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

Detecting normal weights by finite observations

An infinite weight cannot be tested by applying a bounded-functional continuity theorem to it. This unit proves the equivalence between preservation of increasing suprema, arbitrary positive summation, lower semicontinuity, and recovery from dominated normal positive functionals. The algebra and the weight are arbitrary: faithfulness, semifiniteness, separability, and the existence of a faithful normal state on the whole algebra are not hypotheses.

The mathematical target is Takesaki, Theory of Operator Algebras II, VII.1, Theorem 1.11. The proof is organized around three mechanisms: control of a GNS limit by summable energy increments, assembly from corners admitting normal states, and positive separation of hereditary sets. In particular, the GNS argument below does not require an absolute-value operation on the weight's finite linear domain.

OA-MOD-NW-01 — Objects, topologies, and foundation contracts

Let $M$ be a unital von Neumann algebra and $\varphi:M_+\to[0,\infty]$ a weight, with the conventions of OA-MOD-WG-002. Put

The linear map $\Lambda_\varphi:\mathfrak n_\varphi\to H_\varphi$, its null quotient, and its Hilbert completion are those constructed in OA-MOD-WG-003–006. Inner products are linear in the first variable. Write $M_*^+$ for the positive ultraweakly continuous linear functionals on $M$.

The ultraweak topology is $\sigma(M,M_*)$. The sigma-strong topology is generated by

The sigma-strong* topology adds $p_\omega(x^*)$. Throughout this unit, these are locally convex topologies on the entire algebra; when a continuity argument uses a norm bound, that bound is stated.

A family $(a_i)_{i\in I}\subseteq M_+$ is summable here if its finite partial sums have an upper bound in $M_+$. OA-MOD-BK-04 then gives

The partial sums converge sigma-strongly and ultraweakly. Conversely, if the finite partial sums converge sigma-strongly to $b$, then each partial sum is at most $b$, by passing to the limit in its positive differences with all later partial sums. Thus this definition is equivalent to sigma-strong summability of a positive family. Indeed, if $b_F\uparrow b$ and $0\leq b-b_F\leq C1$, then $p_\omega(b-b_F)^2\leq C\omega(b-b_F)\to0$. A scalar sum over an arbitrary set always means the supremum of its finite subsums.

The proofs use the following exact foundation contracts.

• OA-MOD-NW-DEP-DUAL: $M=(M_*)^*$ isometrically; $M_*^+$ separates positive elements and spans $M_*$; the positive cone and norm balls are ultraweakly closed. Every normal positive functional has a support projection $p$, satisfies $\omega(x)=\omega(pxp)$, and is faithful on $pMp$ after restriction. Corners have their inherited ultraweak topology and predual. Fixed multiplication and the adjoint operation are ultraweakly continuous.
• OA-MOD-NW-DEP-TOPO: the continuous linear duals of the sigma-strong and sigma-strong* topologies are $M_*$. Thus a convex subset of $M$ has the same closure for either of these topologies as for the ultraweak topology. The analogous real statement holds on $M_{\mathrm{sa}}$.
• OA-MOD-NW-DEP-CONVEX: Hahn–Banach separation for locally convex spaces; weak* compactness of dual norm balls; weak compactness of Hilbert balls; and Krein–Smulian: a convex subset of a dual Banach space is weak* closed if and only if its intersection with every closed norm ball is weak* closed.

These are bounded functional-analysis and predual prerequisites. They do not assume any characterization theorem for weights. The weight-theoretic arguments below use these precise prerequisites. Bounded functional calculus, support and polar decompositions, inverse order, and monotone operator nets are supplied by OA-MOD-BK. No unbounded spectral theorem, modular theorem, or result about closed positive forms is used.

For later use, bounded multiplication is sigma-strong* continuous. For example, if $x_i\to x$, $y_i\to y$ sigma-strong* and the two nets are norm bounded, then

The adjoints satisfy the analogous estimate. Uniform polynomial approximation therefore makes the square-root operation sigma-strongly continuous on every norm-bounded positive set. It also makes any fixed continuous real function on a common compact spectral interval continuous on bounded self-adjoint sets. The norm of each functional-calculus approximation error controls every $p_\omega$.

OA-MOD-NW-02 — The implications requiring no localization

Consider these four assertions:

Proposition. $\mathrm{P}\Rightarrow\mathrm{L}\Rightarrow\mathrm{N}\Rightarrow\mathrm{A}$.

Proof. Under P, a sublevel set is the intersection of $M_+$ with the closed half-spaces $\{\omega\leq c\}$, proving L. If $a_\alpha\uparrow a$, the net converges ultraweakly by OA-MOD-BK-04. Write $r=\sup_\alpha\varphi(a_\alpha)$. Monotonicity gives $r\leq\varphi(a)$. If $r=\infty$, equality already follows. Otherwise all $a_\alpha$ lie in the closed sublevel set at $r$, so $a$ does too. This proves N. Finally, the finite partial sums of a summable family increase to its sum. N and finite additivity give A. $\square$

L is exactly lower semicontinuity of the extended-valued function on $M_+$ with its relative ultraweak topology. Equivalently, $\varphi(a)\leq\liminf_\alpha\varphi(a_\alpha)$ for every ultraweakly convergent net $a_\alpha\to a$ in $M_+$. The net formulation includes infinite values and does not require the original net to be norm bounded.

OA-MOD-NW-03 — Corners and the countability they actually provide

Call a projection $p$ sigma-finite if $pMp$ has a faithful normal state; include $0$ in this class. Equivalently, a nonzero such $p$ is the support of a normal state on $M$: extend a state of $pMp$ by $x\mapsto\omega(pxp)$, or restrict a state with support $p$.

Let $\mathcal P$ be these projections and define

Projection facts. Subprojections and countable joins of members of $\mathcal P$ again belong to $\mathcal P$. Equivalent projections have this property simultaneously. Consequently $J$ is a two-sided *-ideal, and the ultraweak limit of a sequence in $J$ belongs to $J$.

Proof. Restrict a faithful corner state to a nonzero subprojection and normalize it. For $p_n\in\mathcal P$, choose normal states $\omega_n$ supported on the nonzero $p_n$. A sum with strictly positive summable coefficients has support $\bigvee_n p_n$. To check this last assertion, put $q=\bigvee_n p_n$. The sum vanishes on $1-q$. If $a\in(qMq)_+$ and every $\omega_n(a)=0$, corner faithfulness gives $p_nap_n=0$, hence $a^{1/2}p_n=0$. Since the ranges of the $p_n$ span a dense subspace of $qH$, this forces $a=0$. Thus the normalized sum is faithful on $qMq$.

If $u^*u=p$, $uu^*=q$, a faithful state on $pMp$ transports to $qMq$ by $a\mapsto\omega(u^*au)$. This proves invariance under equivalence.

An element belongs to $J$ precisely when its left and right support projections belong to $\mathcal P$: one direction follows by taking subprojections; the other uses their finite join. The right support of $ax$ is at most the right support of $x$. The two supports of $ax$ are equivalent by polar decomposition, so $ax\in J$ if $x\in J$. Adjoints give the right ideal property, while finite joins handle sums. Finally, if $x_n\in J$, one countable join $q\in\mathcal P$ satisfies $x_n=qx_nq$ for all $n$. If $x_n\to x$ ultraweakly, fixed multiplication gives $x=qxq$. $\square$

There is an orthogonal family $(p_i)_{i\in I}\subseteq\mathcal P$ with sum $1$. Indeed, every nonzero projection $e$ contains a nonzero member of $\mathcal P$: choose $\omega\in M_*^+$ with $\omega(e)>0$, compress to $eMe$, and take its support. A maximal orthogonal family, obtained by Zorn's lemma, must therefore have supremum $1$. No assertion that $I$ is countable is made.

Metric lemma. If $\omega$ has support $p$, the metric

induces the sigma-strong topology on each norm ball of $Mp$. On a norm ball of $pMp$, the sum $d_\omega(x,y)+d_\omega(x^*,y^*)$ induces the sigma-strong* topology.

Proof. For $x= xp$, $x^*x\in pMp$; faithfulness there shows that $d_\omega$ separates points. Only one direction of the topology assertion needs proof. Suppose $x_\alpha,x\in Mp$, all of norm at most $R$, and $d_\omega(x_\alpha,x)\to0$. Put

For any $\psi\in M_*^+$, failure of $\psi(t_\alpha)\to0$ gives a subnet on which $\psi(t_\alpha)\geq\varepsilon>0$. The order interval $[0,4R^2p]$ is ultraweakly compact. A further subnet has an ultraweak limit $t$ there. Then $\omega(t)=0$, so $t=0$ by faithfulness, whereas $\psi(t)\geq\varepsilon$, a contradiction. Thus every sigma-strong seminorm tends to zero. Apply this argument also to adjoints inside $pMp$ for the second assertion. $\square$

The existence of this metric on a corner does not assert separability of that metric space.

OA-MOD-NW-04 — Passing an energy bound through an unbounded majorant

For $t>0$, define

whenever $h=h^*$ and $1+th$ is strictly positive. Inverse order gives operator monotonicity of $r_t$ on this domain. For $h\geq0$,

The last convergence is even in norm for a fixed bounded $h$.

Lemma. Suppose $\varphi$ satisfies N. Let $y_n\in M_+$ be norm bounded and converge sigma-strongly to $y$. Suppose $b_n\in M_+$ increases, $y_n\leq b_n$, and

The norms of the $b_n$ need not be bounded. Then $\varphi(y)\leq C$.

Proof. Fix $t>0$. The bounded increasing sequence $r_t(b_n)$ has a supremum $c_t\in M_+$. By N,

Monotonicity gives $r_t(y_n)\leq r_t(b_n)\leq c_t$. The bounded functional-calculus continuity from NW-01 gives $r_t(y_n)\to r_t(y)$ sigma-strongly and ultraweakly. The positive cone is ultraweakly closed, so $r_t(y)\leq c_t$. Hence $\varphi(r_t(y))\leq C$. Apply N once more to $r_t(y)\uparrow y$ to obtain the result. $\square$

The order supremum of the $b_n$ themselves was never asserted to exist in $M$. This is why the bounded functions $r_t$ enter the argument.

OA-MOD-NW-05 — Closing a GNS graph by summable increments

Theorem. Suppose $\varphi$ satisfies N. If $x_n\in\mathfrak n_\varphi$ is norm bounded, $x_n\to x$ sigma-strong*, and $\Lambda_\varphi(x_n)\to\xi$ in Hilbert norm, then

This sequential statement holds on an arbitrary von Neumann algebra.

Proof. Choose a subsequence, relabeled $u_k$, so fast that, for $z_k=u_{k+1}-u_k$,

This is possible because the given GNS vectors are Cauchy. Let $\lambda_k=2^{-k}$ and $L_k=\sum_{j=k}^\infty\lambda_j=2^{1-k}$. For $N\geq k$, weighted Cauchy–Schwarz on every vector in a faithful concrete representation gives

Explicitly, $\|\sum z_jv\|^2 \leq(\sum\lambda_j)\sum\lambda_j^{-1}\|z_jv\|^2$; enlarging the first scalar sum to $L_k$ proves the operator inequality.

For fixed $k$, the left side is a norm-bounded sequence converging sigma-strongly to $(x-u_k)^*(x-u_k)$, by the strong* assumption and bounded multiplication. The right side increases and

Here $C_k<\infty$ and $C_k\to0$. NW-04 yields

Thus $x-u_k\in\mathfrak n_\varphi$. Since $u_k$ lies there and the domain is linear, $x\in\mathfrak n_\varphi$. Moreover $\|\Lambda_\varphi(x)-\Lambda_\varphi(u_k)\|^2\leq C_k\to0$. The same subsequence converges to $\xi$, so $\Lambda_\varphi(x)=\xi$. $\square$

This proof neither assumes $|h|\in\mathfrak m_\varphi$ for self-adjoint $h\in\mathfrak m_\varphi$, nor applies the weight to a norm limit using an unproved semicontinuity assertion.

OA-MOD-NW-06 — Lower semicontinuity on a sigma-finite corner

Proposition. On a sigma-finite von Neumann algebra, A implies N.

Proof. Choose a faithful normal state $\omega$. Given $a_\alpha\uparrow a$, inductively choose an increasing sequence of indices $\alpha_n$ such that

Normality of the bounded functional $\omega$, or simply ultraweak convergence of the increasing net, supplies each choice. The sequence $a_{\alpha_n}$ has a supremum $b\leq a$. Ultraweak continuity gives $\omega(b)=\omega(a)$, so faithfulness implies $b=a$. Set $a_{\alpha_0}=0$ and $d_n=a_{\alpha_n}-a_{\alpha_{n-1}}\geq0$. Then $\sum_nd_n=a$, and A gives

No infinite subtraction occurs. $\square$

Proposition. On a sigma-finite von Neumann algebra, N implies L.

Proof. For finite $r,s\geq0$, consider the bounded part of the graph

It is convex. The product of sigma-strong* topology in $M$ and norm topology in $H_\varphi$ is metrizable on the containing norm balls, by NW-03. A point of its closure is therefore the limit of a sequence in $G_{r,s}$. NW-05 proves that the point belongs to the graph; the two norm bounds persist. Thus $G_{r,s}$ is closed for that product topology.

By the compatible-dual contract and Hahn–Banach separation, the closure of this convex set is the same for the product of ultraweak and weak Hilbert topologies. It is consequently a closed subset of the product of two compact balls, hence compact.

Its projection to $M$ is

This set is ultraweakly compact and therefore closed. The set $E_s=\{x:\varphi(x^*x)\leq s^2\}$ is convex, because it is the inverse image of a Hilbert ball under the linear GNS map on its linear domain. Krein–Smulian now shows that $E_s$ is ultraweakly closed.

Fix $c\geq0$. In a norm ball, if $a_\alpha\geq0$, $\varphi(a_\alpha)\leq c$, and $a_\alpha\to a$ sigma-strongly, then $a_\alpha^{1/2}\to a^{1/2}$ sigma-strongly and ultraweakly. Every square root lies in $E_{\sqrt c}$, hence so does the limit. Therefore $\varphi(a)\leq c$. The bounded slice of this sublevel set is convex, so compatible duals make it ultraweakly closed. A second use of Krein–Smulian proves that the entire sublevel set is ultraweakly closed. This proves L, including $c=0$. $\square$

The sequence in this proof exists because the algebra is sigma-finite. The next two items remove that local hypothesis.

OA-MOD-NW-07 — A bounded gluing lemma for hereditary sets

Let $F\subseteq M_+$ be convex and hereditary, meaning

Assume $F\cap pMp$ is ultraweakly closed for every $p\in\mathcal P$.

Lemma. If $a_\alpha\in F\cap J$ is norm bounded and converges sigma-strongly to $a\in J$, then $a\in F$.

Proof. The assertion is empty if $F=\varnothing$, so assume $F\neq\varnothing$. Put

This set is convex: for $0\leq t\leq1$,

and then convexity and heredity of $F$ apply. It is also invariant under left multiplication by contractions, since $(vx)^*(vx)\leq x^*x$.

For $q\in\mathcal P$, each bounded slice of $E\cap qMq$ is sigma-strong* closed: it is the inverse image of the closed set $F\cap qMq$ under $x\mapsto x^*x$, using bounded multiplication. Convexity and compatible duals make these slices ultraweakly closed; Krein–Smulian gives ultraweak closedness of $E\cap qMq$.

Fix $p\in\mathcal P$. We claim that $E^*p$, and therefore $pE$, is ultraweakly closed in $M$. Consider a sigma-strong limit point $z$ of a bounded slice of $E^*p$. It belongs to the same norm ball of $Mp$. The metric in NW-03 gives a sequence $z_n\in E^*p$ converging sigma-strongly to $z$. Choose a countable join $q\in\mathcal P$ containing both supports of every $z_n$. Since $z_n\to z$ ultraweakly, $z=qzq$ too. Write $z_n=e_n^*p$ with $e_n\in E$. Then $z_n^*=pe_n\in E$, by contraction invariance. Hence $z_n^*\in E\cap qMq$, whose ultraweak closedness gives $z^*\in E$. Also $zp=z$, so $z=(z^*)^*p\in E^*p$.

Thus each bounded slice of the convex set $E^*p$ is sigma-strongly closed. Compatible duals and Krein–Smulian prove the claim; the adjoint homeomorphism gives it for $pE$.

Return to the given positive net. Its limit is positive. For $p=s(a)\in\mathcal P$, bounded square-root continuity gives

sigma-strongly and ultraweakly. Since $a_\alpha^{1/2}\in E$, the net lies in $pE$. Closedness gives $a^{1/2}\in pE\subseteq E$, so $a\in F$. $\square$

The lemma asserts exactly the bounded convergence needed below. It does not infer a statement about unbounded nets from an argument using bounded square-root continuity.

OA-MOD-NW-08 — Arbitrary positive sums imply global lower semicontinuity

Theorem. A implies L on every von Neumann algebra.

Proof. The zero algebra is immediate. Assume otherwise and put $F=\{a\in M_+:\varphi(a)\leq1\}$. It is convex and hereditary. On any sigma-finite corner, the restricted weight satisfies A: the sums computed in the corner are the same positive operator sums computed in $M$. NW-06 proves that its sublevel set is ultraweakly closed. Thus $F$ meets the hypothesis of NW-07.

Take a norm-bounded net $a_\alpha\in F$ converging sigma-strongly to $a\geq0$. Choose an orthogonal family $(p_i)_{i\in I}\subseteq\mathcal P$ with sum $1$. For a finite subset $K\subseteq I$, write $q_K=\sum_{i\in K}p_i\in\mathcal P$. The positive net

is norm bounded and converges sigma-strongly. Every term and the limit lie in $J$, since $J$ is an ideal. Moreover

so the terms belong to $F$. NW-07 gives $\varphi(a^{1/2}q_Ka^{1/2})\leq1$. The family $(a^{1/2}p_ia^{1/2})_{i\in I}$ is summable, with sum $a$. Apply A to obtain

Thus every bounded slice of $F$ is sigma-strongly closed. Convexity, compatible duals, and Krein–Smulian imply that $F$ is ultraweakly closed. For $c>0$, its scalar multiple $cF$ is the sublevel set at $c$. The zero sublevel set is $\bigcap_{n\geq1}n^{-1}F$. All are closed, proving L. $\square$

The finite subsets $K$ form a directed set. Replacing them with a sequence would lose this argument when $I$ is uncountable.

OA-MOD-NW-09 — Positive separation needs a downward closure

Theorem. If $E\subseteq M_+$ is a nonempty ultraweakly closed hereditary convex set, then

Consequently, for each $a\in M_+\setminus E$, there is $\omega\in M_*^+$ such that

Proof of the closure identity. Work in the real space $M_{\mathrm{sa}}$, and write $D=E-M_+$. This is convex and downward closed: $h\in D$, $k\leq h$ imply $k\in D$. Since $0\in E$, it contains $-M_+$.

Let $G$ consist of the self-adjoint $h$ such that

Scalar functional calculus gives $r_t(h)\leq h$. Hence $D\subseteq G$. Also $r_t(h)\to h$ in norm as $t\downarrow0$, so $G\subseteq\overline D^{\,\mathrm{uw}}$.

We first show that every bounded slice of $G$ is sigma-strongly closed. Suppose $h_\alpha\in G$, $\|h_\alpha\|\leq R$, and $h_\alpha\to h$ sigma-strongly, where $R>0$. Fix $0<t<1/(2R)$. Choose $e_\alpha\in E$ with $r_t(h_\alpha)\leq e_\alpha$. The domain bounds allow a second application of $r_t$, giving

Here $0\leq r_t(e_\alpha)\leq e_\alpha$, so these elements belong to $E$, and they are bounded above by $t^{-1}1$. Choose an ultraweakly convergent subnet, with limit $e_t\in E$. Bounded functional-calculus continuity and closedness of the positive cone give $r_{2t}(h)\leq e_t$. Thus $r_{2t}(h)\in D$.

Given any $\beta>0$ for which $1+\beta h$ is strictly positive, choose the preceding $t$ small enough that $2t<\beta$. The scalar function $r_s(\lambda)$ decreases with $s$ throughout its domain, so $r_\beta(h)\leq r_{2t}(h)\in D$. Downward closedness gives $r_\beta(h)\in D$. Thus $h\in G$. A ball of radius zero is trivial.

Next, if $0\leq r<s$ and $B$ denotes the self-adjoint unit ball, then

For the forward inclusion, approximate $h\in G\cap rB$ in norm by $r_t(h)\in D$; for small $t$, these approximants belong to $sB$. Conversely, $D\cap sB\subseteq G\cap sB$, and the latter is sigma-strongly closed by the preceding paragraph. The right side is convex. Hence each bounded slice of $G$ is convex, and $G$ itself is convex.

Compatible duals now make each bounded slice of $G$ ultraweakly closed. Krein–Smulian makes $G$ ultraweakly closed. Since $D\subseteq G\subseteq\overline D^{\,\mathrm{uw}}$, this proves $G=\overline D^{\,\mathrm{uw}}$.

If $a\in G\cap M_+$, then $r_t(a)\in D\cap M_+$. It is positive and dominated by an element of $E$, so heredity gives $r_t(a)\in E$. Their norm limit $a$ belongs to $E$. The reverse inclusion is immediate, proving the identity.

Proof of positive separation. If $a\geq0$ is outside $E$, the identity puts it outside the closed convex set $\overline D^{\,\mathrm{uw}}$. Real Hahn–Banach separation supplies a continuous real linear functional $f$ with

The number $c$ is finite and nonnegative, since $0\in D$. Because $-tb\in D$ for all $b\geq0$ and $t\geq0$, boundedness above forces $f(b)\geq0$. Complexifying $f$ gives a positive normal linear functional. If $c>0$, divide by $c$. If $c=0$, multiply by a positive scalar making its value at $a$ exceed one. In either case the resulting $\omega$ has the asserted properties. $\square$

Subtracting the positive cone before separating is what forces the separating functional to be positive. An arbitrary real functional separating $a$ from $E$ need not have that property.

OA-MOD-NW-10 — Recovering every value from dominated normal functionals

Theorem. L implies P.

Proof. Put $E=\{b\geq0:\varphi(b)\leq1\}$. This is nonempty, ultraweakly closed, hereditary, and convex. If $\omega\in M_*^+$ satisfies $\omega(e)\leq1$ for every $e\in E$, then $\omega\leq\varphi$ on all of $M_+$. Indeed, when $0<\varphi(b)<\infty$, apply the bound to $b/\varphi(b)$. When $\varphi(b)=0$, every $tb$, $t>0$, lies in $E$, forcing $\omega(b)=0$. At an infinite value the domination inequality is automatic.

Fix $a\geq0$. Every member of $\mathcal F_\varphi$ has value at most $\varphi(a)$. Conversely, if $0<r<\varphi(a)$, then $a/r\notin E$. NW-09 gives a positive normal $\omega$ with $\omega|_E\leq1$ and $\omega(a/r)>1$. By the preceding paragraph $\omega\in\mathcal F_\varphi$, and $\omega(a)>r$. Let $r$ increase to $\varphi(a)$ when that value is finite, or let $r\to\infty$ when it is infinite. If $\varphi(a)=0$, equality follows from positivity and the zero functional. This proves P in all cases. $\square$

The set $\mathcal F_\varphi$ need not be asserted to be upward directed for this theorem. The supremum is taken separately at each positive element.

OA-MOD-NW-11 — The full characterization

Theorem. For an arbitrary weight on an arbitrary von Neumann algebra, N, A, L, and P of NW-02 are equivalent.

Proof. NW-02 gives $\mathrm{P}\Rightarrow\mathrm{L}\Rightarrow\mathrm{N}\Rightarrow\mathrm{A}$. NW-08 gives $\mathrm{A}\Rightarrow\mathrm{L}$, and NW-10 gives $\mathrm{L}\Rightarrow\mathrm{P}$. These implications cover all four assertions. $\square$

Thus normality as defined by monotone nets permits a finite-observation test: whenever $a\geq0$ and $r<\varphi(a)$ with $r$ finite and nonnegative, some positive normal functional $\omega\leq\varphi$ satisfies $\omega(a)>r$. This remains meaningful when $\varphi(a)=\infty$.

A faithful normal state appeared only inside sigma-finite corners. Neither the original weight nor the auxiliary restrictions were assumed faithful or semifinite.

OA-MOD-NW-12 — Consequences for GNS maps and sums of weights

Closed graph for general nets. If $\varphi$ is normal, the graph of $\Lambda_\varphi$ is closed for the product of the sigma-strong topology in $M$ and the weak topology in $H_\varphi$. Consequently, a net $x_\alpha\in\mathfrak n_\varphi$ with $x_\alpha\to x$ sigma-strongly and $\Lambda_\varphi(x_\alpha)\to\xi$ weakly satisfies $x\in\mathfrak n_\varphi$, $\Lambda_\varphi(x)=\xi$.

Proof. For $\omega\in\mathcal F_\varphi$, let $(H_\omega,\pi_\omega,\Omega_\omega)$ be its bounded-functional GNS construction, with $\|\pi_\omega(x)\Omega_\omega\|^2=\omega(x^*x)$. The comparison construction OA-MOD-DW-04, identifying the bounded weight's GNS vector with $\pi_\omega(x)\Omega_\omega$, gives a contraction $C_\omega:H_\varphi\to H_\omega$ satisfying

Sigma-strong convergence gives $\pi_\omega(x_\alpha)\Omega_\omega\to\pi_\omega(x)\Omega_\omega$ in norm. Weak convergence through $C_\omega$ therefore implies $C_\omega\xi=\pi_\omega(x)\Omega_\omega$. Taking norms and using NW-11,

Thus $x\in\mathfrak n_\varphi$ and $C_\omega(\xi-\Lambda_\varphi(x))=0$ for every $\omega$.

These contractions separate vectors of $H_\varphi$. To see this, NW-11 gives $\sup_\omega\|C_\omega\Lambda_\varphi(y)\| =\|\Lambda_\varphi(y)\|$ for every GNS vector. For a general $\eta$, approximate by $\Lambda_\varphi(y)$ and use that all $C_\omega$ are contractions; this proves $\sup_\omega\|C_\omega\eta\|=\|\eta\|$. Apply it to $\xi-\Lambda_\varphi(x)$ to finish the proof. $\square$

This last argument does not assume that a weakly convergent net is norm bounded. The finiteness bound comes from the single limit vector and the comparison contractions.

Arbitrary sums of normal weights. If $(\varphi_j)_{j\in J_0}$ is any family of normal weights, then

is a normal weight. Empty sums give the zero weight.

Proof. Nonnegative finite-subsums commute with addition and nonnegative scalar multiplication, proving the weight axioms. If $a_\alpha\uparrow a$, normality of the finitely many weights in each finite set gives

For finite sums of increasing extended nonnegative numbers, the interchange follows by approximating each finite target from below; the same argument covers infinite values. Thus $\psi$ is normal. $\square$

The converse representation of every normal weight as one fixed sum of normal functionals is a further theorem. NW-11's pointwise supremum is not silently replaced by that stronger assertion.

OA-MOD-NW-13 — Models that test the generality

An infinite part with no finite vectors. On $M=\mathbb C\oplus\mathbb C$, set

This is a normal weight. Directly, if a positive increasing net has positive second-coordinate supremum, some term already has positive second coordinate; otherwise only the first coordinate matters. It is faithful but not semifinite: its finite linear domain is $\mathbb C\oplus0$. The positive normal functionals dominated by it are exactly

Their supremum is $a$ when $b=0$, and infinite when $b>0$. Thus the representation theorem detects the part invisible to the finite GNS domain. Bounded comparison operators alone would not reconstruct that part.

Uncountable coordinates. For any set $I$, let $\varphi(a)=\sum_{i\in I}a_i$ on $\ell^\infty(I)_+$. For finite $F\subseteq I$, the normal functional $\omega_F(a)=\sum_{i\in F}a_i$ is dominated by $\varphi$, and $\sup_F\omega_F(a)=\varphi(a)$. NW-11 proves normality. If $I$ is uncountable, no positive normal functional is faithful: its coordinate masses have a finite sum and therefore only countably many can be nonzero. The theorem nonetheless applies.

OA-MOD-NW-14 — Problems with complete solutions

Problem 1: a sequence cannot test every corner. Let $I$ be uncountable and define a weight on $\ell^\infty(I)$ by

Show that it preserves bounded increasing sequential suprema, but is not normal.

Solution. The union of the supports of two positive functions is the support of their sum, so the zero set of $\chi$ is an additive hereditary cone; scalar homogeneity also holds. Thus $\chi$ is a weight. For an increasing sequence, if every support is countable, the support of the pointwise supremum is contained in their countable union and the values remain zero. If a term already has uncountable support, that term and the supremum have value infinity. This proves sequential order continuity. On the other hand, finite coordinate projections $1_F$ increase as a net to $1$, while $\chi(1_F)=0$ and $\chi(1)=\infty$. Hence it is not normal. It also fails A on the summable family of all coordinate projections.

Problem 2: finite linear domains need not survive absolute values. Let $M=\prod_{n\geq1}M_2(\mathbb C)$, with bounded coordinate norm, and define

Set $v_n=(n^{-1},n^{-3})^\mathsf T$, $w_n=(n^{-1},-n^{-3})^\mathsf T$, $a_n=v_nv_n^*$, $b_n=w_nw_n^*$, and $h=a-b$. Prove that $\varphi$ is normal, $h=h^*\in\mathfrak m_\varphi$, but $|h|^{1/2}\notin\mathfrak n_\varphi$.

Solution. Each coordinate formula is a positive normal functional on $M$, and NW-12 makes their sum normal. The two positive families are bounded, and

Thus $h\in\mathfrak m_\varphi$ and is self-adjoint. Direct matrix calculation gives

Therefore

The defining condition for $|h|^{1/2}\in\mathfrak n_\varphi$ fails. This explains why a polar-decomposition proof must check the finite domain rather than infer it from self-adjointness.

Problem 3: domination at a zero value. Suppose $\varphi$ is normal and $\varphi(a)=0$ for $a\geq0$. Show that every $\omega\in\mathcal F_\varphi$ vanishes on $a$, and conversely that simultaneous vanishing of these functionals implies $\varphi(a)=0$.

Solution. Domination gives $0\leq\omega(a)\leq\varphi(a)=0$. Conversely NW-11 identifies $\varphi(a)$ with their supremum, which is zero. This does not imply $a=0$ unless the weight is faithful.

Problem 4: separating in the wrong set. In the ordered real vector space $\mathbb R^2$ with positive cone $\mathbb R_+^2$, put $E=[0,1]\times\{0\}$. Show that $E-\mathbb R_+^2$ need not equal $\operatorname{conv}(E\cup(-\mathbb R_+^2))$, although their polars at level one agree.

Solution. The point $(1,-1)$ belongs to the difference set. A convex combination from the stated union has the form $t(u,0)+(1-t)(-v,-w)$, where $0\leq t,u\leq1$, $v,w\geq0$. Its first coordinate can equal one only if $t=u=1$; then its second coordinate is zero. Thus $(1,-1)$ is outside the convex hull. For a linear functional $f$, boundedness by one on the difference set is equivalent to positivity of $f$ on the positive cone and $f|_E\leq1$: necessity follows by testing $E$ and all negative rays, and sufficiency follows from $f(e-b)\leq f(e)$. Exactly the same tests characterize the polar of the union and hence of its convex hull. A polar identity therefore does not require the incorrect set identity.

OA-MOD-NW-15 — Exact scope and further dependencies

NW-11 gives all four clauses of the full arbitrary-weight characterization, with complete proofs relative to the three foundation contracts in NW-01. NW-05 proves a sequential GNS graph result without sigma-finiteness; NW-06 upgrades it locally by compactness; NW-12 gives the resulting graph closedness for general nets through dominated functionals. The supplied examples and solutions test failure of sequential normality, nonsemifinite weights, uncountable algebras, and finite-domain hazards.

The separate predual-valued completely positive map on $\mathfrak m_\varphi$, its self-adjoint norm formula, and its closedness are not claimed proved merely because this unit bypasses them. The abstract ordered-space equivalence in Takesaki's Lemma 1.16 and the full relative-closure formulation of Lemma 1.15 also remain separately accountable; NW-07 and NW-09 prove the exact forms used here. A fixed-sum decomposition into normal functionals, modular covariance, weight-to-Hilbert-algebra reconstruction, and spatial derivatives remain further course work.

For comparison, Hiai states this characterization and additional sum representations as Theorem 7.2 in the pinned arXiv version 2004.02383v1, without a proof there. Lurie's Lecture 34 takes ultraweak lower semicontinuity as the definition of a normal weight.

Editable source · Proof dependencies and component terms