Original text: CC0 1.0. Prerequisite proofs and component terms.

Countable amplification and the ultraweak bicommutant closure

OA-MOD-BA-01 — Statement and conventions

Let $K$ be an arbitrary complex Hilbert space, with inner products linear in the first variable. Let $A\subseteq B(K)$ be a complex unital *-subalgebra, without any initial closure assumption. Then $$\overline A^{\,\mathrm{ultraweak}}=A''. \tag{BA.1}$$ In particular, if $A$ is ultraweakly closed, then $A=A''$, so $A$ is WOT closed and SOT closed. More precisely, each $T\in A''$ lies in the sigma-strong closure of $A$: for every finite family of square-summable vector sequences $(u_{k,n})_n$ and positive tolerances $\varepsilon_k$, an $a\in A$ can be chosen with $$\left(\sum_n\|(T-a)u_{k,n}\|^2\right)^{1/2}<\varepsilon_k \quad\text{for every }k. \tag{BA.2}$$ The proof of (BA.1) below directly tests the ultraweak functionals, so it does not require a further convex-closure theorem.

A unital algebra on the zero Hilbert space is included; all statements there are immediate. For a possibly degenerate represented algebra with identity projection $p$, the theorem is applied on $pK$.

Exact prerequisites

The proof uses the Hilbert completion, orthogonal projection and Cauchy–Schwarz results in Hilbert spaces and compact operators; the elementary commutant and bounded weak-operator facts of BK-02 and BK-03; and the concrete-predual vector-series representation of CP-03, CP-04 and CP-06. These are bounded Hilbert/operator facts. It does not use a standard form, a normal weight, the normal-map converse, an unbounded spectral theorem, or a Kaplansky density theorem.

The Hilbert direct sum needed here can be defined directly as $$\ell^2(K)=\{(v_n):\ \sum_n\|v_n\|^2<\infty\}, \qquad \langle v,w\rangle=\sum_n\langle v_n,w_n\rangle.$$ Cauchy–Schwarz makes the pairing absolutely convergent. Finite-support vectors are dense by truncation. Completeness follows by taking coordinate limits of a Cauchy sequence, bounding every finite partial sum by the same Cauchy estimate, and then taking the supremum of these finite sums. Thus no dimension restriction on $K$ is introduced by the countable amplification. A finite family of countable index sets can equally be used as its coordinate set.

For $x\in B(K)$, write $D(x)$ for coordinatewise multiplication by $x$ on $\ell^2(K)$. The inequalities $\sum_n\|xv_n\|^2\leq\|x\|^2\sum_n\|v_n\|^2$ show that $D(x)$ is bounded, and coordinate pairings give $D(x)^*=D(x^*)$.

The orbit projection

Fix $\Xi\in\ell^2(K)$, and let $$L=\overline{\{D(a)\Xi:a\in A\}}^{\,\|\cdot\|}. \tag{BA.3}$$ The set inside the closure is linear because $A$ is linear. Multiplicativity gives $D(a)L\subseteq L$ for every $a\in A$. The same is true of $D(a)^*=D(a^*)$. Thus $L$ reduces every $D(a)$, and its orthogonal projection $P$ commutes with each $D(a)$.

Let $\iota_j:K\to\ell^2(K)$ insert a vector into coordinate $j$, and put $$P_{ij}=\iota_i^*P\iota_j\in B(K).$$ Taking matrix entries of $PD(a)=D(a)P$ gives $$P_{ij}a=aP_{ij}\qquad(a\in A).$$ Consequently $P_{ij}\in A'$ for every $i,j$.

Now let $T\in A''$. It commutes with every $P_{ij}$. For a finite-support vector $v$, the $i$-th coordinates of $PD(T)v$ and $D(T)Pv$ are respectively $$\sum_jP_{ij}Tv_j,\qquad \sum_jTP_{ij}v_j.$$ Both sums here are finite and are equal. Equality of every coordinate proves equality of the two vectors. The operators $PD(T)$ and $D(T)P$ are bounded, so density of finite-support vectors extends this equality to all of $\ell^2(K)$. Therefore $D(T)$ commutes with $P$.

Because $1\in A$, we have $\Xi=D(1)\Xi\in L$. It follows that $$P D(T)\Xi=D(T)P\Xi=D(T)\Xi.$$ Thus $D(T)\Xi\in L$. By the definition of $L$, for each $\delta>0$ there is $a\in A$ such that $$\|D(T-a)\Xi\|<\delta. \tag{BA.4}$$ This is approximation for the actual chosen square-summable family; no operator-norm bound on $a$ is claimed or used. Concatenating any finite collection of vector sequences into $\Xi$, and taking $\delta$ smaller than each prescribed tolerance, proves (BA.2).

Ultraweak approximation

Let $f_1,\ldots,f_m$ be a finite family of ultraweakly continuous complex linear functionals on $B(K)$, with positive tolerances $\varepsilon_1,\ldots,\varepsilon_m$. By CP-03, CP-04 and CP-06, each functional has a vector-series representation $$f_k(x)=\sum_{n=1}^\infty\langle x u_{k,n},v_{k,n}\rangle, \qquad \sum_n\|u_{k,n}\|\,\|v_{k,n}\|<\infty. \tag{BA.5}$$ Terms with a zero vector may be omitted. Rescale the two vectors in each remaining term by reciprocal positive factors, leaving the coefficient unchanged, so that both vector norms equal $\sqrt{\|u_{k,n}\|\,\|v_{k,n}\|}$. After this balancing, both $(u_{k,n})_n$ and $(v_{k,n})_n$ are square summable.

Use coordinates $(k,n)$ in a single Hilbert direct sum and put $$\Xi_{k,n}=u_{k,n}.$$ The finite number of functionals makes $\Xi$ square summable. Let $\Eta^{(k)}$ have entries $v_{k,n}$ in the $k$-th row and zero in the other rows. Then (BA.5) is exactly $$f_k(x)=\langle D(x)\Xi,\Eta^{(k)}\rangle.$$ Choose $$0<\delta<\min_{1\leq k\leq m} \frac{\varepsilon_k}{1+\|\Eta^{(k)}\|}.$$ Apply (BA.4) to find one $a\in A$ with $\|D(T-a)\Xi\|<\delta$. Cauchy–Schwarz gives, simultaneously for every $k$, $$|f_k(T-a)| \leq\|D(T-a)\Xi\|\,\|\Eta^{(k)}\| <\varepsilon_k.$$ If the family of tests is empty there is nothing to check. Thus every ultraweak neighborhood of $T$ meets $A$, proving $A''\subseteq\overline A^{\,\mathrm{ultraweak}}$.

Conversely, $A''$ is WOT closed: its commutation equations are closed under matrix-coefficient convergence, because fixed left and right multiplication are WOT continuous. It is therefore ultraweakly closed, since every vector coefficient is ultraweakly continuous. Since $A\subseteq A''$, we have $\overline A^{\,\mathrm{ultraweak}}\subseteq A''$. This proves (BA.1). A bicommutant is SOT closed as well, by the same fixed-multiplication argument or BK-02. $\square$

Application to WH-02 and the OA-FLOW GNS contract

At the 48,230-byte WH manuscript, SHA-256 43274e05ac747aab3b3e5b422fe99b5a05383c8ec451778c53185d242aeea7de, WH-02 lines 30–38 prove isometry and ultraweak closure of the faithful normal representation image. Its identity is $p=\pi(1)$. Compression and zero extension identify its inherited ultraweak topology with that on $B(pH)$: the coefficient of the zero extension at $\xi,\eta\in H$ equals the corner coefficient at $p\xi,p\eta$, and the same observation applies to the square-summable vector series. Thus the image is an ultraweakly closed unital *-subalgebra of $B(pH)$.

Apply (BA.1) on $pH$. This proves precisely the WOT/bicommutant-closed concrete von Neumann algebra assertion in WH-02 line 38. WH-02 line 40 can then apply NP-04 and NP-06 to both directions of the positive *-isomorphism, giving the stated ultraweak and sigma-strong-star homeomorphism. For the unital GNS representation, $p=I$.

Together with the GNS construction, this proves the concrete-image assertion used in OA-FLOW.DW.IMPORT.GNS. It does not establish any natural cone, relative Tomita operator, modular automorphism theorem or Connes derivative.

Proof boundary

The theorem uses bounded Hilbert-space, commutant and predual results. The normal-map, weight and unbounded-operator conclusions require their separate proofs.

Editable source · Proof dependencies and component terms