# nLab bicommutant theorem

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 $A\subseteq B\left(H\right)$ 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 $A\prime$ 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

Revised on July 27, 2011 20:40:30 by Toby Bartels (64.89.62.147)