# 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.

<a id="oa-flow.wh04.5"></a>

## 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

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

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](OA-FLOW-WR.md#wr-3), [WR-4](OA-FLOW-WR.md#wr-4), and [WR-5](OA-FLOW-WR.md#wr-5); [WR-6](OA-FLOW-WR.md#wr-6) establishes the canonical opposite finite cone for faithful weights.