Finite domains, null directions, and support corners

Self-checked by the writing AI. Original text: CC0.

A general normal weight has two potentially different obstructions: some directions have no finite-weight approximation, while others have weight zero. Separating these obstructions explains why the support convention for an arbitrary normal weight needs two projections. For a semifinite normal weight the first obstruction disappears, and the usual faithful support-corner reduction follows.

Free comparisons are Brent Nelson, Tomita–Takesaki Theory, the support paragraph after Definition 3.19, page 32, and François Combes, Poids associé à une algèbre hilbertienne à gauche, the opening paragraph of printed page 51. Nelson uses $e-f$; Combes uses $1-f$. Here both projections are constructed and distinguished, and each asserted corner property is proved. They coincide for a semifinite weight. The compression criterion in WS-03 gives a further characterization of semifiniteness for normal weights. Central support results for traces and support results for bounded normal functionals are separate specializations.

Conventions and exact inputs

Let $M\subseteq B(H)$ be a concrete unital von Neumann algebra on any Hilbert space. Use the definitions of a weight $\varphi$, its finite cone $F_\varphi$, its finite left ideal $\mathfrak n_\varphi$, its definition algebra $\mathfrak m_\varphi$, and its null left ideal $N_\varphi$ from WG002, WG003, WG004 and WG005. Inner products are linear in the first variable.

OA-MOD-WS-02 and the corner-restriction proposition in OA-MOD-WS-03 allow an arbitrary weight. The compression criterion in WS-03 assumes normality and uses NW-11. All later weight assertions explicitly assume normality, in the sense of preservation of bounded increasing positive suprema.

The written inputs are WG003 for the finite-domain algebra, BK01 for square roots and positive-operator estimates, BK03 for bounded strong and ultraweak convergence, BK04 for monotone nets, BK05 for inverse order, BK06 for support cutoffs, and WG008 for the finite-cutoff characterization of semifiniteness. The compression criterion alone additionally uses NW11. These proofs apply in arbitrary Hilbert dimension; no modular theorem or spatial derivative is used.

For clarity, the concrete sigma-strong topology is given by the seminorms

x\longmapsto\left(\sum_{n\geq1}\|x\xi_n\|^2\right)^{1/2}, \qquad \sum_n\|\xi_n\|^2<\infty.

A norm-bounded strongly convergent net converges in this topology: make the tail uniformly small by the common norm bound, and use strong convergence for the remaining finitely many vectors. This does not assume that $H$ has a countable basis. We use the corresponding square-summable vector-pair definition of the ultraweak topology from BK-03.

The projection of the finite domain

Theorem. For every weight $\varphi$, there is a unique projection $e\in M$ such that

\overline{\mathfrak n_\varphi}^{\,\mathrm{SOT}} =\overline{\mathfrak n_\varphi}^{\,\sigma\text{-strong}} =\overline{\mathfrak n_\varphi}^{\,\mathrm{ultraweak}} =M e.

Moreover,

\overline{\mathfrak m_\varphi}^{\,\mathrm{SOT}} =\overline{\mathfrak m_\varphi}^{\,\sigma\text{-strong}} =\overline{\mathfrak m_\varphi}^{\,\mathrm{ultraweak}} =eMe.

The closures are taken inside $M$. Every $a\in F_\varphi$ satisfies $a=eae$. In particular, $\varphi$ is semifinite if and only if $e=1$.

Proof. Direct $F_\varphi$ by its positive order; addition gives a common upper bound. For each $a\in F_\varphi$, put

u_a=a(1+a)^{-1}.

As in WG-008, these are increasing finite positive contractions. Let $e\in M$ be their strong supremum. At this stage $e$ is only known to be a positive contraction.

Fix $a\in F_\varphi$. All $t a$, $t>0$, are indices, and the bounded support-cutoff theorem gives

u_{ta}\uparrow s(a).

Thus $e\geq s(a)$. A positive contraction dominating a projection acts as the identity on its range: if $q\leq e\leq1$, then

q(1-e)q=0,\qquad (1-e)^{1/2}q=0,

so $eq=q$.

Let $K$ be the closed linear span of the ranges of all $a\in F_\varphi$. The preceding observation shows that $e$ is the identity on $K$. Every $u_a$ vanishes on $K^\perp$, and therefore its strong limit $e$ does as well. Hence $e=P_K$ is a projection. It follows that $a=eae$ for every finite positive $a$.

If $x\in\mathfrak n_\varphi$, then $x^*x\in F_\varphi$, and

\|x(1-e)\xi\|^2 =\langle x^*x(1-e)\xi,(1-e)\xi\rangle=0.

Thus $x=xe$. Conversely, if $x=xe$, then $u_a\in\mathfrak n_\varphi$, since $u_a^2\leq u_a$ and its weight is finite. The left-ideal property gives $x u_a\in\mathfrak n_\varphi$. These operators are norm bounded and converge strongly to $xe=x$, hence sigma-strongly and ultraweakly as well.

The set $Me=\{x\in M:x=xe\}$ is closed in each stated topology: fixed right multiplication is continuous in each, and the defining equality is closed. This proves the three left-ideal closure identities. Uniqueness also follows: $Me=Me'$ for projections implies $e=ee'$ and $e'=e'e$, and taking adjoints yields both projection inequalities.

Since $\mathfrak m_\varphi=\operatorname{span}_{\mathbb C}F_\varphi$, it lies in $eMe$. For $x\in eMe$,

u_a x u_a=u_a^*(x u_a)\in\mathfrak m_\varphi.

Both factors in the last product belong to $\mathfrak n_\varphi$. The operators $u_a x u_a$ are norm bounded and converge strongly to $exe=x$. The same topology arguments prove all three definition-algebra closure identities. Semifiniteness, defined by ultraweak density of $\mathfrak m_\varphi$, is consequently equivalent to $eMe=M$, or $e=1$. $\square$

The projection $e$ need not be central. The proof does not turn the finite left ideal into a two-sided ideal.

A semifinite restriction exists before normality

Proposition. With $e$ as above, the restriction of $\varphi$ to $(eMe)_+$ is a semifinite weight. If $\varphi$ is normal, so is this restriction. For $a\in M_+$,

a\ne eae\quad\Longrightarrow\quad\varphi(a)=+\infty.

Proof. Every finite positive element of $M$ belongs to $eMe$, by WS-02. Hence the finite cone of the restricted weight is exactly $F_\varphi$, now viewed inside the corner. Its complex span is ultraweakly dense in $eMe$, again by WS-02. This is precisely semifiniteness of the restriction. Increasing positive suprema in the corner agree with those in $M$, so normality is inherited. Finally, if $\varphi(a)<\infty$, then $a\in F_\varphi$ and $a=eae$; taking the contrapositive gives the assertion. $\square$

The implication is about all finite elements. It does not say that every positive element of $eMe$ has finite weight.

Compression detects semifiniteness

By the cutoff theorem WG-008, a weight of any kind is semifinite exactly when some increasing net of finite positive contractions has supremum $1$. The criterion below tests semifiniteness in another way. It compresses an arbitrary positive element by finite-weight positive contractions that act as a unit on the norm closure of the definition algebra, and it asks whether infinite weight always shows in the compressed weights. For a normal weight this happens exactly when the weight is semifinite. Normality enters through the net form of ultraweak lower semicontinuity in NW11.

Let $A_\varphi$ be the norm closure of $\mathfrak m_\varphi$ in $M$. It lies in $eMe$ by WS-02, and it is self-adjoint because $\mathfrak m_\varphi$ is. In this subsection a positive contractive approximate unit is a net $(a_i)_{i\in I}$ in $F_\varphi$ with $0\leq a_i\leq1$ such that

\|a_iz-z\|\to0 \quad\text{and}\quad \|za_i-z\|\to0 \qquad(z\in A_\varphi).

Both limits are operator-norm limits for elements of $A_\varphi$. The net need not increase, and nothing is assumed about convergence to $1$ in $M$.

Lemma. Let $\varphi$ be an arbitrary weight, with finite-domain projection $e$.

(a) The net $u_a=a(1+a)^{-1}$ of WS-02, indexed by $a\in F_\varphi$ with the operator order, is a positive contractive approximate unit. More precisely, if $b\in F_\varphi$, $t>0$ and $a\geq tb$, then

\|(1-u_a)b\|=\|b(1-u_a)\|\leq\left(\frac{\|b\|}{t}\right)^{1/2}.

(b) Every positive contractive approximate unit converges strongly to $e$, and hence sigma-strongly and ultraweakly.

Proof. (a) By WS-02 the index set is directed, and each $u_a$ is a positive contraction of finite weight. Fix an index $a$ and put $c=(1+a)^{-1/2}$. Functional calculus gives $0\leq c\leq1$, $1-u_a=c^2$ and $cac=a(1+a)^{-1}=u_a\leq1$. Conjugation by $c$ preserves order: if $p\leq q$, then $c(q-p)c=\bigl((q-p)^{1/2}c\bigr)^*(q-p)^{1/2}c\geq0$. Now suppose $a\geq tb$. Then $0\leq t\,cbc\leq cac\leq1$, so $\|cbc\|\leq1/t$, and the identity $\|y\|^2=\|y^*y\|$ with $y=b^{1/2}c$ gives $\|b^{1/2}c\|^2=\|cbc\|\leq1/t$. Factor $(1-u_a)b=c\,(cb^{1/2})\,b^{1/2}$. Since $\|c\|\leq1$ and $\|cb^{1/2}\|=\|b^{1/2}c\|$, this product has norm at most $t^{-1/2}\|b\|^{1/2}$. The element $b(1-u_a)$ is its adjoint and has the same norm. Because $tb\in F_\varphi$ is itself an index and $t>0$ is arbitrary, $u_ab\to b$ and $bu_a\to b$ in norm. Linearity extends both limits to $\mathfrak m_\varphi=\operatorname{span}_{\mathbb C}F_\varphi$ (WG003). For $z\in A_\varphi$ and $z'\in\mathfrak m_\varphi$, the bound $\|1-u_a\|\leq1$ gives $\|(1-u_a)z\|\leq\|z-z'\|+\|(1-u_a)z'\|$, and similarly on the right; choosing $z'$ close to $z$ completes (a). When $F_\varphi=\{0\}$, the net has the single member $u_0=0$, and $A_\varphi=\{0\}$.

(b) Let $(a_i)$ be a positive contractive approximate unit, $\xi\in H$ and $\varepsilon>0$. The net $(u_a)$ increases strongly to $e$ by WS-02 and BK04, so some $a\in F_\varphi$ has $\|(e-u_a)\xi\|\leq\varepsilon$. Each $a_i$ lies in $F_\varphi$, so WS-02 gives $a_i=ea_ie$, and therefore $a_ie=a_i$. Hence

(a_i-e)\xi=a_i(e-u_a)\xi+(a_iu_a-u_a)\xi+(u_a-e)\xi .

The first and last terms have norm at most $\varepsilon$. The middle term tends to zero, since $u_a\in F_\varphi\subseteq A_\varphi$. Thus $\limsup_i\|(a_i-e)\xi\|\leq2\varepsilon$ for every $\varepsilon>0$, which is strong convergence to $e$. The net is bounded by $1$, so the topology facts in WS-01 and BK03 give the sigma-strong and ultraweak limits. $\square$

Proposition. For a normal weight $\varphi$, the following conditions are equivalent:

$\varphi$ is semifinite, so $e=1$.
For every positive contractive approximate unit $(a_i)$ and every $x\in M_+$, one has $\varphi(x)\leq\liminf_i\varphi(a_ixa_i)$.
For every positive contractive approximate unit $(a_i)$ and every $x\in M_+$ with $\varphi(x)=+\infty$, the net $\varphi(a_ixa_i)$ tends to $+\infty$.
For every positive contractive approximate unit $(a_i)$ and every $x\in M_+$ with $\varphi(x)=+\infty$, one has $\sup_i\varphi(a_ixa_i)=+\infty$.

Every compressed weight is finite. Indeed, $x\leq\|x\|1$ gives $a_ixa_i\leq\|x\|a_i^2$, and $a_i^2\leq a_i$ because $0\leq a_i\leq1$. Hence $\varphi(a_ixa_i)\leq\|x\|\varphi(a_i)<\infty$. Conditions 3 and 4 therefore say that finite numbers exceed every bound, eventually or at some index; no arithmetic with $+\infty$ occurs.

Proof. We prove $1\Rightarrow2\Rightarrow3\Rightarrow4\Rightarrow1$.

1 implies 2. Part (b) of the lemma, with $e=1$, gives $a_i\to1$ strongly. For $\xi\in H$,

\|(a_ixa_i-x)\xi\| \leq\|a_ix(a_i-1)\xi\|+\|(a_i-1)x\xi\| \leq\|x\|\,\|(a_i-1)\xi\|+\|(a_i-1)x\xi\|,

and the right side tends to zero. The positive elements $a_ixa_i$ have norm at most $\|x\|$, so they converge ultraweakly to $x$ by BK03. Lower semicontinuity, NW11, gives $\varphi(x)\leq\liminf_i\varphi(a_ixa_i)$.

2 implies 3. If $\varphi(x)=+\infty$, condition 2 makes the lower limit infinite. Thus for every finite $K$ there is an index $i_0$ with $\varphi(a_ixa_i)>K$ for all $i\geq i_0$, which is condition 3.

3 implies 4. A net that eventually exceeds every finite bound has supremum $+\infty$.

4 implies 1. Use the approximate unit $(u_a)$ of part (a) and the element $x=1-e$. Each $u_a$ lies in $F_\varphi$, so WS-02 gives $u_a=u_ae$; every compression $u_a(1-e)u_a$ is therefore zero, and the supremum of the compressed weights is $0$. Condition 4 thus rules out $\varphi(1-e)=+\infty$. Hence $1-e\in F_\varphi$, and WS-02 gives $1-e=e(1-e)e=0$. So $e=1$, which is semifiniteness by WS-02. $\square$

Normality was used only in the first step. The steps $2\Rightarrow3\Rightarrow4\Rightarrow1$ hold for every weight, so for an arbitrary weight each of conditions 2, 3 and 4 already implies semifiniteness.

For every normal weight, semifinite or not, and every positive contractive approximate unit, elements of finite weight satisfy the inequality of condition 2. If $x\in F_\varphi$, then $x\in A_\varphi$, and

\|a_ixa_i-x\| \leq\|a_i(xa_i-x)\|+\|a_ix-x\| \leq\|xa_i-x\|+\|a_ix-x\|\to0 .

Norm convergence implies ultraweak convergence, so NW11 applies again. The criterion is therefore a statement about elements of infinite weight. The compressions $a_ixa_i$ need not increase with $i$ and need not lie below $x$. For this reason the proof uses lower semicontinuity rather than the monotone form of normality.

The implication from 1 to 4 fails without normality. Take the semifinite nonnormal weight $\theta$ on $\ell^\infty(\mathbb N)$ from WS-08, Problem 4. Its finite cone is $c_0(\mathbb N)_+$. Every element of $c_0(\mathbb N)$ is a combination of four elements of this cone, so $\mathfrak m_\theta=c_0(\mathbb N)$ and $A_\theta=c_0(\mathbb N)$. The projections $q_n$ onto the first $n$ coordinates have weight zero, and

\|q_nz-z\|=\|zq_n-z\|=\sup_{k>n}|z_k|\longrightarrow0 \qquad(z\in c_0(\mathbb N)).

They therefore form a positive contractive approximate unit. For $x=1$, every compression has weight $\theta(q_n1q_n)=\theta(q_n)=0$, although $\theta(1)=+\infty$. Condition 4 fails, and conditions 2 and 3 fail with it. This agrees with WG-008: the same projections increase to $1$ and are finite cutoffs of the kind required there.

The same criterion, for approximate units of the definition algebra, is treated in Matt Daws, Some notes on weights, September 2024, Section 2.1.

Normality produces a largest null projection

Assume from now on that $\varphi$ is normal.

Theorem. There is a largest projection $f\in M$ with $\varphi(f)=0$. It satisfies $f\leq e$, and

N_\varphi=M f.

For $a\in M_+$, the following are equivalent:

\varphi(a)=0,\qquad s(a)\leq f,\qquad a=faf.

Proof. If $a\geq0$ and $\varphi(a)=0$, its cutoff

v_t=t a(1+t a)^{-1}

satisfies $0\leq v_t\leq t a$, and hence $\varphi(v_t)=0$. The cutoffs increase to $s(a)$. Normality yields $\varphi(s(a))=0$.

If projections $q_1,q_2$ have weight zero, then $q_1+q_2$ has weight zero, so $s(q_1+q_2)$ does also. This support is their join: its kernel is $\ker q_1\cap\ker q_2$, as follows from

\langle(q_1+q_2)\xi,\xi\rangle =\|q_1\xi\|^2+\|q_2\xi\|^2.

Therefore finite joins of null projections are null. Their increasing net has a strong supremum $f\in M$. A bounded strong limit of increasing projections is a projection: it fixes the range of every net member and vanishes on the orthogonal complement of the closed union of their ranges. Normality gives $\varphi(f)=0$. By construction every null projection lies below $f$, so it is the largest one.

If $\varphi(a)=0$, the first paragraph gives $s(a)\leq f$, or equivalently $a=faf$. Conversely, $a=faf\geq0$ implies $a\leq\|a\|f$, hence $\varphi(a)=0$. This proves the positive-cone assertions. Since $f$ itself is finite, WS-02 gives $f=efe$, so $f\leq e$.

If $x\in N_\varphi$, apply the positive-cone assertion to $x^*x$. It gives $x(1-f)=0$, or $x=xf$. Conversely, $x=xf$ implies $x^*x\leq\|x\|^2f$, so $x\in N_\varphi$. Hence $N_\varphi=Mf$. $\square$

The resulting null ideal is ultraweakly and strongly closed. Normality is essential to this conclusion; WS-08 contains a nonnormal counterexample.

Removing null directions without subtracting infinities

Put $r=1-f$.

Theorem. For every $a\in M_+$,

\varphi(a)=\varphi(rar).

The restriction of $\varphi$ to $rMr$ is faithful and normal. It is not automatically semifinite.

Proof. We first record an inequality that remains valid for infinite energies. For arbitrary $x,y\in M$ and $\varepsilon>0$,

(x+y)^*(x+y) \leq(1+\varepsilon)x^*x+(1+\varepsilon^{-1})y^*y.

It follows by expanding the positivity of $(\varepsilon^{1/2}x-\varepsilon^{-1/2}y)^* (\varepsilon^{1/2}x-\varepsilon^{-1/2}y)$. If $y\in N_\varphi$, monotonicity and additivity imply

\varphi((x+y)^*(x+y)) \leq(1+\varepsilon)\varphi(x^*x).

Applying the same inequality to $x=(x+y)-y$ gives the reverse comparison with the same factor. If either weight is infinite, the comparisons force both to be infinite. If they are finite, letting $\varepsilon$ decrease to zero proves equality. No difference of infinite numbers is taken.

For a positive $a$, choose $x=a^{1/2}r$ and $y=a^{1/2}f$. Since $y\in Mf=N_\varphi$, the preceding equality gives

\varphi(a)=\varphi((x+y)^*(x+y)) =\varphi(x^*x)=\varphi(rar).

If $a\in(rMr)_+$ has weight zero, WS-04 gives $a=faf$. Because also $a=rar$ and $fr=0$, we obtain $a=0$. This is faithfulness. Normality is inherited under restriction. No semifiniteness conclusion follows solely from removing the null ideal; the everywhere-infinite example in WS-08 has $r=1$ and is not semifinite. $\square$

We call $r$ the null-carrier projection to keep it distinct from the support convention in the next statement.

The faithful semifinite support corner

Theorem. Let $\varphi$ be a normal weight, let $e,f$ be the projections constructed above, and put

p=e-f.

Then $p$ is a projection, and the restriction $\varphi_p$ to $pMp$ is normal, semifinite and faithful. On all of $M_+$, the exact reconstruction rule is

\varphi(a)= \begin{cases} \varphi_p(pap),&a=eae,\\ +\infty,&a\ne eae. \end{cases}

For a normal semifinite weight, $e=1$, so $p=1-f$ and

\varphi(a)=\varphi_p(pap)\qquad(a\in M_+).

Proof. We know $f\leq e$, hence $p=e-f=e(1-f)=(1-f)e$ is a projection. It lies under $r=1-f$, so the restriction is faithful by WS-05 and normal by restriction.

Use the increasing finite positive contractions $u_a$ from WS-02. The compressions $p u_a p$ increase strongly to $p$, are positive contractions in the corner, and have finite weight. Indeed, $u_a=e u_a e$, so

p u_a p=r u_a r,\qquad \varphi(pu_ap)=\varphi(u_a)<\infty

by WS-05. WG-008 applied inside $pMp$, whose identity is $p$, proves semifiniteness. This argument also covers the zero corner.

For $a=eae$, one has $rar=pap$, so WS-05 gives the finite-corner formula, whether its value is finite or infinite. For $a\ne eae$, WS-03 gives the other line. Finally, semifiniteness is equivalent to $e=1$, by WS-02. $\square$

We use support of the normal weight for $p=e-f$, the finite-domain convention stated above. If a discussion instead defines support as the complement of the maximal null projection, its projection is $r$. The convention must be checked before treating results about arbitrary normal weights as identical. For the normal semifinite numerator in spatial derivative theory, the two conventions agree and WS-06 supplies the required support-corner reduction.

Order information from domination

Proposition. Let $\varphi,\psi$ be normal weights and suppose

\psi(a)\leq c\varphi(a)\qquad(a\in M_+)

for some finite $c>0$. Denote their finite-domain projections by $e_\varphi,e_\psi$, their null projections by $f_\varphi,f_\psi$, and their null carriers by $r_\varphi,r_\psi$. Then

e_\varphi\leq e_\psi,\qquad f_\varphi\leq f_\psi,\qquad r_\psi\leq r_\varphi.

If both weights are semifinite, their support projections satisfy $p_\psi\leq p_\varphi$.

Proof. Finiteness of $\varphi(a)$ implies finiteness of $\psi(a)$, so $F_\varphi\subseteq F_\psi$. The description of the ranges of the finite-domain projections in WS-02 gives $e_\varphi\leq e_\psi$. A $\varphi$-null projection is $\psi$-null by domination, so maximality gives $f_\varphi\leq f_\psi$. Taking complements gives the null-carrier inequality. When both weights are semifinite, $p_\varphi=r_\varphi$ and $p_\psi=r_\psi$, proving the final statement. $\square$

For arbitrary normal weights, the last support conclusion cannot be inferred from domination alone: their finite-domain projections can differ. For example, any nonzero bounded normal positive functional is dominated by the everywhere-infinite weight, but the latter has effective support zero under the $e-f$ convention.

Problems with complete solutions

Problem 1: a faithful weight with effective support zero. On a nonzero von Neumann algebra define $\varphi_\infty(0)=0$ and $\varphi_\infty(a)=+\infty$ for every nonzero $a\in M_+$. Determine all three projections $e,f,p$ and the null carrier $r$.

Solution. A sum of positive elements is zero exactly when both are zero. This proves additivity, and positive homogeneity is immediate with $0\cdot\infty=0$. The weight is normal: if an increasing positive net has nonzero supremum, at least one member is nonzero, so the supremum of its weights is infinite. Its only finite positive element is zero, giving $e=0$. Its only null positive element is zero, giving $f=0$. Thus $p=e-f=0$ while $r=1-f=1$. The weight is faithful and normal but not semifinite. There is no contradiction: effective support and null carrier record different domain information.

Problem 2: a noncentral support and a faithful GNS representation. Let $M=M_3(\mathbb C)$ and $\varphi(a)=a_{22}$ for $a\geq0$. Compute $e,f,p$, and determine whether the semicyclic representation is faithful.

Solution. This is a bounded normal positive functional. Every positive element is finite, so $e=1$. Its null ideal consists of matrices whose second column is zero, since

\varphi(x^*x)=\sum_{j=1}^3|x_{j2}|^2.

Consequently $f=1-E_{22}$ and $p=E_{22}$, a noncentral projection. The GNS quotient is isometric to $\mathbb C^3$ via the second-column map, which is surjective. Left multiplication by $a$ becomes the usual action $v\mapsto av$, so the representation is faithful. Thus the support projection of a weight is not the complement of the kernel of its GNS representation; the latter kernel is a two-sided ideal, while the null ideal here is only a left ideal.

Problem 3: a noncentral finite-domain projection. In $M_3(\mathbb C)$, fix $e_0=E_{11}+E_{22}$ and define

\varphi(a)= \begin{cases} \operatorname{Tr}(a),&a=e_0ae_0,\\ +\infty,&a\ne e_0ae_0 \end{cases} \qquad(a\geq0).

Show that this is a faithful normal weight and compute its projections.

Solution. For positive $a,b$, the sum is supported in $e_0$ if and only if both are. To prove the nontrivial direction, compress $a+b$ by $1-e_0$: positivity forces each compressed diagonal to vanish, and the square-root identity then makes $a(1-e_0)=b(1-e_0)=0$. Additivity follows using finite trace inside the corner and infinite values elsewhere. Homogeneity follows by rescaling.

For normality, if the supremum of an increasing positive net belongs to the corner, every member does, and finite-dimensional trace preserves the supremum. Here is the needed trace argument: write $a_i\uparrow a$. By BK04 the net converges strongly, so for each standard basis vector $v_j$, the numbers $\langle a_i v_j,v_j\rangle$ increase to $\langle a v_j,v_j\rangle$. A finite sum commutes with this supremum: for any positive error choose an index for each of the three summands, then a common upper bound for those indices. Therefore

$\sup_i\operatorname{Tr}(a_i) =\sum_{j=1}^{3}\langle a v_j,v_j\rangle =\operatorname{Tr}(a).$ If the supremum is outside the corner, some member is outside it, since the corner is strongly closed. In that case both sides of the normality identity are infinite. The only positive zero of the weight is zero, so it is faithful. Its finite cone is $(e_0Me_0)_+$; hence $e=e_0$, $f=0$, $p=e_0$, and $r=1$. In particular, $e$ need not be central even in a factor. The weight is not semifinite on $M$, although its restriction to $e_0Me_0$ is a faithful finite trace.

Problem 4: why normality was used. On $\ell^\infty(\mathbb N)$, set $\theta(a)=0$ for $a\in c_0(\mathbb N)_+$ and $\theta(a)=+\infty$ for other positive $a$. Prove that it is a semifinite nonnormal weight without a largest null projection.

Solution. The positive cone of $c_0$ is additive, closed under nonnegative scaling, and hereditary in $\ell^\infty_+$. The auxiliary-cone construction in WG-003 therefore gives a weight. The finite-support coordinate projections $q_n$, equal to one in the first $n$ coordinates, have finite weight and increase to $1$. WG-008 makes the weight semifinite. But $\theta(q_n)=0$ while $\theta(1)=+\infty$, so it is not normal.

Every coordinate projection is null. A projection dominating them all must be $1$, which is not null. Thus no largest null projection exists. Equivalently, its null ideal is $c_0$, which is not ultraweakly closed. Indeed, for a bounded sequence $x=(x_k)$, the condition $x^*x\in c_0$ says $|x_k|^2\to0$, which is equivalent to $x_k\to0$. The null projections $q_n$ converge strongly and ultraweakly to $1$, outside this ideal. This shows precisely why the null-projection construction in WS-04 required normality.

The spatial support contract now supplied

For a normal semifinite weight $\varphi$, this unit proves the existence of its support $p$, the identity $\varphi(a)=\varphi(pap)$ on all positive elements, and the normal semifinite faithful restriction to $pMp$. These are the assertions required by SD-DEP-SUPPORT. They use no countable-decomposability hypothesis and do not require the original representation to contain a cyclic vector.

The result concerns this particular support corner. Semifiniteness of the restriction to every projection corner is a different assertion and is false in general, as the infinite-weight rank-one corner in WG013 shows. Nor does this support reduction establish any spatial form's density, closability, core property, or modular covariance.