Original FLOW-WH04-05 scope retained; both directions now use actual earlier GW/NF/WF and faithful n.s.f. WR3–6 proof bodies. The entire finite-star domain and closed-operator transport are preserved.

Weight/GNS transport after the forward construction

The finite-star GNS algebra of a faithful normal semifinite weight carries a closable involution. Conversely, the weight constructed from a full left Hilbert algebra recovers that algebra's closed involution in its GNS representation. The identifications below preserve the entire domains.

FLOW-WH04-05 — Weight/GNS transport on the whole domain

For an arbitrary n.s.f. weight $\varphi$, GW-1–5 proves the finite ideals, positive linear extension, GNS representation and finite-star density; NF-5 proves its ultraweak continuity. The finite-star vector space below therefore has the precise earlier algebra/GNS construction. The additional whole-domain involution closability, full finite-star algebra and exact reverse correspondence supply the required Hilbert-algebra axioms and domain identifications:

$$\mathcal A_\varphi=\Lambda_\varphi(\mathfrak n_\varphi\cap\mathfrak n_\varphi^*)$$

Its finite-star involution is $\Lambda_\varphi(x)\mapsto\Lambda_\varphi(x^*)$. No use of a GNS vector for $1$ is permitted unless $\varphi(1)<\infty$. FLOW-WH04-02–04 applies to this algebra by the earlier closability proof. The exact fullness proof identifies this entire finite-star algebra with the full algebra; no smaller core is substituted.

Conversely let $\Phi$ be the weight reconstructed from $\mathcal C$, with finite multiplication ideal $\mathfrak n_l$, and let $I:H_\Phi\to H$ be the canonical GNS unitary proved in WF-6. The entire finite ideal and algebra are proved in WF-4, and WF-6 identifies the complete finite-star involution domain. Those actual earlier results give

$$ I\Lambda_\Phi(\lambda_\xi)=\xi\quad(\xi\in\mathcal B_l), \qquad \lambda(\mathcal C)=\mathfrak n_l\cap\mathfrak n_l^*. $$

On the whole initial finite-star domain the transported weight involution is consequently $S|_{\mathcal C}$. Unitary transport preserves graph closure: $I\oplus I$ maps convergent graph pairs to convergent graph pairs and is invertible. Taking closures gives

$$I S_\Phi I^*=S,$$

Consequently the reconstructed weight and the full left Hilbert algebra have the same closed involution under the canonical GNS unitary, including equality of domains. The given-weight direction is supplied by WR-3, WR-4, and WR-5; WR-6 establishes the canonical opposite finite cone for faithful weights.