nLab bicommutant theorem

Idea

The bicommutant theorem characterizes concrete von Neumann algebras as those concrete $C^*$-algebras ($C^*$-algebras of bounded operators on some Hilbert space) that are the commutants of their own commutants.

Statement

The bicommutant theorem (as known as the double commutant theorem, or von Neumann’s double commutant theorem) is the following result:

Theorem

Let $A \subseteq B(H)$ be a sub-$*$-algebra of the algebra of bounded linear operators on a Hilbert space $H$. Then $A$ is a von Neumann algebra (and therefore also a $C^*$-algebra) in $H$ if and only if $A = A''$, where $A'$ denotes the commutant of $A$.

Notice that the condition of $A$ being a von Neumann algebra (being closed in the weak operator topology; “weak” here can be replaced by “strong”, “ultrastrong”, or “ultraweak” as described in operator topology), which is a topological condition, is by this result equivalent to an algebraic condition (being equal to its bicommutant).

References

• Wikipedia article: bicommutant theorem