Original reconstruction by GPT-6 Astra (OpenAI), Ultra. Self-checked by the writing AI.

Faithful normal states: GNS normality and the modular application

Fresh local proof by GPT-6 Astra (OpenAI), Ultra, 2026-10-04. CC0-1.0 to the extent of rights in this exposition.

This bridge proves the faithful-state route needed by L159 without the inherited arbitrary-weight normality theorem. A normal functional here means a continuous linear functional for the concrete ultraweak vector-series topology, as in SF-2. A faithful normal state belongs to the positive cone of that predual and has value one at the identity. We do not replace an order-normal arbitrary-weight theorem by this convention.

The precise earlier inputs are CF Section 1, CF Section 4, CF Sections 6–7, GNS Sections 2–5, SF-0/SB-0, SF-2, and CP-01–06. Those six predual proof bodies are earlier in this reader. Their Hilbert and Hahn–Banach inputs are the displayed fresh CF/SF proofs. No support or arbitrary-weight descendant is an input.

For clarity, the entire CP contract used here is this: the completion of the projective-norm algebraic tensor $H\otimes\overline H$ has dual $B(H)$, via vector pairings; every completed tensor is an absolutely summable pure-tensor series, whose factors may be balanced into two square-summable sequences. Quotienting by the annihilator of a concrete von Neumann algebra $M$ gives a Banach space $M_*$, isometric to the vector-series functionals in $M^*$, with $M=(M_*)^*$. In particular $M_*$ is norm closed in $M^*$. These statements are proved, including tensor norm, completion, quotient and annihilator arguments, in the exact CP range just identified; a source-history judgment is separate from this mathematical proof binding.

The new local analytic providers used below are BD, CI, and HA-R. The final full-algebra application uses WH-04 Sections 2–4 and MF-06's general Hilbert-algebra proof, at the replaced foundation bindings specified here. Their arbitrary-weight applications are not premises of this route. The coefficient/GNS-normality argument used below also appears earlier as ST-3 coefficient proof, with the already ultraweak vector-series state premise and no modular input. The full original ST-3 scope and its modular application remain here. The earlier topology conclusions used below are ST-1 and ST-2.

ST-3. A faithful normal state's GNS representation meets that hypothesis

Let $M\subseteq B(H)$ be a nonzero concrete von Neumann algebra and let $\varphi\in M_*^+$ be a faithful state. GNS Sections 2–5 construct the Hilbert space $K$, contractive representation $\pi$, and cyclic vector $\xi$, with $\langle\pi(x)\pi(a)\xi,\pi(b)\xi\rangle=\varphi(b^*xa). \tag{ST4}$ For fixed $a,b$, the right side is normal: in any vector-series expression for $\varphi$, replace its first vectors by $a u_j$ and its second by $b v_j$. The new vector sequences are square summable. Arbitrary pairs $\eta,\zeta\in K$ are norm limits of cyclic-domain pairs. Contractivity gives the uniform functional-norm estimate $\bigl|\langle\pi(x)\eta,\zeta\rangle- \langle\pi(x)\eta_0,\zeta_0\rangle\bigr| \leq\|x\|\bigl(\|\eta-\eta_0\|\|\zeta\| +\|\eta_0\|\|\zeta-\zeta_0\|\bigr). \tag{ST5}$ Since $M_*$ is norm closed, every vector coefficient of $\pi$ pulls back to $M_*$. A square-summable pair of vector sequences in $K$ gives a norm-convergent sum of those pulled-back functionals, because its tail is bounded in functional norm by the product of the two square-sum tails. Again norm closure puts the sum in $M_*$. This proves ultraweak continuity of $\pi$ for the full vector-series topology, without converting order normality through NW or NP.

The representation is faithful: $\pi(x)=0$ implies $\varphi(x^*x)=\|\pi(x)\xi\|^2=0$, hence $x=0$. It is unital since $\pi(1)$ fixes the dense cyclic domain. ST-2 applies. Its image is a concrete von Neumann algebra, and $\xi$ is separating for it because $\pi(x)\xi=0$ gives the same faithful-state test.

In particular the vector algebra $\mathcal A=\{\pi(x)\xi:x\in M\}$, with product inherited from $M$ and involution $\pi(x)\xi\mapsto\pi(x^*)\xi$, is a left Hilbert algebra. Separatingness makes the definitions unambiguous; its left multipliers are bounded, its inner-product adjoint identity is immediate, and $\mathcal A^2=\mathcal A$ because the unit is present. CI-4 proves closability by the dense commutant orbit. HA-R and the separately reviewed WH-04 then give its full second dual with the identical closed involution, on the identical Hilbert space.

MF-06's general full-Hilbert-algebra proof can therefore be applied with CI/HA-R/BD in place of its affected TC/HA/RD premises. It returns the modular commutant and automorphism conclusions for the original $\pi(M)$ and this same $S$. ST-2 transports the bounded algebra and its normal topology back to $M$. This is the faithful-normal-state route needed in L159, preserving all its separable-predual cases; ST-1–3 themselves require no separability.

ST-4. Exact remaining distinctions

This route uses the actual free-developed CP-01–06 construction at newly specified Hilbert/Hahn–Banach inputs, not its support or arbitrary-weight descendants.

The arbitrary given-weight finite ideals, GNS construction, cutoffs, density and normal faithful representation now have the earlier GW/NF proof bodies. Its finite-star involution closability and full/reverse/opposite-weight correspondence now have the earlier WR-3–WR-6 faithful n.s.f. proofs. WF proves the full forward algebra-to-weight construction and its GNS/involution-domain transport. The earlier extended-valued normality theorem covers every weight, and MW supplies the faithful modular opposite formula. Broader nonfaithful statements retain their precise scopes.