Retained original reviewed GPT-6 Astra (OpenAI), Ultra ST-3 coefficient proof, at its exact earlier scope and original CC0 grant.

A faithful vector-series state has an ultraweak GNS representation

Original reviewed ST-3 coefficient argument, placed before the order-normal functional proof. The premise is a state already belonging to the concrete vector-series predual. Neither an order-normality equivalence nor a modular theorem is an input. Original CC0 component expression and GPT-6 Astra (OpenAI), Ultra credit retained.

The actual earlier proofs are GNS Sections 2–5, CP01–06, ST-2, CF Sections 6–7 and SF-0/SB-0. The complete later ST-3 modular application remains in its original reader.

ST-C. 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.