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2023-08-12T23:33:40Z

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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>.

The bicommutant of ''S'' always contains ''S''. So <math>S^{\prime \prime \prime} = \left(S^{\prime \prime}\right)^{\prime} \subseteq S^{\prime}</math>. On the other hand, <math>S^{\prime} \subseteq \left(S^{\prime}\right)^{\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} = \cdots = S^{2n-1} = \cdots</math>

and

:<math>S \subseteq S^{\prime \prime} = S^{\prime \prime \prime \prime} = S^{\prime \prime \prime \prime \prime \prime} = \cdots = S^{2n} = \cdots</math>

for ''n'' > 1.

It is clear that, if ''S''<sub>1</sub> and ''S''<sub>2</sub> are subsets of a semigroup,

:<math>\left( S_1 \cup S_2 \right)' = 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>\left(S_1' \cup S_2'\right)'' = \left(S_1 '' \cap S_2 ''\right)' = \left(S_1 \cap S_2\right)' .</math>

==See also==
* [[von Neumann double commutant theorem]]

== References ==
*J. Dixmier, ''Von Neumann Algebras'', North-Holland, Amsterdam, 1981.

[[Category:Group theory]]

{{group-theory-stub}}

Bicommutant - Revision history

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