---
tag: BSA6GFT8SQ
kind: statement
role: theorem
name: Von Neumann bicommutant theorem
msc: [46L10]
uses: [HM2HHE3J51, BMDG4JQPEH, 6HSJBJ8TTQ, 1VC7EJ1ZBZ, 4XM1VN1C1N, KHJ2G2R5G7, 0B8KZRT28M]
about: [PSFWQVZ0FS, 1VC7EJ1ZBZ]
sources:
  - url: https://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem
same_as:
  wikidata: Q528987
  wikipedia: Von Neumann bicommutant theorem
---

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] with [[algebra of bounded operators|BMDG4JQPEH]] $B(H)$.
Let $M$ be a subalgebra of $B(H)$ that contains the identity $I$ and is closed under the [[adjoint|6HSJBJ8TTQ]].
Let $M''$ be the [[bicommutant|1VC7EJ1ZBZ]] of $M$.
Then the following are equivalent:

1. $M = M''$;
2. $M$ is closed in the [[weak operator topology|4XM1VN1C1N]];
3. $M$ is closed in the [[strong operator topology|KHJ2G2R5G7]].

In particular, $M''$ is the closure of $M$ in each of these two topologies.

## Remarks

Every commutant is weakly closed, so 1 implies 2, and 2 implies 3 because the strong topology is finer.
For 3 implies 1, let $T \in M''$ and $x \in H$.
The [[orthogonal projection|0B8KZRT28M]] $P$ onto the closure of $Mx$ commutes with $M$ because $M$ is closed under adjoints, so $T$ commutes with $P$ and $Tx \in \overline{Mx}$, using $x = Ix \in Mx$.
Applying this to the diagonal action of $M$ on $H^n$ handles $n$ vectors at once, which shows that $T$ is a strong limit of elements of $M$.

The identity is needed: the algebra $M = \set{0}$ is closed under adjoints and weakly closed, but $M' = B(H)$ and $M'' = \C I \ne M$.
More generally, without the identity the theorem holds in the form $M'' = \overline{M}$ only when $M$ acts nondegenerately.

For example, on $\C^n$ the diagonal matrices $D$ satisfy $D' = D$, so $D'' = D' = D$, as the theorem predicts for a finite-dimensional, hence closed, algebra.

## Explanations

### L1 human

A von Neumann algebra can be described in two very different ways.
One is analytic: it is closed under weak limits of operators.
The other is algebraic: it consists of all operators that commute with everything that commutes with it.
The theorem says these two descriptions agree, as long as the algebra contains the identity and is closed under adjoints.

### L2 human

The theorem lets one produce von Neumann algebras for free.
The commutant of any self-adjoint set $S$ of operators is one, and so is the bicommutant $S''$, the von Neumann algebra generated by $S$.
It turns questions about topological closure into algebraic questions about commuting operators.
This is how one shows, for example, that the spectral projections of a self-adjoint element of $M$ lie in $M$.
Together with the Kaplansky density theorem, it is the basic tool for approximating elements of $M''$ by elements of $M$.