Von Neumann double commutant theorem (${\overline{A}}^{\mathrm{SOT}}={\overline{A}}^{\mathrm{WOT}}=A^{\prime \prime }$)

For a unital star-subalgebra $A$ of the bounded operators on a complex Hilbert space, its strong operator topology and weak operator topology closures equal its double commutant of an operator algebra. To approximate $T\in A^{\prime \prime }$ on finitely many vectors, take their tuple in $H^m$. The projection onto its diagonal-$A$ cyclic subspace has matrix entries in $A^{\prime }$, so it commutes with the diagonal action of $T$. Therefore the image tuple under $T$ lies in that cyclic subspace and can be approximated by a single $a\in A$.

For a unital star-subalgebra $A\subseteq B(H)$, its unit ball is dense in the unit ball of $A^{\prime \prime }$ in the strong operator topology. The same holds for self-adjoint unit balls and for positive unit balls, and the full unit balls may be approximated strongly together with their adjoints. Passing first to the norm closure allows continuous functional calculus; resolvent identities prove strong continuity of continuous functions vanishing at infinity, even for unbounded approximating nets of self-adjoint operators. A cutoff function supplies the norm bound, and a two-by-two self-adjoint matrix removes the self-adjoint restriction.