Von Neumann bicommutant theorem

Statement

Let $H$ be a Hilbert space and let $A\subseteq B(H)$ be a unital self-adjoint subalgebra. The von Neumann bicommutant theorem states that

\overline A^{\,\mathrm{SOT}} =\overline A^{\,\mathrm{WOT}} =A'',

where the closures use the strong operator topology and weak operator topology, and $A''$ is the bicommutant of $A$ . Consequently, such an algebra is a von Neumann algebra exactly when it equals its bicommutant. The result concerns operator topologies, not operator-norm closure.

Why the closures agree

Commutants are weak-operator closed, so both operator-topology closures of $A$ lie in $A''$ . For the converse, fix $\xi\in H$ . The orthogonal projection onto $\overline{A\xi}$ belongs to $A'$ . This makes it possible to approximate $T\xi$ by vectors $a\xi$ , with $a\in A$ , for every $T\in A''$ . Applying the same argument to finite direct sums of $H$ simultaneously approximates $T$ on any finite set of vectors, which is exactly strong-operator approximation.

Hypotheses and consequences

Unitality and closure under adjoints are essential to the stated form. A non-self-adjoint operator algebra can equal neither its weak closure nor the self-adjoint algebra suggested by a bicommutant. The theorem also shows that the von Neumann algebra generated by a unital self-adjoint algebra $A$ can be described without choosing between strong closure, weak closure, or double commutation.

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume I: Elementary Theory, American Mathematical Society, 1997. AMS DOI record. Relevant: Theorem 5.3.1 and the surrounding discussion of operator topologies.
Jacques Dixmier, Von Neumann Algebras, North-Holland, 1981. Publisher record. Relevant: Chapter I on commutants and the density theorem.