Free-developed EW proof by GPT-6 Astra (OpenAI), Ultra. Self-checked by the writing AI.

Order normality and ultraweak lower semicontinuity for every weight

Fresh reconstruction, GPT-6 Astra (OpenAI), Ultra, 2026-10-04. CC0-1.0 to the extent of rights held.

Let $M\subseteq B(K)$ be a concrete von Neumann algebra on an arbitrary Hilbert space. A weight $\varphi:M_+\to[0,\infty]$ is additive and positively homogeneous, with $0\cdot\infty=0$. We prove, without faithfulness, semifiniteness or a countability assumption, that these conditions are equivalent:

$\varphi$ preserves bounded increasing positive suprema.
Every sublevel $E_r=\{a\in M_+:\varphi(a)\leq r\}$, $0\leq r<\infty$, is ultraweakly closed.
On the whole positive cone, including infinite values, $\varphi(a)=\sup\{f(a):f\in M_*^+,\ f\leq\varphi\}. \tag{EW1}$

Here $f\leq\varphi$ means the inequality on every positive element. No finite-cone-only convention is used.

The exact earlier inputs are GW-1–5, NF-5, CF Sections 1 and 6–10, SF-0 and SB-0–6, FF-4 and FF-5 form invariance, CP01–06, BD7, and CV-1–4. In particular NF-5 was proved using only the bounded-functional normality criterion; it did not import any implication of the present theorem. FF-4 permits arbitrary Hilbert dimension and a nondense form domain. The proof below uses its dense-domain case on a reducing subspace constructed explicitly.

The broader theorem is stated in the freely available Hiai notes, §7.1, Theorem 7.2; that statement is context, not a proof input. The argument here is developed from the listed local bodies: a weighted difference form proves bounded GNS graph closedness, and a rational order cutoff proves positivity of the separating functionals.

Actual earlier proof ranges: OA-FLOW.GW.1, OA-FLOW.GW.2, OA-FLOW.GW.3, OA-FLOW.GW.4, OA-FLOW.GW.5, OA-FLOW.NF.5, OA-FLOW.CF.1, OA-FLOW.CF.6, OA-FLOW.CF.7, OA-FLOW.CF.8, OA-FLOW.CF.9, OA-FLOW.CF.10, OA-FLOW.SF.SF0, OA-FLOW.SF.SB0, OA-FLOW.SF.SB1, OA-FLOW.SF.SB2, OA-FLOW.SF.SB3, OA-FLOW.SF.SB4, OA-FLOW.SF.SB5, OA-FLOW.SF.SB6, OA-FLOW.FF.5, OA-FLOW.FF.6, OA-FLOW.CP.1, OA-FLOW.CP.2, OA-FLOW.CP.3, OA-FLOW.CP.4, OA-FLOW.CP.5, OA-FLOW.CP.6, OA-FLOW.BD.5, OA-FLOW.CV.1, OA-FLOW.CV.2, OA-FLOW.CV.3, OA-FLOW.CV.4.

EW-1. The norm-closed GNS graph

Assume condition 1. Let $N=\{x:\varphi(x^*x)<\infty\}$, and write $(H,\Lambda,\pi)$ for GW's quotient GNS construction. The map $\pi$ is ultraweakly continuous by NF-5, even if the weight is not faithful or semifinite.

Suppose $x_n\in N$, $x_n\to x$ in operator norm, and $\Lambda x_n\to\xi$ in Hilbert norm. Choose a subsequence $x_{n_k}$ whose successive differences $d_k=x_{n_{k+1}}-x_{n_k}$ satisfy $\|d_k\|+\|\Lambda d_k\|\leq4^{-k}.$ The positive series $b=\sum_{k\geq1}2^k d_k^*d_k$ converges in operator norm. Its increasing partial sums have supremum $b$, so order normality gives $\varphi(b)=\sum_{k\geq1}2^k\|\Lambda d_k\|^2<\infty. \tag{EW2}$ For a finite sum, Hilbert-space Cauchy–Schwarz applied after evaluation on each vector proves $\left(\sum_{j=k}^{l}d_j\right)^* \left(\sum_{j=k}^{l}d_j\right) \leq\left(\sum_{j=k}^{l}2^{-j}\right) \left(\sum_{j=k}^{l}2^j d_j^*d_j\right).$ Pass to the norm limit $l\to\infty$. Since $z_k=x-x_{n_k}=\sum_{j\geq k}d_j$, $z_k^*z_k\leq2^{1-k}b,\qquad z_k\in N,\qquad \|\Lambda z_k\|^2\leq2^{1-k}\varphi(b)\longrightarrow0. \tag{EW3}$ Thus $x=x_{n_k}+z_k\in N$ and $\Lambda x=\xi$. The linear graph of $\Lambda$ is norm closed in $M\oplus H$. This argument uses no assertion about continuity of the weight in the operator norm.

EW-2. Bounded adjoint-strong graph closedness

Fix $C<\infty$, and let a net satisfy $x_i\in N,\quad \|x_i\|\leq C,\quad x_i^*v\longrightarrow x^*v\ (v\in K),\quad \Lambda x_i\longrightarrow\xi\text{ in }H. \tag{EW4}$ We prove $x\in N$, $\Lambda x=\xi$. The uniform norm bound in (EW4) is a hypothesis, not a consequence asserted for arbitrary convergent nets.

Fix a finite set $F\subset K$. Let $p$ be the orthogonal projection onto $K_F=\overline{\operatorname{span}}\{y'v:y'\in M',\ v\in F\}.$ This subspace reduces $M'$, so $p\in M$ by the bicommutant/unitary test in SF-0. Choose successive indices $i_n$ so that, writing $z_n=x_{i_n}$, $\|\Lambda z_n-\xi\|\leq2^{-3n},\qquad \|(z_n^*-x^*)v\|\leq2^{-3n}\quad(v\in F). \tag{EW5}$ Directedness permits this selection. We do not assert that the sequence is cofinal in the original net. Because $z_n^*-x^*$ commutes with $M'$ and is uniformly bounded, (EW5) implies $z_n^*v\longrightarrow x^*v\qquad(v\in K_F). \tag{EW6}$

On $K_F$ define $q(v)=\sum_{n\geq1}4^n\|(z_{n+1}-z_n)^*v\|^2. \tag{EW7}$ Its finite domain is linear. It contains $F$, by (EW5), and is invariant under $M'$; indeed every unitary of $M'$ preserves $q$, and every element of $M'$ is a linear combination of its unitaries. Hence the finite domain is dense in $K_F$. The form is closed: if $v_j$ is Cauchy in its graph norm and $v_j\to v$ in $K_F$, each finite partial sum at $v_j-v$ is bounded by the graph-Cauchy tail, after taking the limit in the second index. Supremizing those partial sums gives $q(v_j-v)\to0$; one $v_j$ then shows $v$ has finite form value.

FF-4 gives a positive self-adjoint $T$ on $K_F$, with $q(v)=\|T^{1/2}v\|^2$. Form invariance and the uniqueness proved there imply that its spectral projections commute with the restrictions of all unitaries of $M'$. Extend $e_m=1_{[0,m]}(T)$ by zero on $K_F^\perp$. Then $e_m\in M$, $e_m\uparrow p$, and (EW7) gives $\|(z_{n+1}-z_n)^*e_m\|\leq\sqrt m\,2^{-n},\qquad \|e_m(z_{n+1}-z_n)\|\leq\sqrt m\,2^{-n}. \tag{EW8}$ Consequently $e_m z_n$ converges in operator norm. By (EW6) its adjoint limit on every vector is $x^*e_m$; thus its norm limit is exactly $e_m x$. Also $\Lambda(e_m z_n)=\pi(e_m)\Lambda z_n\longrightarrow\pi(e_m)\xi.$ EW-1 proves $e_m x\in N,\qquad \Lambda(e_m x)=\pi(e_m)\xi. \tag{EW9}$ Since $x^*e_m x\uparrow x^*px$, order normality yields $\varphi(x^*px)=\sup_m\|\pi(e_m)\xi\|^2\leq\|\xi\|^2.$ Thus $px\in N$. Finite additivity inside this finite value gives $\|\Lambda((p-e_m)x)\|^2=\varphi(x^*px)-\varphi(x^*e_mx)\longrightarrow0.$ NF-5 gives $\pi(e_m)\to\pi(p)$ strongly, so (EW9) implies $\Lambda(px)=\pi(p)\xi$.

Now let $F$ run over all finite subsets of $K$, ordered by inclusion. The corresponding $p_F$ increase to $1$, since their ranges contain those finite subsets. We have just proved $\varphi(x^*p_Fx)=\|\pi(p_F)\xi\|^2\leq\|\xi\|^2.$ Order normality again gives $x\in N$, and the same finite-value difference argument gives $\Lambda x=\xi$. This proves (EW4) for arbitrary nets on arbitrary Hilbert spaces. Only the sequence on the explicitly chosen reducing subspace was used.

EW-3. Ultraweak lower semicontinuity on the whole cone

The set $G_C=\{(x,\Lambda x):x\in N,\ \|x\|\leq C\}\subset M\times H$ is convex and closed for adjoint-strong times Hilbert norm by EW-2; the norm bound passes to adjoint-strong limits. CV-2 makes $G_C$ closed for ultraweak times Hilbert weak topology.

Fix $r,C<\infty$, $r,C\geq0$. Suppose $a_i\in E_r$, $\|a_i\|\leq C$, and $a_i\to a$ strongly. Positivity and the norm bound pass to the limit. The square roots converge strongly: approximate $t^{1/2}$ uniformly on $[0,C]$ by the actual CF polynomials, and use bounded strong continuity of multiplication in each polynomial. The vectors $\Lambda(a_i^{1/2})$ lie in the radius-$\sqrt r$ Hilbert ball. CV-3 supplies a convergent subnet for the weak topology (or a cluster point of the tail filter), of weak limit $\eta$ and norm at most $\sqrt r$. Bounded strong convergence implies ultraweak convergence by BD7. Closedness of $G_{\sqrt C}$ therefore gives $a^{1/2}\in N,\qquad\Lambda(a^{1/2})=\eta,\qquad\varphi(a)\leq r.$ Thus $E_r\cap C\overline B_M$ is strongly closed. It is convex by weight additivity, so CV-2 makes it ultraweakly closed.

CP01–06 identifies $M=(M_*)^*$ isometrically with its concrete ultraweak topology and proves completeness of $M_*$. Apply the full CV-4 criterion to the convex subset $E_r$ of this dual space. All its bounded slices have just been checked; hence $E_r$ itself is ultraweakly closed. This proves $1\Rightarrow2$, including the unbounded parts of the positive cone.

EW-4. A positive downward hull and its exact closure

We prove the separation step needed for (EW1); a separator of a positive sublevel alone need not be positive.

Put $E=E_1$ and $D=\overline{E-M_+}^{\,\|\cdot\|}\subset M_{\rm sa}.$ This is convex, norm closed, contains $0$ and $-M_+$, and is downward closed under subtraction of $M_+$. First, $D\cap M_+=E. \tag{EW10}$ Indeed let $a_n-b_n\to x\geq0$ in norm with $a_n\in E,b_n\geq0$, and choose $\epsilon_n>0$ tending to zero with $x\leq a_n+\epsilon_n1$. Set $v_n=\left(a_n(a_n+\epsilon_n1)^{-1}\right)^{1/2},\qquad c_n=v_n x v_n.$ Then $0\leq c_n\leq a_n$, so $c_n\in E$. Moreover $\begin{split} \|(1-v_n)x^{1/2}\|^2 &=\|(1-v_n)x(1-v_n)\|\\ &\leq\|(1-v_n)(a_n+\epsilon_n1)(1-v_n)\| \leq\epsilon_n. \end{split} \tag{EW11}$ The last inequality is the scalar identity $(t+\epsilon)(1-\sqrt{t/(t+\epsilon)})^2=(\sqrt{t+\epsilon}-\sqrt t)^2\leq\epsilon$ in continuous calculus. Writing $c_n=(v_nx^{1/2})(v_nx^{1/2})^*$ proves $\|c_n-x\|\leq2\sqrt{\|x\|\epsilon_n}\to0$. EW-3 makes $E$ norm closed, proving (EW10).

Next $D$ is ultraweakly closed. By CV-4 it suffices to check its bounded slices. Suppose $x_i\in D$, $\|x_i\|\leq C$, and $x_i\to x$ ultraweakly. Approximate each $x_i$ in norm by elements of $E-M_+$, on the product of the original directed set and a decreasing positive error parameter. This gives a net $y_j=a_j-b_j\longrightarrow x\text{ ultraweakly},\quad a_j\in E,\ b_j\geq0,\quad\|y_j\|\leq C+1=:L.$ The uniform bound follows by restricting the error parameter to at most $1$. No bound on $a_j$ or $b_j$ is claimed.

Fix $0<\delta<1/L$ when $L>0$, and put $g_\delta(t)=t/(1+\delta t)$. Inversion reverses the order of strictly positive operators: if $0<A\leq B$, conjugate by $A^{-1/2}$, invert its positive spectrum, and conjugate back to get $B^{-1}\leq A^{-1}$. Since $y_j\leq a_j$ and $1+\delta y_j\geq(1-\delta L)1$, this proves $g_\delta(y_j)\leq g_\delta(a_j)=:r_j,\quad 0\leq r_j\leq a_j,\quad \|r_j\|\leq\delta^{-1}.$ Thus $r_j\in E$. On $[-L,L]$, $g_\delta(t)=t-\frac{\delta t^2}{1+\delta t} \geq t-\eta_\delta,\qquad \eta_\delta=\frac{\delta L^2}{1-\delta L}\longrightarrow0. \tag{EW12}$ The bounded slice of $E$ is ultraweakly compact by EW-3 and CV-3. Along a subnet $r_j\to r_\delta\in E$, positivity and (EW12) give $r_\delta\geq x-\eta_\delta1.$ Consequently $x-\eta_\delta1\in E-M_+\subset D$. Norm closedness and $\eta_\delta\to0$ imply $x\in D$. The zero bound case is immediate. This verifies every slice and hence proves ultraweak closedness of $D$.

EW-5. Normal positive minorants, including infinite values

Let $a\geq0$ and $0<t<\varphi(a)$. Then $a/t\notin E$, hence $a/t\notin D$ by (EW10). Apply real ultraweak separation CV-1 to the closed convex set $D$ inside $M_{\rm sa}$. It gives a continuous real linear $L$ with $L(a/t)>c:=\sup_{d\in D}L(d)\geq0. \tag{EW13}$ As $-s b\in D$ for all $s\geq0,b\geq0$, finiteness of $c$ forces $L(b)\geq0$. Its complexification $f(x+iy)=L(x)+iL(y)$, $x,y\in M_{\rm sa}$, is positive and ultraweakly continuous: the self-adjoint real/imaginary part maps are ultraweakly continuous, directly from the concrete vector-series description in CP01–06. Thus $f\in M_*^+$.

If $c>0$, replace $f$ by $f/c$. Then $f(b)\leq1$ for $b\in E$. If $0<\varphi(b)<\infty$, apply this to $b/\varphi(b)$; if $\varphi(b)=0$, apply it to every $s b$, $s>0$, to obtain $f(b)=0$. Infinite weight values impose no upper restriction. Hence $f\leq\varphi$, and (EW13) gives $f(a)>t$.

If $c=0$, then $f$ vanishes on $E$, and therefore on every finite-weight positive element by the same scaling argument. Every positive multiple of $f$ is dominated by $\varphi$; since $f(a)>0$, choose such a multiple with value greater than $t$.

We have proved that for every $0<t<\varphi(a)$ some $f\in M_*^+$, $f\leq\varphi$, satisfies $f(a)>t$. When $\varphi(a)=0$ use the zero functional. When $\varphi(a)=\infty$, let $t$ be arbitrarily large. This proves (EW1) on the entire cone, with no faithfulness or semifiniteness assumption.

Finally, condition 3 implies 2 because each normal positive functional is ultraweakly continuous and a supremum of continuous real functions is lower semicontinuous. Condition 2 implies 1: if $a_i\uparrow a$ boundedly, bounded strong convergence and BD7 give ultraweak convergence. With $s=\sup_i\varphi(a_i)$, the case $s=\infty$ follows from monotonicity; if $s<\infty$, closedness of $E_s$ gives $\varphi(a)\leq s$, and monotonicity gives equality. All three conditions are equivalent.

Exact scope

The equivalence theorem applies to every additive extended-valued weight, allowing zero, nonfaithful, nonsemifinite and non-countably-decomposable cases. The bounded GNS graph closedness statement assumes order normality (condition 1); it requires no additional faithfulness, semifiniteness or countability assumption. They prove neither a sum decomposition into normal functionals nor operator-valued weight comparison. Finite-star involution closability for a faithful normal semifinite weight is a separate consequence developed in the next provider.