# Weight/GNS transport after the forward construction

This is the original FLOW-WH04-05 scope, placed after the complete earlier WF construction. The arbitrary faithful n.s.f. given-weight closability, full/reverse correspondence and canonical opposite finite cone now have the earlier WR proofs, after GW/NF and EW. The constructed-weight direction uses the earlier WF proof.

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

For an arbitrary n.s.f. weight $\varphi$, [GW-1–5](OA-FLOW-GW.md#oa-flow.gw.1) proves the finite ideals, positive linear extension, GNS representation and finite-star density; [NF-5](OA-FLOW-NF.md#oa-flow.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](OA-FLOW-WR.md#wr-3), [full finite-star algebra](OA-FLOW-WR.md#wr-4) and [exact reverse correspondence](OA-FLOW-WR.md#wr-5) 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](OA-FLOW-WH04.md#oa-flow.wh04.2) applies to this algebra by the earlier [closability proof](OA-FLOW-WR.md#wr-3). The exact [fullness proof](OA-FLOW-WR.md#wr-4) 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](OA-FLOW-WF.md#oa-flow.wf.6). The entire finite ideal and algebra are proved in [WF-4](OA-FLOW-WF.md#oa-flow.wf.4), and [WF-6](OA-FLOW-WF.md#oa-flow.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

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

including equality of domains. This is an equality of closed operators, not merely agreement on the original algebra. It proves the domain transport required by the Haagerup construction at the actual earlier WF forward-construction inputs. The first paragraph, beginning from an arbitrary faithful n.s.f. given weight, now uses the complete earlier [closability](OA-FLOW-WR.md#wr-3), [fullness](OA-FLOW-WR.md#wr-4) and [recovery](OA-FLOW-WR.md#wr-5) proofs; [the canonical opposite finite cone](OA-FLOW-WR.md#wr-6) retains its exact faithful scope.