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.

The Von Neumann double commutant theorem states that for a unital star-subalgebra $A\subseteq B(H)$,The commutant of an operator algebra consists of operators commuting with every element. Each commutation equation is closed in the weak operator topology, so $A^{\prime \prime }$ is weakly closed and contains both indicated closures.

Let $T\in A^{\prime \prime }$ and fix ${\xi }_1,\dots ,{\xi }_m\in H$. On $H^m$ let $A$ act diagonally and put $\xi =({\xi }_1,\dots ,{\xi }_m)$. The subspace $K=\overline{\{(a{\xi }_1,\dots ,a{\xi }_m):a\in A\}}$ reduces the diagonal star-algebra. Its orthogonal projection $P$ therefore commutes with every diagonal $a$. Writing $P$ as an $m\times m$ operator matrix shows that each entry belongs to $A^{\prime }$. Since $T\in A^{\prime \prime }$, the diagonal $T$ commutes with $P$. Unitality gives $\xi \in K$, so $P\xi =\xi$ and $P(T\xi )=T\xi$. Thus $T\xi \in K$. By definition of $K$, for any $\epsilon >0$ a single $a\in A$ satisfies ${\sum }_j\Vert (T-a){\xi }_j{\Vert }^2<{\epsilon }^2$. These are precisely the finite-vector neighbourhoods of the strong operator topology. Hence $T$ belongs to the strong closure, completing the proof. Unitality, or the corresponding nondegeneracy hypothesis, is essential to the theorem's stated form.

A linear subalgebra closed under the involution. In bounded operators, this means closure under the adjoint operator; neither norm closure nor operator-topology closure is part of the definition. These distinctions matter in the Von Neumann double commutant theorem and the Kaplansky density theorem.

A unital star-subalgebra of the bounded operators on a Hilbert space that is closed in the weak operator topology, equivalently in the strong operator topology. The Von Neumann double commutant theorem makes these conditions equivalent to equality with the double commutant of an operator algebra. It is norm closed and hence a C-star algebra.