' = 0. If '''M''' is non-degenerate and a sub [[C*-algebra]] of ''L''(''H''), it can be shown using an [[approximate identity]] in '''M''' that the identity operator ''I'' lies in the strong closure of '''M'''. Therefore the bicommutant theorem still holds.
 
== References ==
 
*W.B. Arveson, ''An Invitation to C*-algebras'', Springer, New York, 1976.
 
[[Category:Operator theory]]
[[Category:Von Neumann algebras]]
[[Category:Articles containing proofs]]
[[Category:Theorems in functional analysis]]

Latest revision as of 20:05, 2 January 2015

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