Countable amplification and the ultraweak bicommutant closure

OA-MOD-BA-01. Exact programme bicommutant import and local weight-representation application.

This programme proof import supplies the precise implication used after ultraweak image closure in OA-MOD-WH-02: an ultraweakly closed unital *-subalgebra of bounded operators is a concrete von Neumann algebra. It does not assume a uniform operator bound on a strongly convergent approximating net.

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

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

The exact programme proof of (BA.1) directly tests the ultraweak functionals; it does not require an additional 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 programme proof and countable amplification

Read Lemma 4.1 and Theorem 4.2 of Spatial tensor products: complete proof supplement through the commutation theorem, original text by Claude Opus 5.5 (Anthropic), September 2026, under CC0; current selection and prerequisite bindings by GPT-6.1 Sol (OpenAI), Ultra, October 2026. These complete amplified-density and bicommutant proofs belong to the existing programme supplement. The argument uses the actual square-summable vector family on the Hilbert tensor product; it assumes neither a separable ambient Hilbert space nor a bounded approximating net. Sections 1–2 supply the Hilbert tensor and operator-matrix facts; CP-03, CP-04 and CP-06 give the concrete vector-series representation used in the local ultraweak-test application.

Lemma 4.1 states that for \(T\in A''\), every square-summable family \((u_n)\) and every \(\delta>0\), there is \(a\in A\) with \(\sum_n\|(T-a)u_n\|^2<\delta^2\). Theorem 4.2 then gives the strong, weak and ultraweak closures as \(A''\), in particular (BA.1).

Solved finite-family application. Concatenate the finitely many square-summable families \((u_{k,n})_n\) into

Let \(D(x)\) act coordinatewise by \(x\) on this direct sum. Applying the programme lemma to this single family, with \(0<\delta<\min_k\varepsilon_k\), gives

Each separate family norm is at most this concatenated norm, proving the simultaneous inequalities (BA.2). If the family of tests is empty there is nothing to prove. No operator-norm bound on \(a\) is asserted. This is a specialization of the existing density theorem, not a second proof of it.

Solved balanced ultraweak-test application

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

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

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

Choose

Apply the programme Lemma 4.1, as specialized in (BA.4), to find one \(a\in A\) with \(\|D(T-a)\Xi\|<\delta\). Cauchy–Schwarz gives, simultaneously for every \(k\),

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 verifies the ultraweak-test specialization of the imported Theorem 4.2. That theorem owns (BA.1) and its strong/weak closure conclusions. \(\square\)

Application to multiplication representations

The representation-image proof in WH-02 establishes 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. Its following argument 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, \(p=I\).

It does not establish any natural cone, relative Tomita operator, modular automorphism theorem or Connes derivative.

Validation and status

The bicommutant theorem (BA.1) is proved in Lemma 4.1 and Theorem 4.2 of the spatial tensor products supplement cited above; this lesson contributes the solved finite-family and balanced-functional applications and its support-corner/topology argument for WH-02. The OA-FLOW binding keeps that exact consumer consequence.