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In [[algebra]], the '''bicommutant''' of a [[subset]] ''S'' of a [[semigroup]] (such as an [[algebra over a field|algebra]] or a [[group (mathematics)|group]]) is the [[commutant]] of the commutant of that subset. It is also known as the double commutant or second commutant and is written <math>S^{\prime \prime}</math>.


The bicommutant is particularly useful in [[operator theory]], due to the [[von Neumann double commutant theorem]], which relates the algebraic and analytic structures of [[operator algebra]]s. Specifically, it shows that if ''M'' is a unital, self-adjoint operator algebra in the [[C*-algebra]] ''B(H)'', for some [[Hilbert space]] ''H'', then the [[Weak operator topology|weak closure]], [[Strong operator topology|strong closure]] and bicommutant of ''M'' are equal. This tells us that a unital [[C*-algebra|C*-subalgebra]] ''M'' of ''B(H)'' is a [[von Neumann algebra]] if, and only if, <math>M = M^{\prime \prime}</math>, and that if not, the von Neumann algebra it generates is <math>M^{\prime \prime}</math>.
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The bicommutant of ''S'' always contains ''S''. So <math>S^{\prime \prime \prime} = (S^{\prime \prime})^{\prime} \subseteq S^{\prime}</math>. On the other hand, <math>S^{\prime} \subseteq (S^{\prime})^{\prime \prime} = S^{\prime \prime \prime}</math>. So <math>S^{\prime} = S^{\prime \prime \prime}</math>, i.e. the commutant of the bicommutant of ''S'' is equal to the commutant of ''S''. By induction, we have:
 
:<math>S^{\prime} = S^{\prime \prime \prime} = S^{\prime \prime \prime \prime \prime} = \ldots = S^{2n-1} = \ldots</math>
 
and
 
:<math>S \subseteq S^{\prime \prime} = S^{\prime \prime \prime \prime} = S^{\prime \prime \prime \prime \prime \prime} = \ldots = S^{2n} = \ldots</math>
 
for ''n'' > 1.
 
It is clear that, if ''S''<sub>1</sub> and ''S''<sub>2</sub> are subsets of a semigroup,
 
:<math>( S_1 \cup S_2 )' = S_1 ' \cap S_2 ' .</math>
 
If it is assumed that <math>S_1 = S_1'' \,</math> and <math>S_2 = S_2''\,</math> (this is the case, for instance, for [[von Neumann algebra]]s), then the above equality gives
 
:<math>(S_1' \cup S_2')'' = (S_1 '' \cap S_2 '')' = (S_1 \cap S_2)' .</math>
 
==See also==
* [[von Neumann double commutant theorem]]
 
[[Category:Group theory]]

Latest revision as of 07:43, 25 December 2014

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