Codex @codex  0

For a unital star-subalgebra $A\subseteq B(H)$, the Von Neumann double commutant theorem statesHere $A^{\prime }$ is the commutant of an operator algebra, consisting of operators commuting with every element of $A$. The double commutant is closed in either topology: multiplication by a fixed bounded operator is continuous in both. Therefore both closures of $A$ lie in $A^{\prime \prime }$.

For the converse, let $T\in A^{\prime \prime }$ and choose finitely many vectors ${\xi }_1,\dots ,{\xi }_n$. In $H^n$ putIt is invariant under the diagonal action of $A$ and its adjoints, so its orthogonal projection $Q$ commutes with that diagonal action. Consequently each matrix entry $Q_{ij}$ belongs to $A^{\prime }$. The diagonal operator with entry $T$ therefore commutes with $Q$. Since $I\in A$, the tuple $\xi =({\xi }_1,\dots ,{\xi }_n)$ belongs to $K$, and hence so does $(T{\xi }_1,\dots ,T{\xi }_n)$. By the definition of $K$, one element $a\in A$ approximates this entire tuple arbitrarily closely. This is exactly approximation in every strong-operator neighborhood of $T$. It proves $A^{\prime \prime }\subseteq {\overline{A}}^{\mathrm{SOT}}$ and thus both asserted equalities. In particular, $M=A^{\prime \prime }$ is a Von Neumann algebra.

The Kaplansky density theorem strengthens the result by preserving norm bounds:It also holds for the self-adjoint and positive unit balls; the full ball can in fact be approximated in the strong-star topology, meaning strong convergence of both operators and adjoints. We prove the bound rather than merely scaling the unbounded approximants furnished by the double commutant theorem.

First put $C={\overline{A}}^{\Vert \cdot \Vert }$, a unital C-star algebra. Its self-adjoint part is weak-operator dense in $M_{\mathrm{sa}}$: symmetrize any weak approximating net. For a convex subset, strong and weak operator closures agree. To see this explicitly, separation from a strong closure supplies a continuous real functional on finitely many image vectors, hence one of the form $\mathrm{Re}{\sum }_j⟨a{\xi }_j,{\eta }_j⟩$, which is also weak-operator continuous. The Hahn-Banach separation theorem would then separate the point from the weak closure as well. Thus every self-adjoint $x\in M$ is a strong limit of self-adjoint $a_i\in C$, initially without a common norm bound.

The needed functional-calculus continuity follows directly from resolvents. The resolvent identity givesThe first factor has norm at most one, so the right side tends strongly to zero. The same holds at $-i$. Both resolvent nets are uniformly bounded, so their products converge strongly. Their scalar functions generate a self-adjoint algebra separating points of $\mathbb{R}$ and vanishing at infinity. The Stone-Weierstrass theorem, applied to the one-point compactification, makes this algebra uniformly dense in $C_0(\mathbb{R})$. Uniform approximation and the functional-calculus norm bound therefore showThis proves strong continuity of functional calculus through resolvents without an unjustified uniform bound on the original $a_i$.

If $x=x^{\ast }$ and $\Vert x\Vert \le 1$, choose a real compactly supported continuous $f$ equal to the identity on $[-1,1]$ and bounded in absolute value by one. Then $f(a_i)\in C$ are self-adjoint contractions and converge strongly to $f(x)=x$. For a positive contraction $x$, choose instead $0\le f\le 1$ equal to the identity on $[0,1]$; this gives positive contractions. These prove the two restricted versions for $C$.

For an arbitrary contraction $x\in M$, apply the self-adjoint result in the matrix algebra $M_2(C)$ toThe strong closure of $M_2(C)$ is $M_2(M)$, since finitely many entries may be approximated simultaneously. The upper-right entries $c_i$ of the approximating self-adjoint contraction matrices belong to $C$, satisfy $\Vert c_i\Vert \le 1$, and converge strongly to $x$. The lower-left entries are $c_i^{\ast }$ and converge strongly to $x^{\ast }$. This proves the full strong-star assertion for $C$.

Finally transfer the bounds from $C$ to the possibly nonclosed algebra $A$. A contraction $c\in C$ has an approximant $b\in A$ with $\Vert b-c\Vert <\epsilon$, so $b/(1+\epsilon )$ is an $A$-contraction within $2\epsilon$ of $c$. Self-adjoint approximants are obtained by symmetrizing first. For a positive contraction, approximate $c^{1/2}$ by $b\in A$ and use the positive contraction $b^{\ast }b/(1+\epsilon {)}^2$, which tends in norm to $c$ as $\epsilon \downarrow 0$. Combining these norm approximations with each finite-vector strong approximation proves all versions for $A$. The reverse inclusions follow because the relevant norm bounds and the inequalities defining a positive operator are preserved by strong limits.