# Weights and the Hilbert spaces of multiplication ## OA-MOD-WH-01 — Interfaces and analytic prerequisites All Hilbert spaces and index sets are arbitrary. Inner products are linear in the first variable. A von Neumann algebra is concrete and unital; the zero algebra is allowed. A normal weight preserves bounded increasing positive suprema, and semifiniteness means ultraweak density of its finite definition algebra, as in WG-002. We use the following results with their stated hypotheses: - **WG-003–010:** the finite ideals, their positive cone, GNS construction, finite positive contraction nets, GNS density of the finite-star domain, and faithfulness and normality of the GNS representation. - **NW-11 and OW-02–03:** recovery of a normal weight from its dominated normal positive functionals, and the exact implementing vectors for a normal semifinite weight. - **HA-01–08 and RD-02–07:** the two bounded multiplication constructions, closed involutions, right-algebra graph density, commutant generation, product cores and the adjoint intersection of the multiplication ideal. - **BK-02–08 and DW-02:** bicommutants, bounded operator topologies, positive square roots, inverse order, support cutoffs, and polar and Douglas factorizations inside von Neumann algebras, also for rectangular matrices. - **CP and NP-04–06:** concrete Banach preduals and ultraweak continuity of positive normal maps. These do not themselves include the image-closure theorem proved in WH-02. - **OA-MOD-OPEN-CONVEX-ALAOGLU**, **OA-MOD-OPEN-CONVEX-HILBERT-BALL** and **OA-MOD-OPEN-CONVEX-KREIN-SMULIAN**: dual norm balls are weak-star compact; Hilbert norm balls are weakly compact; a convex subset of a Banach dual is weak-star closed if its intersections with all centered norm balls are weak-star closed. The last statement is used only in WH-02. The direct normality proof in WH-07 uses Hilbert weak compactness, with subnets rather than a sequential compactness assertion. The unbounded spectral contract remains a transitive prerequisite of RD's graph-density theorem; this unit does not replace that contract with a bounded spectral theorem. We shall use strong continuity of positive square roots on a uniformly bounded positive set. Here is the needed argument. If \(0\leq a_i,a\leq C1\) and \(a_i\to a\) strongly, fixed powers and therefore fixed polynomials converge strongly, by induction and the uniform operator bounds. Uniform polynomial approximation of \(t^{1/2}\) on \([0,C]\), followed by bounded continuous functional calculus, makes the errors in both square roots uniformly small in operator norm. Applying this to each vector proves \(a_i^{1/2}\to a^{1/2}\) strongly. This works for arbitrary nets. ## OA-MOD-WH-02 — The image of a faithful normal representation **Lemma.** If \(\pi:M\to B(H)\) is a faithful normal *-representation of a von Neumann algebra, then \(\pi(M)\) is an ultraweakly closed *-subalgebra with identity \(p=\pi(1)\). It acts as a von Neumann algebra on \(pH\), and as zero on \((I-p)H\). The map \(\pi\) is an ultraweak and sigma-strong* homeomorphism onto its image, with the inherited corner topologies. When \(\pi\) is unital, \(p=I_H\). **Proof.** First \(\pi\) is isometric. Contractivity follows from positivity and the C*-identity. To check the reverse inequality without assuming that its image is closed, let \(b\in M_+\). If \(\|\pi(b)\|<\|b\|\), choose a continuous function on \([0,\|b\|]\) vanishing on \([0,\|\pi(b)\|]\) but nonzero at \(\|b\|\). Continuous functional calculus gives \(f(b)\ne0\), since the norm of a positive element belongs to its spectrum, but \(\pi(f(b))=f(\pi(b))=0\), contradicting faithfulness. Applying this to \(b=x^*x\) proves isometry. For every \(r>0\), the closed ball \(rB_M\) is compact for \(\sigma(M,M_*)\), by Alaoglu and the concrete predual. Normality makes \(\pi\) ultraweakly continuous, by NP-04. Its image \(\pi(rB_M)\) is therefore ultraweakly compact and closed in \(B(H)\). Isometry gives \[ \pi(rB_M)=\pi(M)\cap rB_{B(H)}. \tag{WH.1} \] The linear subspace \(\pi(M)\) is convex. Apply Krein–Smulian in the Banach dual \(B(H)\) to (WH.1): \(\pi(M)\) is ultraweakly closed. Multiplicativity gives \(\pi(x)=p\pi(x)p\), with identity \(p=\pi(1)\). Thus the image is a von Neumann algebra on \(pH\); the inherited ultraweak and sigma-strong* topologies are exactly the corner topologies, since compressing the test vectors by \(p\) gives the same seminorms and functionals. In the nonunital case the continuous function used above vanishes at zero, so its functional-calculus identity also holds in this corner. Both \(\pi\) and its inverse are now positive *-isomorphisms between von Neumann algebras. They transport order bounds in both directions, so preserve existing bounded increasing positive suprema. NP-04 and NP-06 give ultraweak and sigma-strong* continuity in both directions. The identity assertion follows by evaluating \(\pi(1)\). \(\square\) The weak-star closed-ball argument is essential here. A merely bounded positive normal map can have a nonclosed image, as NP-08 shows. ## OA-MOD-WH-03 — Completing the two multiplication domains Let \(\mathcal A\subseteq H\) be any left Hilbert algebra. Retain \[ M=L(\mathcal A)'',\qquad S=\overline{\sharp},\qquad F=S^*, \] and the right-bounded space \(\mathcal B_r\), its operators \(R_\eta\), and the right algebra \[ \mathcal A_r=\mathcal B_r\cap D(F). \tag{WH.2} \] By RD-04–05, \(\mathcal A_r\) is a right Hilbert algebra, its closed involution is \(F\), and \(R(\mathcal A_r)''=M'\). Define the **left-bounded space relative to this right algebra** by \[ \mathcal B_l= \left\{\xi\in H:\ \exists C<\infty\ \forall\eta\in\mathcal A_r,\ \|R_\eta\xi\|\leq C\|\eta\|\right\}. \tag{WH.3} \] For \(\xi\in\mathcal B_l\), let \(\lambda_\xi\) be the unique bounded operator satisfying \[ \lambda_\xi\eta=R_\eta\xi\qquad(\eta\in\mathcal A_r). \tag{WH.4} \] Put \[ \mathfrak n_l=\{\lambda_\xi:\xi\in\mathcal B_l\}, \qquad \mathcal A_l=\mathcal B_l\cap D(S). \tag{WH.5} \] **Theorem.** The map \(\lambda:\mathcal B_l\to M\) is linear and injective, and \[ x\xi\in\mathcal B_l,\qquad \lambda_{x\xi}=x\lambda_\xi \quad(x\in M,\ \xi\in\mathcal B_l). \tag{WH.6} \] Thus \(\mathfrak n_l\) is a left ideal in \(M\). Moreover, \[ \mathcal A\subseteq\mathcal A_l,\qquad \lambda_a=L_a\quad(a\in\mathcal A). \tag{WH.7} \] The space \(\mathcal A_l\) is a left Hilbert algebra with \[ \xi\zeta=\lambda_\xi\zeta,\qquad \xi^\sharp=S\xi,\qquad \lambda_{S\xi}=\lambda_\xi^*, \tag{WH.8} \] and \[ \overline{S|_{\mathcal A_l}}=S,\qquad \lambda(\mathcal A_l)''=M,\qquad \lambda(\mathcal A_l)=\mathfrak n_l\cap\mathfrak n_l^*. \tag{WH.9} \] **Proof.** Apply HA and RD to the opposite algebra of \(\mathcal A_r\), whose product is \(\eta\circ\zeta=\zeta\eta\). Its left multiplication is \(R_\eta\), its generated algebra is \(M'\), its closed involution is \(F\), and the adjoint involution is \(F^*=S\). The right-bounded-vector construction in that application is exactly (WH.3–4): the test operator for \(\xi\) sends \(\eta\) to \(R_\eta\xi\). HA-05 therefore proves linearity, injectivity, membership in \((M')'=M\), and (WH.6). For \(a\in\mathcal A\) and \(\eta\in\mathcal A_r\), \(R_\eta a=L_a\eta\), so \(a\in\mathcal B_l\) and \(\lambda_a=L_a\). Also \(a\in D(S)\), proving (WH.7). The right algebra in the application to \(\mathcal A_r^{\mathrm{op}}\) has product \(\xi\circ\zeta=\lambda_\zeta\xi\). Taking its opposite makes the product \(\xi\zeta=\lambda_\xi\zeta\), as asserted. HA-08 and RD-04 give the Hilbert algebra axioms and closed involution \(S\). RD-05 gives the generated algebra \((M')'=M\). RD-07 gives precisely the ideal intersection in (WH.9). These are applications to a fully established right Hilbert algebra, not an assumption that \(\mathcal A\) was already full. \(\square\) ## OA-MOD-WH-04 — Mixed bounded vectors and fullness **Mixed-product lemma.** For every \(\xi\in\mathcal B_l\) and every \(\eta\in\mathcal B_r\), \[ \lambda_\xi\eta=R_\eta\xi. \tag{WH.10} \] The right algebra obtained by dualizing \(\mathcal A_l\) is exactly \(\mathcal A_r\), with the same products, involution and right operators. **Proof.** Choose the positive contractions \(e_i\in R(\mathcal A_r)\) of RD-05, with \(e_i\to I\) strongly. Because each \(e_i\) is self-adjoint, it has the form \(R_{\alpha_i}^*\) with \(\alpha_i\in\mathcal A_r\). For \(\eta\in\mathcal B_r\), HA-08 gives \[ \eta_i=e_i\eta\in\mathcal A_r,\qquad R_{\eta_i}=e_iR_\eta. \] Thus \(\eta_i\to\eta\) in norm and \(R_{\eta_i}\to R_\eta\) strongly. For \(\xi\in\mathcal B_l\), (WH.4) gives \(\lambda_\xi\eta_i=R_{\eta_i}\xi\); the limits prove (WH.10). The closed involution of \(\mathcal A_l\) is \(S\), so the adjoint-domain condition for its right algebra is membership in \(D(F)\). If \(\eta\) belongs to that right algebra, restricting its right-boundedness inequality to \(\mathcal A\subseteq\mathcal A_l\) puts \(\eta\) in \(\mathcal B_r\). Therefore \(\eta\in\mathcal A_r\). Conversely, if \(\eta\in\mathcal A_r\), then for every \(\xi\in\mathcal A_l\), \[ \|\lambda_\xi\eta\|=\|R_\eta\xi\| \leq\|R_\eta\|\|\xi\|. \] Hence it is right bounded for \(\mathcal A_l\); it is already in \(D(F)\), and its right operator is \(R_\eta\). The involution and products consequently agree. \(\square\) We call a left Hilbert algebra **full** when it equals the left algebra obtained by this two-step dualization. Since the right algebra of \(\mathcal A_l\) is \(\mathcal A_r\), its next left dual is again \(\mathcal A_l\). Thus \(\mathcal A_l\) is full. The original \(\mathcal A\) is a graph core in this completion, because both closed involutions are \(S\); equality need not hold. ## OA-MOD-WH-05 — Measuring square roots by vectors Keep an arbitrary \(\mathcal A\) and the completed multiplication spaces of WH-03. For \(a\in M_+\), define \[ \psi(a)= \begin{cases} \|\xi\|^2,&a^{1/2}=\lambda_\xi\text{ for some }\xi\in\mathcal B_l,\\ +\infty,&a^{1/2}\notin\mathfrak n_l. \end{cases} \tag{WH.11} \] Injectivity of \(\lambda\) makes the finite value unambiguous. In the finite case, \(\lambda_\xi\) is self-adjoint, so (WH.9) implies \(\xi\in\mathcal A_l\) and \(S\xi=\xi\). **Finite-cone theorem.** The set \[ P=\{a\in M_+:a^{1/2}\in\mathfrak n_l\} \tag{WH.12} \] is additive, positively homogeneous and hereditary. Its finite value in (WH.11) is additive and positively homogeneous, and is increasing for the positive order. **Proof of heredity.** If \(0\leq b\leq a\in P\), Douglas factorization in \(M\) gives \(c\in M\), \(\|c\|\leq1\), with \(b^{1/2}=ca^{1/2}\). If \(a^{1/2}=\lambda_\xi\), covariance gives \(b^{1/2}=\lambda_{c\xi}\). Thus \(b\in P\) and \[ \psi(b)=\|c\xi\|^2\leq\|\xi\|^2=\psi(a). \tag{WH.13} \] **Proof of additivity.** Suppose \(a,b\in P\), with square-root vectors \(\xi,\eta\). Write \(t=(a+b)^{1/2}\). The polar decomposition of the column operator \[ \binom{a^{1/2}}{b^{1/2}}=\binom{u}{v}t \tag{WH.14} \] lies in the rectangular matrix algebra over \(M\). Its initial projection is \(p=s(t)\), so \[ u^*u+v^*v=p,\qquad up=u,\quad vp=v,\qquad a^{1/2}=ut,\quad b^{1/2}=vt. \] In particular \(u^*a^{1/2}+v^*b^{1/2}=pt=t\). Define \(\zeta=u^*\xi+v^*\eta\). Covariance and linearity yield \[ \lambda_\zeta=t. \] Thus \(a+b\in P\). Applying \(\lambda\) to \(u\zeta\), \(v\zeta\) and \(p\zeta\), then using injectivity, gives \[ u\zeta=\xi,\qquad v\zeta=\eta,\qquad p\zeta=\zeta. \] Consequently \[ \psi(a)+\psi(b)=\|u\zeta\|^2+\|v\zeta\|^2 =\langle p\zeta,\zeta\rangle=\|\zeta\|^2 =\psi(a+b). \tag{WH.15} \] This norm identity uses the polar support; a generic pair of contractions would not suffice. For \(r>0\), the square-root vector of \(ra\) is \(\sqrt r\,\xi\). This proves homogeneity and preservation of the finite cone under nonzero rescaling. The zero operator has the unique vector zero. Thus \(\psi(0)=0\), and homogeneity at zero uses \(0\cdot\infty=0\). \(\square\) If a positive sum belongs to \(P\), heredity puts each summand in \(P\). Therefore, if either summand has infinite value, the sum does also. The finite and infinite cases together prove that (WH.11) is a weight. ## OA-MOD-WH-06 — The exact finite ideals and GNS norm **Theorem.** The weight in (WH.11) satisfies \[ \mathfrak n_\psi=\mathfrak n_l,\qquad \psi(x^*x)=\|\xi\|^2\quad\text{when }x=\lambda_\xi. \tag{WH.16} \] Its definition algebra is \[ \mathfrak m_\psi=\operatorname{span}\mathfrak n_l^*\mathfrak n_l, \qquad (\mathfrak m_\psi)_+=P. \tag{WH.17} \] For \(x=\lambda_\xi\), \(y=\lambda_\eta\), \[ \widetilde\psi(y^*x)=\langle\xi,\eta\rangle. \tag{WH.18} \] **Proof.** Let \(x=\lambda_\xi\in\mathfrak n_l\), and write its bounded polar decomposition \(x=w|x|\). Covariance gives \[ |x|=w^*x=\lambda_{w^*\xi}. \] Thus \(x^*x\in P\), so \(x\in\mathfrak n_\psi\). The final support \(q=ww^*\) satisfies \(qx=x\), and injectivity gives \(q\xi=\xi\). Hence \[ \psi(x^*x)=\|w^*\xi\|^2=\langle q\xi,\xi\rangle=\|\xi\|^2. \] Conversely, if \(\psi(x^*x)<\infty\), then \(|x|=\lambda_\zeta\) for some \(\zeta\). The same polar decomposition gives \(x=\lambda_{w\zeta}\), proving (WH.16). The algebra and positive-cone identities in (WH.17) are now precisely WG-003 applied to the weight already proved in WH-05. Polarization of (WH.16), using linearity of \(\lambda\), gives (WH.18). All evaluations of \(\widetilde\psi\) lie in its finite algebra; no complex extension of \(\psi\) to arbitrary elements of \(M\) is used. \(\square\) These statements also prove the hereditary-cone and ideal assertions of source Lemma VII.2.1. Applying the same construction to the opposite right algebra gives the corresponding right-ideal assertions. ## OA-MOD-WH-07 — Arbitrary-net normality and semifiniteness **Theorem.** The weight \(\psi\) is faithful, normal and semifinite. **Proof of normality.** Suppose \(0\leq a_i\uparrow a\) is a bounded increasing net in \(M\). Set \(L=\sup_i\psi(a_i)\). Monotonicity already gives \(L\leq\psi(a)\); if \(L=\infty\), equality follows. Assume \(L<\infty\). Then \[ a_i^{1/2}=\lambda_{\xi_i},\qquad \|\xi_i\|^2=\psi(a_i)\leq L. \] The closed Hilbert ball of radius \(\sqrt L\) is weakly compact. Therefore this net has a subnet \(\xi_{i(j)}\) converging weakly to some \(\xi\) in the same ball. For each \(\eta\in\mathcal A_r\), \[ a_{i(j)}^{1/2}\eta=R_\eta\xi_{i(j)}. \tag{WH.19} \] The left side converges in norm to \(a^{1/2}\eta\), by WH-01. The right side converges weakly to \(R_\eta\xi\), since \(R_\eta\) is bounded. Thus \[ R_\eta\xi=a^{1/2}\eta\qquad(\eta\in\mathcal A_r). \] This is the boundedness criterion (WH.3), with constant \(\|a^{1/2}\|\). Hence \(\xi\in\mathcal B_l\), \(\lambda_\xi=a^{1/2}\), and \(\psi(a)=\|\xi\|^2\leq L\). Combining with the opposite inequality proves normality. Compactness was used for the actual arbitrary net; no separability or countable exhaustion was inserted. **Proof of faithfulness.** If \(\psi(a)=0\), its square-root vector is zero, so \(a^{1/2}=\lambda_0=0\). Thus \(a=0\). **Proof of semifiniteness.** Let \(\mathcal C=L(\mathcal A)\). It is a nondegenerate *-algebra by HA-02, and lies in \(\mathfrak n_\psi\) by WH-03 and WH-06. For a finite subset \(E\subseteq\mathcal C\) and \(\varepsilon>0\), put \[ b_E=\sum_{r\in E}r^*r,\qquad c_{E,\varepsilon}=b_E(b_E+\varepsilon I)^{-1}. \tag{WH.20} \] Finite additivity makes \(\psi(b_E)<\infty\). Since \(0\leq c_{E,\varepsilon}\leq\varepsilon^{-1}b_E\), these are finite-weight positive contractions. Order the pairs by enlarging \(E\) and decreasing \(\varepsilon\). Inverse order proves that \(c_{E,\varepsilon}\) increases. For fixed \(E\), its limit as \(\varepsilon\downarrow0\) is \(s(b_E)\). The common kernel of the \(b_E\)'s is the common kernel of \(\mathcal C\), which is zero by nondegeneracy and *-closure. Their support ranges therefore span \(H\). The strong supremum \(c\leq I\) of the contraction net dominates every \(s(b_E)\), so it acts as the identity on those ranges and hence equals \(I\). We have obtained an increasing finite positive contraction net with supremum \(I\). WG-008 proves semifiniteness. \(\square\) Normality was proved before invoking any normal-weight characterization for \(\psi\). In particular, the proof does not assume the source's predual-valued map or supremum formula in order to prove that \(\psi\) is a weight. ## OA-MOD-WH-08 — Recovering the representation and the full algebra Let \((H_\psi,\pi_\psi,\Lambda_\psi)\) be the weight's GNS triple. Define on its dense range \[ U\Lambda_\psi(\lambda_\xi)=\xi\qquad(\xi\in\mathcal B_l). \tag{WH.21} \] **Theorem.** The map \(U\) extends uniquely to a unitary \(H_\psi\to H\), and \[ U\pi_\psi(x)U^*=x\quad(x\in M). \tag{WH.22} \] It identifies the finite-star algebra exactly: \[ U\Lambda_\psi(\mathfrak n_\psi\cap\mathfrak n_\psi^*) =\mathcal A_l. \tag{WH.23} \] Under this identification its product is \(\lambda_\xi\eta\), its involution is \(S\), and its closed involution has domain \(D(S)\). **Proof.** Equation (WH.18) makes (WH.21) a well-defined isometry. Its range is \(\mathcal B_l\), which contains the dense space \(\mathcal A\). Thus it extends to a surjective isometry. For \(x\in M\), covariance gives \[ U\pi_\psi(x)\Lambda_\psi(\lambda_\xi) =U\Lambda_\psi(x\lambda_\xi)=x\xi, \] proving (WH.22) by density. By WH-06 and (WH.9), \(\mathfrak n_\psi\cap\mathfrak n_\psi^*=\lambda(\mathcal A_l)\). This proves (WH.23). If \(\xi,\eta\in\mathcal A_l\), then \(\lambda_\xi\lambda_\eta=\lambda_{\lambda_\xi\eta}\), by covariance; hence the product transports as stated. The equality \(\lambda_\xi^*=\lambda_{S\xi}\) transports the involution. Its closure is \(S\), by WH-03. \(\square\) The construction recovers the full completion \(\mathcal A_l\). It recovers the original algebra exactly when that algebra was full. A proper algebra core and its full completion can therefore determine the same weight. ## OA-MOD-WH-09 — From a faithful normal semifinite weight to an algebra Now start with a faithful normal semifinite weight \(\varphi\) on \(M\). Write its GNS triple as \((H,\pi,\Lambda)\), and put \[ \mathfrak a_\varphi=\mathfrak n_\varphi\cap\mathfrak n_\varphi^*, \qquad \mathcal A_\varphi=\Lambda(\mathfrak a_\varphi). \tag{WH.24} \] Faithfulness makes \(\Lambda\) injective. Consequently \[ \Lambda(x)\Lambda(y)=\Lambda(xy),\qquad \Lambda(x)^\sharp=\Lambda(x^*) \quad(x,y\in\mathfrak a_\varphi) \tag{WH.25} \] are well-defined algebra operations. The domain \(\mathfrak a_\varphi\) is a *-algebra: it is closed under linear combinations and adjoints, and \(xy\in\mathfrak n_\varphi\), \((xy)^*=y^*x^*\in\mathfrak n_\varphi\), by the left-ideal property. **Proposition.** The algebra \(\mathcal A_\varphi\) is dense in \(H\), has bounded left multiplication \[ L_{\Lambda(x)}=\pi(x), \tag{WH.26} \] satisfies the left adjoint identity, and has dense product span. Its generated von Neumann algebra is \(\pi(M)\). **Proof.** Density is WG-009. The GNS module identity gives (WH.26) on the dense domain and the bound \(\|L_{\Lambda(x)}\|\leq\|x\|\). The adjoint identity follows from \(\pi(x)^*=\pi(x^*)\). Choose the finite positive contractions \(e_i\uparrow1\) of WG-008. They belong to \(\mathfrak a_\varphi\), since \(e_i^2\leq e_i\). Normality gives \(\pi(e_i)\to I\) strongly. For \(x\in\mathfrak a_\varphi\), \[ \Lambda(e_i)\Lambda(x)=\pi(e_i)\Lambda(x)\longrightarrow\Lambda(x). \] Thus products are dense. WG-007 and WG-010 make \(\pi\) normal and faithful. WH-02 makes its image a von Neumann algebra. For any \(a\in M\), \(e_i a e_i\in\mathfrak m_\varphi\subseteq\mathfrak a_\varphi\) and \(e_i a e_i\to a\) sigma-strong*, by bounded strong* convergence. NP-06 implies \(\pi(e_i a e_i)\to\pi(a)\) strongly. Thus \(\pi(\mathfrak a_\varphi)\) generates \(\pi(M)\), proving the final assertion. \(\square\) Only closability remains among the four Hilbert algebra axioms. It requires more than the normality of \(\pi\). ## OA-MOD-WH-10 — Closability on the full finite-star domain Let \[ \Phi_\varphi=\{\omega\in M_*^+:\omega\leq\varphi\}. \] For each \(\omega\in\Phi_\varphi\), OW-02–03 give \[ 0\leq h_\omega\leq I,\quad h_\omega\in\pi(M)',\quad h_\omega^{1/2}\Lambda(x)=\pi(x)\eta_\omega \quad(x\in\mathfrak n_\varphi), \tag{WH.27} \] and \[ \omega(a)=\langle\pi(a)\eta_\omega,\eta_\omega\rangle \quad(a\in M). \tag{WH.28} \] **Two totality facts.** One has \[ \sup_{\omega\in\Phi_\varphi}\|h_\omega^{1/2}\xi\|=\|\xi\| \quad(\xi\in H), \tag{WH.29} \] and \[ \overline{\operatorname{span} \{b\eta_\omega:b\in\pi(M)',\ \omega\in\Phi_\varphi\}}=H. \tag{WH.30} \] **Proof.** On \(\xi=\Lambda(x)\), NW-11 and (WH.27) give \[ \|\Lambda(x)\|^2=\varphi(x^*x) =\sup_\omega\omega(x^*x) =\sup_\omega\|h_\omega^{1/2}\Lambda(x)\|^2. \] The supremum in (WH.29) is a 1-Lipschitz function of \(\xi\), since every operator involved is a contraction. Density of the GNS range extends the equality to every vector. In particular, the \(h_\omega^{1/2}\)'s have common kernel zero. Let \(K\) be the closed subspace in (WH.30). It reduces \(\pi(M)'\); its projection \(P\) lies in \(\pi(M)''=\pi(M)\). By WH-02 there is a projection \(p\in M\) with \(P=\pi(p)\). Each \(\eta_\omega\) belongs to \(K\), so \[ \omega(1-p)=\|(I-P)\eta_\omega\|^2=0. \] NW-11 gives \(\varphi(1-p)=0\). Faithfulness implies \(p=1\), hence \(K=H\). \(\square\) **Closability theorem.** The conjugate-linear operator \[ s_0:\mathcal A_\varphi\to H,\qquad s_0\Lambda(x)=\Lambda(x^*) \tag{WH.31} \] is closable. Therefore \(\mathcal A_\varphi\) is a left Hilbert algebra. **Proof.** For \(x\in\mathfrak a_\varphi\), \(b\in\pi(M)'\), and \(\omega,\rho\in\Phi_\varphi\), equations (WH.27–28) give \[ \begin{aligned} \langle\Lambda(x^*),h_\omega^{1/2}b\eta_\rho\rangle &=\langle\pi(x^*)\eta_\omega,b\eta_\rho\rangle\\ &=\langle\eta_\omega,b\pi(x)\eta_\rho\rangle\\ &=\langle h_\rho^{1/2}b^*\eta_\omega,\Lambda(x)\rangle. \end{aligned} \tag{WH.32} \] All operators in this calculation are bounded; the two appearances of \(\Lambda\) are permitted because both \(x\) and \(x^*\) are in \(\mathfrak n_\varphi\). Suppose \(\Lambda(x_n)\to0\) and \(\Lambda(x_n^*)\to\zeta\) in Hilbert norm. Equation (WH.32) implies \[ \langle\zeta,h_\omega^{1/2}b\eta_\rho\rangle=0 \quad\text{for every }\omega,\rho,b. \] Fix \(\omega\). Totality (WH.30) implies \(h_\omega^{1/2}\zeta=0\). The common-kernel consequence of (WH.29) then gives \(\zeta=0\). This is exactly the graph criterion for closability of a conjugate-linear operator on a Hilbert space. It uses sequences only to test closure in the metrizable Hilbert graph norm; it does not assume a separable Hilbert space. WH-09 supplies the other three axioms. \(\square\) Write \(S_\varphi=\overline{s_0}\). Its domain is, exactly, the set of norm limits of \(\Lambda(x_n)\) for which \(\Lambda(x_n^*)\) also converges, and the latter limit is \(S_\varphi\) of the former. HA-04 and TC then give the closed involution and its polar data with their stated spectral prerequisites. The fundamental modular theorem remains separate. ## OA-MOD-WH-11 — Fullness and recovery of the original weight Apply WH-03 to \(\mathcal A_\varphi\), identifying the generated algebra with \(\pi(M)\). Use its spaces \(\mathcal B_l,\mathcal B_r,\mathcal A_r\) and its map \(\lambda\). **Theorem.** One has the exact bijection \[ \mathcal B_l=\Lambda(\mathfrak n_\varphi),\qquad \lambda_{\Lambda(x)}=\pi(x)\quad(x\in\mathfrak n_\varphi). \tag{WH.33} \] Moreover \(\mathcal A_\varphi\) is full, and the reconstructed weight satisfies \[ \psi(\pi(a))=\varphi(a)\qquad(a\in M_+), \tag{WH.34} \] including infinite values. **Proof.** Equation (WH.27) implies that \(\eta_\omega\) is right bounded for \(\mathcal A_\varphi\), with \(R_{\eta_\omega}=h_\omega^{1/2}\). This operator is self-adjoint. RD-07, applied to the pair \(\eta_\omega,\eta_\omega\), gives \[ \eta_\omega\in\mathcal A_r,\qquad F\eta_\omega=\eta_\omega. \tag{WH.35} \] Take \(\xi\in\mathcal B_l\). Since \(\lambda_\xi\in\pi(M)\), there is a unique \(x\in M\) with \(\lambda_\xi=\pi(x)\). Using NW-11, (WH.4), (WH.29) and (WH.35), \[ \begin{aligned} \varphi(x^*x) &=\sup_{\omega\in\Phi_\varphi}\|\pi(x)\eta_\omega\|^2\\ &=\sup_{\omega\in\Phi_\varphi}\|R_{\eta_\omega}\xi\|^2 =\|\xi\|^2<\infty. \end{aligned} \tag{WH.36} \] Thus \(x\in\mathfrak n_\varphi\). For each \(\omega\), (WH.27) and (WH.4) give \[ h_\omega^{1/2}\Lambda(x) =\pi(x)\eta_\omega =h_\omega^{1/2}\xi. \] Their common kernel is zero, so \(\Lambda(x)=\xi\). Conversely, let \(x\in\mathfrak n_\varphi\) and write \(x=u|x|\). The self-adjoint element \(|x|\) lies in \(\mathfrak a_\varphi\), since \(\varphi(|x|^2)=\varphi(x^*x)<\infty\). Therefore \(\Lambda(|x|)\in\mathcal A_\varphi\subseteq\mathcal B_l\). Covariance under \(\pi(u)\) gives \[ \Lambda(x)=\pi(u)\Lambda(|x|)\in\mathcal B_l,\qquad \lambda_{\Lambda(x)}=\pi(u)\pi(|x|)=\pi(x). \] This proves (WH.33). The ideal intersection (WH.9) now reads \[ \lambda(\mathcal A_l) =\pi(\mathfrak n_\varphi)\cap\pi(\mathfrak n_\varphi)^* =\pi(\mathfrak a_\varphi) =\lambda(\mathcal A_\varphi). \] Injectivity gives \(\mathcal A_l=\mathcal A_\varphi\), proving fullness. Finally, \(\psi(\pi(a))<\infty\) if and only if \(\pi(a^{1/2})\in\lambda(\mathcal B_l)=\pi(\mathfrak n_\varphi)\). Faithfulness of \(\pi\) makes this equivalent to \(a^{1/2}\in\mathfrak n_\varphi\), or \(\varphi(a)<\infty\). In that case the square-root vector is \(\Lambda(a^{1/2})\), whose squared norm is \(\varphi(a)\). Otherwise both values are infinite. \(\square\) Thus the passage from a full left Hilbert algebra to its weight and back is inverse, up to the specified canonical unitary, to the passage from a faithful normal semifinite weight to its finite-star algebra. ## OA-MOD-WH-12 — The right weight and the variational formula Return to an arbitrary left Hilbert algebra \(\mathcal A\), with full completion \(\mathcal A_l\), right algebra \(\mathcal A_r\), and reconstructed weight \(\psi\) on \(M\). The symmetric construction on \(\mathcal A_r^{\mathrm{op}}\) defines a faithful normal semifinite weight \(\chi\) on \(M'\): \[ \chi(b)= \begin{cases} \|\eta\|^2,&b^{1/2}=R_\eta,\ \eta\in\mathcal B_r,\\ +\infty,&\text{otherwise}, \end{cases} \qquad b\in(M')_+. \tag{WH.37} \] Its finite ideal is \(\mathfrak n_r=R(\mathcal B_r)\), and \[ \widetilde\chi(R_\eta^*R_\zeta)=\langle\zeta,\eta\rangle. \tag{WH.38} \] These are WH-05–08 applied to the opposite right algebra. That application has the same right-bounded space on the other side: \(\mathcal A\subseteq\mathcal A_l\), while (WH.10) proves boundedness on \(\mathcal A_l\) for every original \(\eta\in\mathcal B_r\); restriction gives the converse. **Theorem.** Under the GNS unitary of WH-08, \(\chi\) is the opposite weight \(\psi^{\mathrm{opp}}\) of OW. Furthermore, for every \(a\in M_+\), \[ \psi(a)= \sup_{\substack{\eta\in\mathcal A_r\\\|R_\eta\|\leq1}} \langle a\eta,\eta\rangle = \sup_{\substack{\eta\in\mathcal A_r\\\|R_\eta\|<1}} \langle a\eta,\eta\rangle. \tag{WH.39} \] The analogous formula for \(\chi\) uses \(\xi\in\mathcal A_l\) with \(\|\lambda_\xi\|\leq1\) or \(<1\). **Proof of the opposite identification.** Identify the GNS space of \(\psi\) with \(H\), so \(\Lambda_\psi(\lambda_\xi)=\xi\) and \(\pi_\psi(x)=x\). For \(\eta\in\mathcal B_r\), the vector functional \(\omega_\eta(x)=\langle x\eta,\eta\rangle\) is normal and positive. If \(x=\lambda_\xi\in\mathfrak n_\psi\), then (WH.10) gives \[ \omega_\eta(x^*x)=\|R_\eta\xi\|^2 \leq\|R_\eta\|^2\psi(x^*x). \tag{WH.40} \] Testing finite positive elements with their square roots, and using any strictly larger positive constant for infinite values if necessary, proves domination by a finite multiple of \(\psi\). Its comparison operator is \[ h_{\omega_\eta}=R_\eta^*R_\eta, \tag{WH.41} \] by polarization on all \(\xi=\Lambda_\psi(x)\). If \(\chi(b)<\infty\), take \(b^{1/2}=R_\eta\). Equation (WH.41) gives \(h_{\omega_\eta}=b\), and \[ \psi^{\mathrm{opp}}(b)=\|\omega_\eta\|=\|\eta\|^2=\chi(b). \] Conversely, suppose \(b=h_\omega\) for a normal positive functional dominated by a finite multiple of \(\psi\). OW-03 supplies \(\eta_\omega\in H\) with \[ b^{1/2}\Lambda_\psi(x)=x\eta_\omega \quad(x\in\mathfrak n_\psi),\qquad \|\eta_\omega\|^2=\|\omega\|. \] Taking \(x=L_a\) with \(a\in\mathcal A\) gives \(b^{1/2}a=L_a\eta_\omega\). Thus \(\eta_\omega\in\mathcal B_r\), \(R_{\eta_\omega}=b^{1/2}\), and \(\chi(b)=\|\omega\|=\psi^{\mathrm{opp}}(b)\). The finite cones and their values agree, so the infinite values agree as well. **Proof of the variational formula.** Equation (WH.40) shows that every \(\omega_\eta\) with \(\|R_\eta\|\leq1\) is at most \(\psi\) on \(M_+\). Conversely, if \(0\leq\omega\leq\psi\), OW-02 gives \(h_\omega\leq I\), and the implementing vector in the preceding paragraph has \(R_{\eta_\omega}=h_\omega^{1/2}\). Because this operator is self-adjoint, RD-07 gives \(\eta_\omega\in\mathcal A_r\). Thus every dominated normal positive functional is one of the displayed vector functionals with right norm at most one. NW-11, applied to the already proved normal weight \(\psi\), gives the first equality in (WH.39). For \(0 ## OA-MOD-WH-13 — Every von Neumann algebra has such a weight **Theorem.** Every von Neumann algebra admits a faithful normal semifinite weight. No sigma-finiteness assumption is required. **Proof.** In a faithful concrete representation, normal positive functionals separate nonzero positive elements: if \(a\ne0\), choose \(\xi\) with \(\langle a\xi,\xi\rangle>0\). The vector functional is normal. For a nonzero normal positive functional \(\omega\), denote by \(s(\omega)\) its support projection, using WS-04–06 in the bounded case. In particular, \[ \omega(x^*x)=0\quad\Longleftrightarrow\quad xs(\omega)=0. \tag{WH.44} \] Choose, by Zorn's lemma, a maximal family of normal states \((\omega_i)_{i\in I}\) whose nonzero supports \(p_i=s(\omega_i)\) are pairwise orthogonal. For a chain of such families, its union is again such a family, so the maximality argument applies. The strong sum \(p=\sum_i p_i\) equals \(1\). Otherwise choose a unit vector in \((1-p)H\). Its vector state has a nonzero support below \(1-p\), since it vanishes on \(p\), contradicting maximality. Define \[ \varphi(a)=\sum_{i\in I}\omega_i(a) =\sup_{F\subseteq I\text{ finite}}\sum_{i\in F}\omega_i(a), \qquad a\in M_+. \tag{WH.45} \] Suprema of finite nonnegative subsums commute with addition and nonnegative scalar multiplication; for addition, combine the two finite sets used to approximate the two separate suprema. Thus (WH.45) is a weight. If \(a_j\uparrow a\), normality of each finite sum and commutation of the two suprema give \[ \varphi(a)=\sup_F\sup_j\sum_{i\in F}\omega_i(a_j) =\sup_j\varphi(a_j). \] This proves arbitrary-net normality. If \(\varphi(a)=0\), then every \(\omega_i(a)=0\). Equation (WH.44) gives \(a^{1/2}p_i=0\) for every \(i\). Their finite sums converge strongly to \(1\), so \(a^{1/2}=0\); the weight is faithful. For a finite \(F\subseteq I\), let \(p_F=\sum_{i\in F}p_i\). Supportedness and orthogonality give \(\omega_i(p_F)=1\) for \(i\in F\) and zero otherwise. Hence \(\varphi(p_F)=|F|<\infty\). The projections \(p_F\uparrow1\), so WG-008 proves semifiniteness. For the zero algebra, the zero weight has all three properties and the empty family suffices. \(\square\) The projections in this proof form a specially chosen orthogonal family. It does not assert that the set of all finite projections of an arbitrary weight is directed.