Original text: CC0 1.0. Prerequisite proofs and component terms.

Constructing spatial energy from finite observations

OA-MOD-SC-01. Conventions and the actual prerequisites

Fix a concrete unital von Neumann algebra $M\subseteq B(H)$, put $N=M'$, and let $\psi$ be a normal semifinite faithful weight on $N$. Both $H$ and the algebras are arbitrary. Inner products are linear in the first variable. Write $$(H_\psi,\pi_\psi,\Lambda_\psi),\qquad \mathfrak n_\psi=\{x\in N:\psi(x^*x)<\infty\}$$ for the semicyclic construction. We use the direct commutant convention: the denominator is a weight on $M'$, not on an unnamed opposite algebra.

For $\xi\in H$, say $\xi\in D_\psi$ when $$\|x\xi\|\le C_\xi\|\Lambda_\psi(x)\| \qquad(x\in\mathfrak n_\psi) \tag{SC.1}$$ for some finite constant. Then $R(\xi):H_\psi\to H$ is the bounded extension of $\Lambda_\psi(x)\mapsto x\xi$, and $$\theta(\xi,\eta)=R(\xi)R(\eta)^*\in M. \tag{SC.2}$$ The extension, the intertwining identity, and $M$-covariance are proved in OA-MOD-SD-02. In particular $D_\psi$ is linear and invariant under $M$, and $R(a\xi)=aR(\xi)$. The coefficient ideal and its positive cone are proved in SD-04; the finite energy polarization is SD-05.

Here are the further exact inputs.

• CP-06–08: the concrete Banach predual; its identification with ultraweakly continuous functionals; the positive norm formula; and domination of a positive normal functional by a square-summable vector sum. The exact positive vector-sum representation needed here is proved in SC-02 below.
• NW-11: every normal weight $\chi$ satisfies $\chi(a)=\sup_{\omega\in N_*^+,\,\omega\le\chi}\omega(a)$ on its whole positive cone. Its proof uses the convex-analysis foundations stated in NW-01.
• WG-003–006 and WG-008: finite-domain algebra, GNS construction, and finite positive contraction nets for semifinite weights. WS-02–06 provide the finite-domain projection $e_\varphi$, the largest null projection $f_\varphi\le e_\varphi$, and their exact corner conventions for a normal numerator $\varphi$.
• BK-02–06: bicommutants, bounded monotone nets, bounded strong-to-ultraweak convergence, continuous functional calculus, inverse order and support cutoffs. The positive-operator square root, bounded-form representation, completion and orthogonal projection inputs are the course's stated Hilbert and bounded-calculus prerequisites.
• FC-02–05: the completion test for closability, relative lower semicontinuity and exact form cores. QF-03 represents a closed positive form on the closure of its domain. The required arbitrary-dimensional spectral calculus is supplied by SK-04–09, including exact integral domains, square roots and spectral transport. These proofs use their stated scalar, Hilbert-space and bounded-calculus prerequisites.

No standard form, modular conjugation, relative Tomita map, normal-representation image theorem or weight on a GNS commutant is used in the proof. In particular, the bounded normal-map criterion is not substituted for NW's infinite-valued weight theorem. The arguments below use these specified mathematical prerequisites.

OA-MOD-SC-02. Positive observations can be implemented by vector sums

Lemma. If $A\subseteq B(L)$ is a concrete unital von Neumann algebra and $\omega\in A_*^+$, there are vectors $v_j\in L$, indexed by positive integers, such that $$\sum_j\|v_j\|^2=\omega(1),\qquad \omega(a)=\sum_j\langle av_j,v_j\rangle \quad(a\in A). \tag{SC.3}$$ The series is absolutely convergent. Every component functional $\omega_{v_j}$ is positive normal and satisfies $\omega_{v_j}\le\omega$. No separability hypothesis on $L$ is needed.

Proof. CP-08 gives square-summable vectors $\zeta=(\zeta_j)\in\ell^2(\mathbb N,L)$ such that $$\omega(a)\le\vartheta(a):= \sum_j\langle a\zeta_j,\zeta_j\rangle \qquad(a\ge0).$$ Let $\rho(a)=\operatorname{diag}(a,a,\ldots)$, a bounded *-representation on $\ell^2(\mathbb N,L)$, and put $K=\overline{\rho(A)\zeta}$. The subspace is invariant under $\rho(A)$ and its adjoints, so it reduces this representation. Also $\zeta\in K$, because $A$ is unital.

On the dense subspace $\rho(A)\zeta\subseteq K$, define $$B(\rho(a)\zeta,\rho(b)\zeta)=\omega(b^*a).$$ The inequality $\omega(a^*a)\le\vartheta(a^*a)=\|\rho(a)\zeta\|^2$, together with the positive-functional Cauchy–Schwarz inequality, proves that this rule is well-defined and $$|B(\rho(a)\zeta,\rho(b)\zeta)| \le\|\rho(a)\zeta\|\,\|\rho(b)\zeta\|.$$ It extends to a positive bounded sesquilinear form on $K$. The bounded-form theorem gives a unique positive contraction $h\in B(K)$ with $B(u,v)=\langle hu,v\rangle$.

For $c\in A$, the equality $$B(\rho(c)\rho(a)\zeta,\rho(b)\zeta) =B(\rho(a)\zeta,\rho(c^*)\rho(b)\zeta)$$ holds because both sides equal $\omega(b^*ca)$. Extension from the dense cyclic subspace shows that $h$ commutes with $\rho(c)|_K$. Its square root commutes as well. Set $v=h^{1/2}\zeta\in K$, and write its components as $v_j\in L$. For every $a\in A$, $$\omega(a)=B(\rho(a)\zeta,\zeta) =\langle\rho(a)v,v\rangle =\sum_j\langle av_j,v_j\rangle.$$ Evaluation at $1$ gives the asserted sum of squared norms. Absolute convergence follows from $|\langle av_j,v_j\rangle|\le\|a\|\|v_j\|^2$. For positive $a$, each nonnegative summand is at most the sum, proving $\omega_{v_j}\le\omega$. Single-vector functionals are ultraweakly continuous by CP. If $\omega=0$, the construction may be replaced by the zero sequence. ∎

The use of a countable series concerns one predual functional. The family of all normal functionals can be arbitrarily large. We have not chosen a countable family which detects the entire algebra.

OA-MOD-SC-03. The exact closure of bounded vectors

For this result alone let $\chi$ be any normal weight on a concrete unital von Neumann algebra $A\subseteq B(L)$. Do not assume semifiniteness or faithfulness. Define $D_\chi(L)$ by the estimate (SC.1), using $\chi$, its finite left ideal, and the original action on $L$. Let $f_\chi$ be the largest $\chi$-null projection from WS-04.

Theorem. $$\overline{D_\chi(L)}^{\,\|\cdot\|} =(1-f_\chi)L. \tag{SC.4}$$ In particular, a faithful normal weight has dense bounded vectors. For a normal semifinite weight, $1-f_\chi$ is its support in the WS-06 convention.

Proof. The bounded-vector space is linear. If $b\in A'$ and $\xi$ is bounded, then $$\|x b\xi\|=\|b x\xi\| \le\|b\|C_\xi\chi(x^*x)^{1/2},$$ so $b\xi$ is bounded. The closed subspace $\overline{D_\chi(L)}$ therefore reduces $A'$, and its projection $P$ lies in $A''=A$.

Since $\chi(f_\chi)=0$, the projection $f_\chi$ lies in $\mathfrak n_\chi$. Testing the bounded-vector estimate with $x=f_\chi$ gives $f_\chi\xi=0$. Consequently $P\le1-f_\chi$.

For the reverse inequality, let $\omega\in A_*^+$ satisfy $\omega\le\chi$, and implement it by the vector sum in SC-02. For every $x\in\mathfrak n_\chi$, $$\|xv_j\|^2=\omega_{v_j}(x^*x) \le\omega(x^*x)\le\chi(x^*x).$$ Thus each $v_j\in D_\chi(L)$, with constant one. Hence $$\omega(1-P)=\sum_j\|(1-P)v_j\|^2=0.$$ NW-11, applied at the positive element $1-P$, gives $\chi(1-P)=0$. Maximality of $f_\chi$ implies $1-P\le f_\chi$, or $1-f_\chi\le P$. Combining the inequalities proves (SC.4). ∎

Faithfulness gives $f_\chi=0$, so the denominator $\psi$ of SC-01 has $D_\psi$ dense in $H$. Semifiniteness is still needed elsewhere: it makes the bounded-vector map $\xi\mapsto R(\xi)$ injective by finite cutoffs. SC-03 alone does not assert that injectivity for an arbitrary normal weight.

OA-MOD-SC-04. A coefficient ideal with an explicit approximate identity

Return to the setting of SC-01, and set $$\mathcal J=\operatorname{span}\{\theta(\xi,\eta):\xi,\eta\in D_\psi\}.$$ By SD-04 this is an algebraic two-sided *-ideal of $M$, and $$\mathcal J_+=\mathcal J\cap M_+ =\left\{\sum_{j=1}^m\theta(\xi_j,\xi_j): \xi_j\in D_\psi,\ m<\infty\right\}. \tag{SC.5}$$

Proposition. The ideal is ultraweakly dense. More precisely, direct $\mathcal J_+$ by the usual positive order and put $$u_a=a(1+a)^{-1}\qquad(a\in\mathcal J_+). \tag{SC.6}$$ Then $u_a\in\mathcal J_+$, $0\le u_a\le1$, and $u_a\uparrow1$ strongly. For any $x\in M_+$, $$x^{1/2}u_a x^{1/2}\in\mathcal J_+, \qquad x^{1/2}u_a x^{1/2}\uparrow x. \tag{SC.7}$$

Proof. Addition makes the index set directed. Functional calculus and the ideal property put $u_a$ in $\mathcal J_+$. Inverse order gives $a\le b\Rightarrow u_a\le u_b$, so there is a positive contraction $u\in M$ which is their strong supremum.

For every fixed $a\in\mathcal J_+$, all $ta$, $t>0$, are indices, and the support-cutoff theorem gives $u_{ta}\uparrow s(a)$. Thus $u\ge s(a)$, and a positive contraction dominating a projection is the identity on its range.

These support ranges span $H$. Indeed, a vector $v$ orthogonal to all of them is annihilated by every $\theta(\xi,\xi)$. Therefore $R(\xi)^*v=0$. Choose the finite positive contraction net $e_i\uparrow1$ for $\psi$. Each $e_i\in\mathfrak n_\psi$, and $$\langle e_i\xi,v\rangle =\langle R(\xi)\Lambda_\psi(e_i),v\rangle=0.$$ Strong convergence gives $v\perp\xi$ for every $\xi\in D_\psi$; SC-03 gives $v=0$. Thus $u$ acts as the identity on a dense subspace, and $u=1$.

The ideal property gives $xu_a\in\mathcal J$ for every $x\in M$. This norm-bounded net converges strongly and ultraweakly to $x$, proving density. Finally, compression of the increasing net by $x^{1/2}$ preserves positivity and order, gives the strong limit in (SC.7), and stays in the ideal. ∎

This proof supplies actual increasing positive approximants. Ultraweak density of a linear space, by itself, would not supply (SC.7) for passing limits through an infinite-valued normal weight.

OA-MOD-SC-05. Finite energy and its Hilbert-space closure

Let $\varphi$ be any normal weight on $M$. Write $e=e_\varphi$ and $f=f_\varphi$ for its finite-domain and null projections, so $f\le e$. Define $$F_\varphi(\xi)=\varphi(\theta(\xi,\xi)) \quad(\xi\in D_\psi),\qquad E_\varphi=\{\xi\in D_\psi:F_\varphi(\xi)<\infty\}. \tag{SC.8}$$ SD-05 proves that $E_\varphi$ is linear and that $$q_\varphi^0(\xi,\eta) =\widetilde\varphi(\theta(\xi,\eta)) \quad(\xi,\eta\in E_\varphi) \tag{SC.9}$$ is a nonnegative sesquilinear form. Here $\widetilde\varphi$ is the finite linear extension on $\mathfrak m_\varphi$; its argument is in that domain by SD-05. The proof there needs no semifiniteness of the numerator, so it applies with the present hypotheses.

Proposition. $$\overline{E_\varphi}^{\,\|\cdot\|}=eH. \tag{SC.10}$$ Consequently this form is densely defined on $H$ exactly when $\varphi$ is semifinite.

Proof. If $\xi\in E_\varphi$, the positive operator $\theta(\xi,\xi)$ has finite weight. WS-02 therefore gives $\theta(\xi,\xi)=e\theta(\xi,\xi)e$. For every $v\in H$, $$\|R(\xi)^*(1-e)v\|^2 =\langle\theta(\xi,\xi)(1-e)v,(1-e)v\rangle=0.$$ Thus $(1-e)R(\xi)=0$. Testing at $\Lambda_\psi(e_i)$, with the finite denominator cutoffs $e_i\uparrow1$, gives $(1-e)e_i\xi=0$. Since $e\in M$ commutes with $e_i\in N$, passage to the strong limit gives $\xi=e\xi$. Hence $E_\varphi\subseteq eH$.

Conversely, let $a\in M_+$ have finite $\varphi$-weight, and let $\xi\in D_\psi$. Covariance and the bound $\theta(\xi,\xi)\le\|R(\xi)\|^2 1$ give $$0\le\theta(a^{1/2}\xi,a^{1/2}\xi) =a^{1/2}\theta(\xi,\xi)a^{1/2} \le\|R(\xi)\|^2a.$$ Its weight is finite, so $a^{1/2}\xi\in E_\varphi$. Density of $D_\psi$ shows that the closure of $E_\varphi$ contains $a^{1/2}H$, and hence its closure $s(a)H$. By WS-02, the ranges of all these finite positive elements span $eH$. This proves the reverse inclusion and (SC.10). Finally $e=1$ is exactly semifiniteness by WS-02. ∎

No invariance of $E_\varphi$ under arbitrary elements of $M$ has been asserted. The displayed estimate uses a finite positive element on the outside of a bounded coefficient; a general weight does not satisfy a tracial conjugation bound.

OA-MOD-SC-06. Lower semicontinuity on bounded vectors

Theorem. The function $F_\varphi:D_\psi\to[0,\infty]$ is lower semicontinuous for the topology inherited from the Hilbert norm of $H$. The form $q_\varphi^0$ on $E_\varphi$ is closable.

Proof. For each $\omega\in M_*^+$ with $\omega\le\varphi$, choose a representation (SC.3). Since $R(\xi)$ is bounded, $$\omega(\theta(\xi,\xi)) =\sum_j\|R(\xi)^*v_j\|^2 \qquad(\xi\in D_\psi). \tag{SC.11}$$ For a fixed vector $v\in H$, the norm of $R(\xi)^*v$ has the variational description $$\|R(\xi)^*v\|^2 =\sup_{x\in\mathfrak n_\psi} \left(2\operatorname{Re}\langle v,x\xi\rangle -\|\Lambda_\psi(x)\|^2\right). \tag{SC.12}$$ Indeed, the expression equals $2\operatorname{Re}\langle R(\xi)^*v,\Lambda_\psi(x)\rangle-\|\Lambda_\psi(x)\|^2$, and $\Lambda_\psi(\mathfrak n_\psi)$ is dense in $H_\psi$. Completing the square gives the supremum of the squared norm.

Each expression inside the supremum in (SC.12) is norm continuous in $\xi$ on all of $H$: $x$ is one fixed bounded operator. Thus this squared norm is lower semicontinuous on $D_\psi$. Finite sums of these nonnegative lower semicontinuous functions are lower semicontinuous. Their increasing supremum over finite initial segments is (SC.11), so that function is also lower semicontinuous.

Finally NW-11 gives, including infinite values, $$F_\varphi(\xi) =\sup_{\omega\in M_*^+,\,\omega\le\varphi} \omega(\theta(\xi,\xi)). \tag{SC.13}$$ A supremum of lower semicontinuous functions is lower semicontinuous. Restricting to $E_\varphi$ proves relative lower semicontinuity of the diagonal of $q_\varphi^0$; FC-04 then proves closability. ∎

There is no assumption that $\|R(\xi_n)\|$ stays bounded when $\xi_n\to\xi$. Formula (SC.12) was used precisely to make every test continuous without such a bound. Also, the normal-functionals supremum need not be a directed or countable supremum.

OA-MOD-SC-07. Completion gives the exact core and energy formula

Let $q_\varphi$ be the canonical closure of $q_\varphi^0$, constructed by completing its form norm as in FC-02. Write $V_\varphi=D(q_\varphi)\subseteq eH$.

Theorem. The form $q_\varphi$ is closed and densely defined on $eH$. Its original domain is exactly $$E_\varphi=D_\psi\cap V_\varphi. \tag{SC.14}$$ It is a form core: every $\xi\in V_\varphi$ is the limit of a sequence $\xi_n\in E_\varphi$ in the norm $(\|\xi\|^2+q_\varphi[\xi])^{1/2}$. On all bounded vectors the exact extended energy identity is $$\varphi(\theta(\xi,\xi))= \begin{cases} q_\varphi[\xi],&\xi\in V_\varphi,\\ +\infty,&\xi\notin V_\varphi, \end{cases} \qquad\xi\in D_\psi. \tag{SC.15}$$ These properties determine the closed form uniquely.

Proof. Closability is SC-06. FC-02 supplies the closed extension and makes $E_\varphi$ a form core by construction. Hilbert limits of its form-Cauchy sequences stay in $eH$ by SC-05, and its domain contains $E_\varphi$, which is dense there. Thus it is densely defined on $eH$.

The inclusion $E_\varphi\subseteq D_\psi\cap V_\varphi$ and equality of its old and new energies follow from extension. Conversely, let $\xi\in D_\psi\cap V_\varphi$, and choose the core sequence $\xi_n\in E_\varphi$. Its energies converge to $q_\varphi[\xi]$. One way to see this is to apply Cauchy–Schwarz for the form to the difference: $|q_\varphi[\xi_n]^{1/2}-q_\varphi[\xi]^{1/2}|\le q_\varphi[\xi_n-\xi]^{1/2}$. Relative lower semicontinuity from SC-06 gives $$\varphi(\theta(\xi,\xi)) \le\liminf_n\varphi(\theta(\xi_n,\xi_n)) =q_\varphi[\xi]<\infty.$$ Therefore $\xi\in E_\varphi$. This proves (SC.14); agreement of the extension then gives equality, rather than merely the preceding inequality. The remaining bounded vectors have infinite original energy by the definition of $E_\varphi$, proving (SC.15).

If another closed form has the same diagonal on $E_\varphi$ and has $E_\varphi$ as a form core, polarization gives the same form there. Both forms are the completion of that same form norm with the same inclusion into $H$, so FC-02 identifies them. ∎

The sequence in this statement approximates one vector in a Hilbert norm. It does not impose a countable exhaustion of $M$ or a separable Hilbert space. The core condition is necessary for the uniqueness statement; agreement on a merely Hilbert-dense subspace is insufficient.

OA-MOD-SC-08. All and only the null directions survive at zero energy

Theorem. $$\{\xi\in V_\varphi:q_\varphi[\xi]=0\}=fH. \tag{SC.16}$$

Proof of the inclusion from the null projection. For $\xi\in D_\psi$, covariance gives $$\theta(f\xi,f\xi)=f\theta(\xi,\xi)f.$$ This positive element has $\varphi$-weight zero by WS-04. Thus $fD_\psi\subseteq E_\varphi$ and its energy vanishes. Since $D_\psi$ is dense, $fD_\psi$ is dense in $fH$. For any $\eta\in fH$, a Hilbert-norm approximating sequence from this space is also form-Cauchy, because all its differences have zero energy. The closure construction puts $\eta$ in $V_\varphi$ with zero energy.

Proof that there are no further null directions. Choose one vector-sum implementation $(v_{\omega,j})_j$ for every $\omega\in M_*^+$ dominated by $\varphi$, and let $$L=\overline{\operatorname{span}} \{y v_{\omega,j}:y\in N,\ j\ge1,\ \omega\le\varphi\}.$$ This space reduces $N$, so its projection $g$ lies in $M$. For every such $\omega$, $$0\le\omega(f)\le\varphi(f)=0, \qquad \omega(f)=\sum_j\|fv_{\omega,j}\|^2.$$ Thus every implementing vector is in $(1-f)H$, and this space is $N$-invariant because $f\in M$. Hence $g\le1-f$. On the other hand, all implementing vectors lie in $gH$, so $\omega(1-g)=0$ for every dominated $\omega$. NW-11 gives $\varphi(1-g)=0$, hence $1-g\le f$. Therefore $$L=(1-f)H. \tag{SC.17}$$

Now let $\xi\in V_\varphi$ have zero energy. Choose $\xi_n\in E_\varphi$ converging to it in form norm. Then $q_\varphi^0[\xi_n]\to0$. For fixed $\omega,j$ and $x\in\mathfrak n_\psi$, (SC.11) gives $$|\langle x\xi_n,v_{\omega,j}\rangle| \le\|\Lambda_\psi(x)\|\, \|R(\xi_n)^*v_{\omega,j}\| \le\|\Lambda_\psi(x)\|\,q_\varphi^0[\xi_n]^{1/2} \longrightarrow0.$$ The fixed bounded operator $x$ preserves Hilbert-norm convergence, so $\langle x\xi,v_{\omega,j}\rangle=0$. For any $y\in N$, the denominator's finite cutoffs satisfy $ye_i\in\mathfrak n_\psi$ and $ye_i\xi\to y\xi$. It follows that $\langle y\xi,v_{\omega,j}\rangle=0$ for all $y\in N$. Replacing $y$ by its adjoint shows that $\xi$ is orthogonal to every generator of $L$. Equation (SC.17) gives $\xi\in fH$, as required. ∎

The argument uses $1-f$ as a null-carrier projection during testing. The effective support of the closed form on its actual Hilbert space $eH$ is $e-f$. Equating these two projections before imposing semifiniteness would lose the infinite-energy directions.

OA-MOD-SC-09. The representing operator and its domains

Apply QF-03 to $q_\varphi$ on $K=eH$. There is a unique nonnegative self-adjoint operator $A_\varphi$ on $K$ such that $$D(A_\varphi^{1/2})=V_\varphi, \qquad q_\varphi(\xi,\eta) =\langle A_\varphi^{1/2}\xi,A_\varphi^{1/2}\eta\rangle. \tag{SC.18}$$ Its operator domain is exactly $$\begin{split} D(A_\varphi)=\{\xi\in V_\varphi:\ &\text{there is }z\in K\text{ with}\\ &q_\varphi(\xi,\eta)=\langle z,\eta\rangle \text{ for every }\eta\in V_\varphi\}, \end{split} \qquad A_\varphi\xi=z. \tag{SC.19}$$ Uniqueness of $z$ follows from density in $K$. The initial finite-energy domain $E_\varphi$ is an operator core for $A_\varphi^{1/2}$, because its graph norm is exactly the form norm. Thus the restriction of $A_\varphi^{1/2}$ to $E_\varphi$ is essentially self-adjoint on $K$.

The kernel of $A_\varphi$ is $fH$, and its support, viewed as a projection on $H$, is $e-f$. To verify the kernel without confusing the two domains, if $q_\varphi[\xi]=0$, form Cauchy–Schwarz gives $q_\varphi(\xi,\eta)=0$ for every $\eta$; (SC.19) then gives $\xi\in D(A_\varphi)$ and $A_\varphi\xi=0$. Conversely, an operator-null vector has zero form by (SC.18), or by pairing (SC.19) with itself. Now apply SC-08. For a self-adjoint operator on $K$, the closure of its range is the orthogonal complement in $K$ of its kernel, which gives $(e-f)H$.

When $\varphi$ is semifinite, $e=1$. We denote this operator on all of $H$ by $$A_\varphi=\frac{d\varphi}{d\psi}. \tag{SC.20}$$ Equations (SC.14)–(SC.19) give its full finite-coefficient construction and exact square-root core. Its support is $1-f=s(\varphi)$, and if $\varphi$ is also faithful, it has zero kernel. These conclusions do not assert that $A_\varphi$ is affiliated with $M$; a spatial derivative generally is not.

When $e\ne1$, retain the pair $(eH,A_\varphi)$, or equivalently the closed form extended by infinity outside $V_\varphi$. Adding the zero operator on $(1-e)H$ would give finite zero energy there and violate (SC.15). We make no such extension. This construction for arbitrary normal numerators is stronger in domain generality than the ordinary semifinite-numerator operator statement, while its relative modular identification remains a separate obligation.

OA-MOD-SC-10. Weight order is exactly closed-form order

For any normal numerator, extend the diagonal of $q_\varphi$ by $+\infty$ off $V_\varphi$. For two such forms, write $q_\varphi\le q_\rho$ when $$V_\rho\subseteq V_\varphi, \qquad q_\varphi[\xi]\le q_\rho[\xi] \quad(\xi\in V_\rho). \tag{SC.21}$$ This is the order of these extended diagonals on $H$.

Theorem. For arbitrary normal weights $\varphi,\rho$ on $M$, with the same denominator $\psi$, $$\varphi\le\rho\quad\Longleftrightarrow\quad q_\varphi\le q_\rho. \tag{SC.22}$$ For semifinite numerators this is the usual form order of their spatial derivatives.

Proof of the forward implication. Pointwise weight order gives $E_\rho\subseteq E_\varphi$ and $q_\varphi^0[\xi]\le q_\rho^0[\xi]$ on $E_\rho$. Let $\xi\in V_\rho$, and approximate it by $\xi_n\in E_\rho$ in the $q_\rho$-norm. The inequality applied to differences shows that this sequence is Cauchy in the closed form norm of $q_\varphi$. Completeness gives a limit in $V_\varphi$, whose Hilbert-space image must be $\xi$. Passing to the limits of the two energies proves $q_\varphi[\xi]\le q_\rho[\xi]$, and also proves the needed domain inclusion.

Proof of the reverse implication. Let $\xi\in D_\psi$. If $\rho(\theta(\xi,\xi))$ is finite, (SC.14) puts $\xi\in V_\rho$. The form-order assumption and (SC.15) give $$\varphi(\theta(\xi,\xi))\le\rho(\theta(\xi,\xi)).$$ If the right side is infinite, this inequality holds automatically. Thus it holds for every bounded vector. The positive-cone identity (SC.5) and finite additivity of weights extend it to every element of $\mathcal J_+$, including infinite values. For $a\in M_+$, use the increasing approximants (SC.7) and normality: $$\begin{aligned} \varphi(a) &=\sup_b\varphi(a^{1/2}u_ba^{1/2})\\ &\le\sup_b\rho(a^{1/2}u_ba^{1/2}) =\rho(a). \end{aligned}$$ This proves the weight inequality on the whole positive cone. ∎

The construction is therefore injective on normal weights: equal closed spatial forms imply equal weights. An energy comparison on a smaller test set would need an additional core or positive approximation argument; that requirement is met here by SC-04 and SC-07.

Editable source · Proof dependencies and component terms