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

Finite domains, null directions, and support corners

OA-MOD-WS-01 — 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 OA-MOD-WG-002 through OA-MOD-WG-005. Inner products are linear in the first variable.

OA-MOD-WS-02 and OA-MOD-WS-03 allow an arbitrary weight. All later weight assertions explicitly assume normality, in the sense of preservation of bounded increasing positive suprema.

The proofs use the finite-domain algebra from WG-003, bounded inverse order, monotone nets, support cutoffs and topology facts from OA-MOD-BK-03 through OA-MOD-BK-06, and the finite-cutoff characterization of semifiniteness from WG-008. These are bounded-operator and Hilbert prerequisites; 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.

OA-MOD-WS-02 — 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.

OA-MOD-WS-03 — 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.

OA-MOD-WS-04 — 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.

OA-MOD-WS-05 — 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.

OA-MOD-WS-06 — 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 $p=e-f$ when following the stated Takesaki convention. 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.

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