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

Weights and the Hilbert spaces of multiplication

OA-MOD-WH-01 — Interfaces and analytic prerequisites

All Hilbert spaces and index sets are arbitrary. Inner products are linear in the first variable. A von Neumann algebra is concrete and unital; the zero algebra is allowed. A normal weight preserves bounded increasing positive suprema, and semifiniteness means ultraweak density of its finite definition algebra, as in WG-002.

We use the following results with their stated hypotheses:

• WG-003–010: the finite ideals, their positive cone, GNS construction, finite positive contraction nets, GNS density of the finite-star domain, and faithfulness and normality of the GNS representation.
• NW-11 and OW-02–03: recovery of a normal weight from its dominated normal positive functionals, and the exact implementing vectors for a normal semifinite weight.
• HA-01–08 and RD-02–07: the two bounded multiplication constructions, closed involutions, right-algebra graph density, commutant generation, product cores and the adjoint intersection of the multiplication ideal.
• BK-02–08 and DW-02: bicommutants, bounded operator topologies, positive square roots, inverse order, support cutoffs, and polar and Douglas factorizations inside von Neumann algebras, also for rectangular matrices.
• CP and NP-04–06: concrete Banach preduals and ultraweak continuity of positive normal maps. These do not themselves include the image-closure theorem proved in WH-02.
• OA-MOD-OPEN-CONVEX-ALAOGLU, OA-MOD-OPEN-CONVEX-HILBERT-BALL and OA-MOD-OPEN-CONVEX-KREIN-SMULIAN: dual norm balls are weak-star compact; Hilbert norm balls are weakly compact; a convex subset of a Banach dual is weak-star closed if its intersections with all centered norm balls are weak-star closed.

The last statement is used only in WH-02. The direct normality proof in WH-07 uses Hilbert weak compactness, with subnets rather than a sequential compactness assertion. The unbounded spectral contract remains a transitive prerequisite of RD's graph-density theorem; this unit does not replace that contract with a bounded spectral theorem.

We shall use strong continuity of positive square roots on a uniformly bounded positive set. Here is the needed argument. If $0\leq a_i,a\leq C1$ and $a_i\to a$ strongly, fixed powers and therefore fixed polynomials converge strongly, by induction and the uniform operator bounds. Uniform polynomial approximation of $t^{1/2}$ on $[0,C]$, followed by bounded continuous functional calculus, makes the errors in both square roots uniformly small in operator norm. Applying this to each vector proves $a_i^{1/2}\to a^{1/2}$ strongly. This works for arbitrary nets.

OA-MOD-WH-02 — The image of a faithful normal representation

Lemma. If $\pi:M\to B(H)$ is a faithful normal *-representation of a von Neumann algebra, then $\pi(M)$ is an ultraweakly closed -subalgebra with identity $p=\pi(1)$. It acts as a von Neumann algebra on $pH$, and as zero on $(I-p)H$. The map $\pi$ is an ultraweak and sigma-strong homeomorphism onto its image, with the inherited corner topologies. When $\pi$ is unital, $p=I_H$.

Proof. First $\pi$ is isometric. Contractivity follows from positivity and the C*-identity. To check the reverse inequality without assuming that its image is closed, let $b\in M_+$. If $\|\pi(b)\|<\|b\|$, choose a continuous function on $[0,\|b\|]$ vanishing on $[0,\|\pi(b)\|]$ but nonzero at $\|b\|$. Continuous functional calculus gives $f(b)\ne0$, since the norm of a positive element belongs to its spectrum, but $\pi(f(b))=f(\pi(b))=0$, contradicting faithfulness. Applying this to $b=x^*x$ proves isometry.

For every $r>0$, the closed ball $rB_M$ is compact for $\sigma(M,M_*)$, by Alaoglu and the concrete predual. Normality makes $\pi$ ultraweakly continuous, by NP-04. Its image $\pi(rB_M)$ is therefore ultraweakly compact and closed in $B(H)$. Isometry gives $$\pi(rB_M)=\pi(M)\cap rB_{B(H)}. \tag{WH.1}$$ The linear subspace $\pi(M)$ is convex. Apply Krein–Smulian in the Banach dual $B(H)$ to (WH.1): $\pi(M)$ is ultraweakly closed. Multiplicativity gives $\pi(x)=p\pi(x)p$, with identity $p=\pi(1)$. Thus the image is a von Neumann algebra on $pH$; the inherited ultraweak and sigma-strong* topologies are exactly the corner topologies, since compressing the test vectors by $p$ gives the same seminorms and functionals. In the nonunital case the continuous function used above vanishes at zero, so its functional-calculus identity also holds in this corner.

Both $\pi$ and its inverse are now positive -isomorphisms between von Neumann algebras. They transport order bounds in both directions, so preserve existing bounded increasing positive suprema. NP-04 and NP-06 give ultraweak and sigma-strong continuity in both directions. The identity assertion follows by evaluating $\pi(1)$. $\square$

The weak-star closed-ball argument is essential here. A merely bounded positive normal map can have a nonclosed image, as NP-08 shows.

OA-MOD-WH-03 — Completing the two multiplication domains

Let $\mathcal A\subseteq H$ be any left Hilbert algebra. Retain $$M=L(\mathcal A)'',\qquad S=\overline{\sharp},\qquad F=S^*,$$ and the right-bounded space $\mathcal B_r$, its operators $R_\eta$, and the right algebra $$\mathcal A_r=\mathcal B_r\cap D(F). \tag{WH.2}$$ By RD-04–05, $\mathcal A_r$ is a right Hilbert algebra, its closed involution is $F$, and $R(\mathcal A_r)''=M'$.

Define the left-bounded space relative to this right algebra by $$\mathcal B_l= \left\{\xi\in H:\ \exists C<\infty\ \forall\eta\in\mathcal A_r,\ \|R_\eta\xi\|\leq C\|\eta\|\right\}. \tag{WH.3}$$ For $\xi\in\mathcal B_l$, let $\lambda_\xi$ be the unique bounded operator satisfying $$\lambda_\xi\eta=R_\eta\xi\qquad(\eta\in\mathcal A_r). \tag{WH.4}$$ Put $$\mathfrak n_l=\{\lambda_\xi:\xi\in\mathcal B_l\}, \qquad \mathcal A_l=\mathcal B_l\cap D(S). \tag{WH.5}$$

Theorem. The map $\lambda:\mathcal B_l\to M$ is linear and injective, and $$x\xi\in\mathcal B_l,\qquad \lambda_{x\xi}=x\lambda_\xi \quad(x\in M,\ \xi\in\mathcal B_l). \tag{WH.6}$$ Thus $\mathfrak n_l$ is a left ideal in $M$. Moreover, $$\mathcal A\subseteq\mathcal A_l,\qquad \lambda_a=L_a\quad(a\in\mathcal A). \tag{WH.7}$$ The space $\mathcal A_l$ is a left Hilbert algebra with $$\xi\zeta=\lambda_\xi\zeta,\qquad \xi^\sharp=S\xi,\qquad \lambda_{S\xi}=\lambda_\xi^*, \tag{WH.8}$$ and $$\overline{S|_{\mathcal A_l}}=S,\qquad \lambda(\mathcal A_l)''=M,\qquad \lambda(\mathcal A_l)=\mathfrak n_l\cap\mathfrak n_l^*. \tag{WH.9}$$

Proof. Apply HA and RD to the opposite algebra of $\mathcal A_r$, whose product is $\eta\circ\zeta=\zeta\eta$. Its left multiplication is $R_\eta$, its generated algebra is $M'$, its closed involution is $F$, and the adjoint involution is $F^*=S$. The right-bounded-vector construction in that application is exactly (WH.3–4): the test operator for $\xi$ sends $\eta$ to $R_\eta\xi$. HA-05 therefore proves linearity, injectivity, membership in $(M')'=M$, and (WH.6).

For $a\in\mathcal A$ and $\eta\in\mathcal A_r$, $R_\eta a=L_a\eta$, so $a\in\mathcal B_l$ and $\lambda_a=L_a$. Also $a\in D(S)$, proving (WH.7).

The right algebra in the application to $\mathcal A_r^{\mathrm{op}}$ has product $\xi\circ\zeta=\lambda_\zeta\xi$. Taking its opposite makes the product $\xi\zeta=\lambda_\xi\zeta$, as asserted. HA-08 and RD-04 give the Hilbert algebra axioms and closed involution $S$. RD-05 gives the generated algebra $(M')'=M$. RD-07 gives precisely the ideal intersection in (WH.9). These are applications to a fully established right Hilbert algebra, not an assumption that $\mathcal A$ was already full. $\square$

OA-MOD-WH-04 — Mixed bounded vectors and fullness

Mixed-product lemma. For every $\xi\in\mathcal B_l$ and every $\eta\in\mathcal B_r$, $$\lambda_\xi\eta=R_\eta\xi. \tag{WH.10}$$ The right algebra obtained by dualizing $\mathcal A_l$ is exactly $\mathcal A_r$, with the same products, involution and right operators.

Proof. Choose the positive contractions $e_i\in R(\mathcal A_r)$ of RD-05, with $e_i\to I$ strongly. Because each $e_i$ is self-adjoint, it has the form $R_{\alpha_i}^*$ with $\alpha_i\in\mathcal A_r$. For $\eta\in\mathcal B_r$, HA-08 gives $$\eta_i=e_i\eta\in\mathcal A_r,\qquad R_{\eta_i}=e_iR_\eta.$$ Thus $\eta_i\to\eta$ in norm and $R_{\eta_i}\to R_\eta$ strongly. For $\xi\in\mathcal B_l$, (WH.4) gives $\lambda_\xi\eta_i=R_{\eta_i}\xi$; the limits prove (WH.10).

The closed involution of $\mathcal A_l$ is $S$, so the adjoint-domain condition for its right algebra is membership in $D(F)$. If $\eta$ belongs to that right algebra, restricting its right-boundedness inequality to $\mathcal A\subseteq\mathcal A_l$ puts $\eta$ in $\mathcal B_r$. Therefore $\eta\in\mathcal A_r$. Conversely, if $\eta\in\mathcal A_r$, then for every $\xi\in\mathcal A_l$, $$\|\lambda_\xi\eta\|=\|R_\eta\xi\| \leq\|R_\eta\|\|\xi\|.$$ Hence it is right bounded for $\mathcal A_l$; it is already in $D(F)$, and its right operator is $R_\eta$. The involution and products consequently agree. $\square$

We call a left Hilbert algebra full when it equals the left algebra obtained by this two-step dualization. Since the right algebra of $\mathcal A_l$ is $\mathcal A_r$, its next left dual is again $\mathcal A_l$. Thus $\mathcal A_l$ is full. The original $\mathcal A$ is a graph core in this completion, because both closed involutions are $S$; equality need not hold.

OA-MOD-WH-05 — Measuring square roots by vectors

Keep an arbitrary $\mathcal A$ and the completed multiplication spaces of WH-03. For $a\in M_+$, define $$\psi(a)= \begin{cases} \|\xi\|^2,&a^{1/2}=\lambda_\xi\text{ for some }\xi\in\mathcal B_l,\\ +\infty,&a^{1/2}\notin\mathfrak n_l. \end{cases} \tag{WH.11}$$ Injectivity of $\lambda$ makes the finite value unambiguous. In the finite case, $\lambda_\xi$ is self-adjoint, so (WH.9) implies $\xi\in\mathcal A_l$ and $S\xi=\xi$.

Finite-cone theorem. The set $$P=\{a\in M_+:a^{1/2}\in\mathfrak n_l\} \tag{WH.12}$$ is additive, positively homogeneous and hereditary. Its finite value in (WH.11) is additive and positively homogeneous, and is increasing for the positive order.

Proof of heredity. If $0\leq b\leq a\in P$, Douglas factorization in $M$ gives $c\in M$, $\|c\|\leq1$, with $b^{1/2}=ca^{1/2}$. If $a^{1/2}=\lambda_\xi$, covariance gives $b^{1/2}=\lambda_{c\xi}$. Thus $b\in P$ and $$\psi(b)=\|c\xi\|^2\leq\|\xi\|^2=\psi(a). \tag{WH.13}$$

Proof of additivity. Suppose $a,b\in P$, with square-root vectors $\xi,\eta$. Write $t=(a+b)^{1/2}$. The polar decomposition of the column operator $$\binom{a^{1/2}}{b^{1/2}}=\binom{u}{v}t \tag{WH.14}$$ lies in the rectangular matrix algebra over $M$. Its initial projection is $p=s(t)$, so $$u^*u+v^*v=p,\qquad up=u,\quad vp=v,\qquad a^{1/2}=ut,\quad b^{1/2}=vt.$$ In particular $u^*a^{1/2}+v^*b^{1/2}=pt=t$. Define $\zeta=u^*\xi+v^*\eta$. Covariance and linearity yield $$\lambda_\zeta=t.$$ Thus $a+b\in P$. Applying $\lambda$ to $u\zeta$, $v\zeta$ and $p\zeta$, then using injectivity, gives $$u\zeta=\xi,\qquad v\zeta=\eta,\qquad p\zeta=\zeta.$$ Consequently $$\psi(a)+\psi(b)=\|u\zeta\|^2+\|v\zeta\|^2 =\langle p\zeta,\zeta\rangle=\|\zeta\|^2 =\psi(a+b). \tag{WH.15}$$ This norm identity uses the polar support; a generic pair of contractions would not suffice.

For $r>0$, the square-root vector of $ra$ is $\sqrt r\,\xi$. This proves homogeneity and preservation of the finite cone under nonzero rescaling. The zero operator has the unique vector zero. Thus $\psi(0)=0$, and homogeneity at zero uses $0\cdot\infty=0$. $\square$

If a positive sum belongs to $P$, heredity puts each summand in $P$. Therefore, if either summand has infinite value, the sum does also. The finite and infinite cases together prove that (WH.11) is a weight.

OA-MOD-WH-06 — The exact finite ideals and GNS norm

Theorem. The weight in (WH.11) satisfies $$\mathfrak n_\psi=\mathfrak n_l,\qquad \psi(x^*x)=\|\xi\|^2\quad\text{when }x=\lambda_\xi. \tag{WH.16}$$ Its definition algebra is $$\mathfrak m_\psi=\operatorname{span}\mathfrak n_l^*\mathfrak n_l, \qquad (\mathfrak m_\psi)_+=P. \tag{WH.17}$$ For $x=\lambda_\xi$, $y=\lambda_\eta$, $$\widetilde\psi(y^*x)=\langle\xi,\eta\rangle. \tag{WH.18}$$

Proof. Let $x=\lambda_\xi\in\mathfrak n_l$, and write its bounded polar decomposition $x=w|x|$. Covariance gives $$|x|=w^*x=\lambda_{w^*\xi}.$$ Thus $x^*x\in P$, so $x\in\mathfrak n_\psi$. The final support $q=ww^*$ satisfies $qx=x$, and injectivity gives $q\xi=\xi$. Hence $$\psi(x^*x)=\|w^*\xi\|^2=\langle q\xi,\xi\rangle=\|\xi\|^2.$$ Conversely, if $\psi(x^*x)<\infty$, then $|x|=\lambda_\zeta$ for some $\zeta$. The same polar decomposition gives $x=\lambda_{w\zeta}$, proving (WH.16).

The algebra and positive-cone identities in (WH.17) are now precisely WG-003 applied to the weight already proved in WH-05. Polarization of (WH.16), using linearity of $\lambda$, gives (WH.18). All evaluations of $\widetilde\psi$ lie in its finite algebra; no complex extension of $\psi$ to arbitrary elements of $M$ is used. $\square$

These statements also prove the hereditary-cone and ideal assertions of source Lemma VII.2.1. Applying the same construction to the opposite right algebra gives the corresponding right-ideal assertions.

OA-MOD-WH-07 — Arbitrary-net normality and semifiniteness

Theorem. The weight $\psi$ is faithful, normal and semifinite.

Proof of normality. Suppose $0\leq a_i\uparrow a$ is a bounded increasing net in $M$. Set $L=\sup_i\psi(a_i)$. Monotonicity already gives $L\leq\psi(a)$; if $L=\infty$, equality follows. Assume $L<\infty$. Then $$a_i^{1/2}=\lambda_{\xi_i},\qquad \|\xi_i\|^2=\psi(a_i)\leq L.$$ The closed Hilbert ball of radius $\sqrt L$ is weakly compact. Therefore this net has a subnet $\xi_{i(j)}$ converging weakly to some $\xi$ in the same ball. For each $\eta\in\mathcal A_r$, $$a_{i(j)}^{1/2}\eta=R_\eta\xi_{i(j)}. \tag{WH.19}$$ The left side converges in norm to $a^{1/2}\eta$, by WH-01. The right side converges weakly to $R_\eta\xi$, since $R_\eta$ is bounded. Thus $$R_\eta\xi=a^{1/2}\eta\qquad(\eta\in\mathcal A_r).$$ This is the boundedness criterion (WH.3), with constant $\|a^{1/2}\|$. Hence $\xi\in\mathcal B_l$, $\lambda_\xi=a^{1/2}$, and $\psi(a)=\|\xi\|^2\leq L$. Combining with the opposite inequality proves normality. Compactness was used for the actual arbitrary net; no separability or countable exhaustion was inserted.

Proof of faithfulness. If $\psi(a)=0$, its square-root vector is zero, so $a^{1/2}=\lambda_0=0$. Thus $a=0$.

Proof of semifiniteness. Let $\mathcal C=L(\mathcal A)$. It is a nondegenerate *-algebra by HA-02, and lies in $\mathfrak n_\psi$ by WH-03 and WH-06. For a finite subset $E\subseteq\mathcal C$ and $\varepsilon>0$, put $$b_E=\sum_{r\in E}r^*r,\qquad c_{E,\varepsilon}=b_E(b_E+\varepsilon I)^{-1}. \tag{WH.20}$$ Finite additivity makes $\psi(b_E)<\infty$. Since $0\leq c_{E,\varepsilon}\leq\varepsilon^{-1}b_E$, these are finite-weight positive contractions. Order the pairs by enlarging $E$ and decreasing $\varepsilon$. Inverse order proves that $c_{E,\varepsilon}$ increases.

For fixed $E$, its limit as $\varepsilon\downarrow0$ is $s(b_E)$. The common kernel of the $b_E$'s is the common kernel of $\mathcal C$, which is zero by nondegeneracy and *-closure. Their support ranges therefore span $H$. The strong supremum $c\leq I$ of the contraction net dominates every $s(b_E)$, so it acts as the identity on those ranges and hence equals $I$. We have obtained an increasing finite positive contraction net with supremum $I$. WG-008 proves semifiniteness. $\square$

Normality was proved before invoking any normal-weight characterization for $\psi$. In particular, the proof does not assume the source's predual-valued map or supremum formula in order to prove that $\psi$ is a weight.

OA-MOD-WH-08 — Recovering the representation and the full algebra

Let $(H_\psi,\pi_\psi,\Lambda_\psi)$ be the weight's GNS triple. Define on its dense range $$U\Lambda_\psi(\lambda_\xi)=\xi\qquad(\xi\in\mathcal B_l). \tag{WH.21}$$

Theorem. The map $U$ extends uniquely to a unitary $H_\psi\to H$, and $$U\pi_\psi(x)U^*=x\quad(x\in M). \tag{WH.22}$$ It identifies the finite-star algebra exactly: $$U\Lambda_\psi(\mathfrak n_\psi\cap\mathfrak n_\psi^*) =\mathcal A_l. \tag{WH.23}$$ Under this identification its product is $\lambda_\xi\eta$, its involution is $S$, and its closed involution has domain $D(S)$.

Proof. Equation (WH.18) makes (WH.21) a well-defined isometry. Its range is $\mathcal B_l$, which contains the dense space $\mathcal A$. Thus it extends to a surjective isometry. For $x\in M$, covariance gives $$U\pi_\psi(x)\Lambda_\psi(\lambda_\xi) =U\Lambda_\psi(x\lambda_\xi)=x\xi,$$ proving (WH.22) by density.

By WH-06 and (WH.9), $\mathfrak n_\psi\cap\mathfrak n_\psi^*=\lambda(\mathcal A_l)$. This proves (WH.23). If $\xi,\eta\in\mathcal A_l$, then $\lambda_\xi\lambda_\eta=\lambda_{\lambda_\xi\eta}$, by covariance; hence the product transports as stated. The equality $\lambda_\xi^*=\lambda_{S\xi}$ transports the involution. Its closure is $S$, by WH-03. $\square$

The construction recovers the full completion $\mathcal A_l$. It recovers the original algebra exactly when that algebra was full. A proper algebra core and its full completion can therefore determine the same weight.

OA-MOD-WH-09 — From a faithful normal semifinite weight to an algebra

Now start with a faithful normal semifinite weight $\varphi$ on $M$. Write its GNS triple as $(H,\pi,\Lambda)$, and put $$\mathfrak a_\varphi=\mathfrak n_\varphi\cap\mathfrak n_\varphi^*, \qquad \mathcal A_\varphi=\Lambda(\mathfrak a_\varphi). \tag{WH.24}$$ Faithfulness makes $\Lambda$ injective. Consequently $$\Lambda(x)\Lambda(y)=\Lambda(xy),\qquad \Lambda(x)^\sharp=\Lambda(x^*) \quad(x,y\in\mathfrak a_\varphi) \tag{WH.25}$$ are well-defined algebra operations. The domain $\mathfrak a_\varphi$ is a *-algebra: it is closed under linear combinations and adjoints, and $xy\in\mathfrak n_\varphi$, $(xy)^*=y^*x^*\in\mathfrak n_\varphi$, by the left-ideal property.

Proposition. The algebra $\mathcal A_\varphi$ is dense in $H$, has bounded left multiplication $$L_{\Lambda(x)}=\pi(x), \tag{WH.26}$$ satisfies the left adjoint identity, and has dense product span. Its generated von Neumann algebra is $\pi(M)$.

Proof. Density is WG-009. The GNS module identity gives (WH.26) on the dense domain and the bound $\|L_{\Lambda(x)}\|\leq\|x\|$. The adjoint identity follows from $\pi(x)^*=\pi(x^*)$.

Choose the finite positive contractions $e_i\uparrow1$ of WG-008. They belong to $\mathfrak a_\varphi$, since $e_i^2\leq e_i$. Normality gives $\pi(e_i)\to I$ strongly. For $x\in\mathfrak a_\varphi$, $$\Lambda(e_i)\Lambda(x)=\pi(e_i)\Lambda(x)\longrightarrow\Lambda(x).$$ Thus products are dense.

WG-007 and WG-010 make $\pi$ normal and faithful. WH-02 makes its image a von Neumann algebra. For any $a\in M$, $e_i a e_i\in\mathfrak m_\varphi\subseteq\mathfrak a_\varphi$ and $e_i a e_i\to a$ sigma-strong*, by bounded strong* convergence. NP-06 implies $\pi(e_i a e_i)\to\pi(a)$ strongly. Thus $\pi(\mathfrak a_\varphi)$ generates $\pi(M)$, proving the final assertion. $\square$

Only closability remains among the four Hilbert algebra axioms. It requires more than the normality of $\pi$.

OA-MOD-WH-10 — Closability on the full finite-star domain

Let $$\Phi_\varphi=\{\omega\in M_*^+:\omega\leq\varphi\}.$$ For each $\omega\in\Phi_\varphi$, OW-02–03 give $$0\leq h_\omega\leq I,\quad h_\omega\in\pi(M)',\quad h_\omega^{1/2}\Lambda(x)=\pi(x)\eta_\omega \quad(x\in\mathfrak n_\varphi), \tag{WH.27}$$ and $$\omega(a)=\langle\pi(a)\eta_\omega,\eta_\omega\rangle \quad(a\in M). \tag{WH.28}$$

Two totality facts. One has $$\sup_{\omega\in\Phi_\varphi}\|h_\omega^{1/2}\xi\|=\|\xi\| \quad(\xi\in H), \tag{WH.29}$$ and $$\overline{\operatorname{span} \{b\eta_\omega:b\in\pi(M)',\ \omega\in\Phi_\varphi\}}=H. \tag{WH.30}$$

Proof. On $\xi=\Lambda(x)$, NW-11 and (WH.27) give $$\|\Lambda(x)\|^2=\varphi(x^*x) =\sup_\omega\omega(x^*x) =\sup_\omega\|h_\omega^{1/2}\Lambda(x)\|^2.$$ The supremum in (WH.29) is a 1-Lipschitz function of $\xi$, since every operator involved is a contraction. Density of the GNS range extends the equality to every vector. In particular, the $h_\omega^{1/2}$'s have common kernel zero.

Let $K$ be the closed subspace in (WH.30). It reduces $\pi(M)'$; its projection $P$ lies in $\pi(M)''=\pi(M)$. By WH-02 there is a projection $p\in M$ with $P=\pi(p)$. Each $\eta_\omega$ belongs to $K$, so $$\omega(1-p)=\|(I-P)\eta_\omega\|^2=0.$$ NW-11 gives $\varphi(1-p)=0$. Faithfulness implies $p=1$, hence $K=H$. $\square$

Closability theorem. The conjugate-linear operator $$s_0:\mathcal A_\varphi\to H,\qquad s_0\Lambda(x)=\Lambda(x^*) \tag{WH.31}$$ is closable. Therefore $\mathcal A_\varphi$ is a left Hilbert algebra.

Proof. For $x\in\mathfrak a_\varphi$, $b\in\pi(M)'$, and $\omega,\rho\in\Phi_\varphi$, equations (WH.27–28) give $$\begin{aligned} \langle\Lambda(x^*),h_\omega^{1/2}b\eta_\rho\rangle &=\langle\pi(x^*)\eta_\omega,b\eta_\rho\rangle\\ &=\langle\eta_\omega,b\pi(x)\eta_\rho\rangle\\ &=\langle h_\rho^{1/2}b^*\eta_\omega,\Lambda(x)\rangle. \end{aligned} \tag{WH.32}$$ All operators in this calculation are bounded; the two appearances of $\Lambda$ are permitted because both $x$ and $x^*$ are in $\mathfrak n_\varphi$.

Suppose $\Lambda(x_n)\to0$ and $\Lambda(x_n^*)\to\zeta$ in Hilbert norm. Equation (WH.32) implies $$\langle\zeta,h_\omega^{1/2}b\eta_\rho\rangle=0 \quad\text{for every }\omega,\rho,b.$$ Fix $\omega$. Totality (WH.30) implies $h_\omega^{1/2}\zeta=0$. The common-kernel consequence of (WH.29) then gives $\zeta=0$. This is exactly the graph criterion for closability of a conjugate-linear operator on a Hilbert space. It uses sequences only to test closure in the metrizable Hilbert graph norm; it does not assume a separable Hilbert space. WH-09 supplies the other three axioms. $\square$

Write $S_\varphi=\overline{s_0}$. Its domain is, exactly, the set of norm limits of $\Lambda(x_n)$ for which $\Lambda(x_n^*)$ also converges, and the latter limit is $S_\varphi$ of the former. HA-04 and TC then give the closed involution and its polar data with their stated spectral prerequisites. The fundamental modular theorem remains separate.

OA-MOD-WH-11 — Fullness and recovery of the original weight

Apply WH-03 to $\mathcal A_\varphi$, identifying the generated algebra with $\pi(M)$. Use its spaces $\mathcal B_l,\mathcal B_r,\mathcal A_r$ and its map $\lambda$.

Theorem. One has the exact bijection $$\mathcal B_l=\Lambda(\mathfrak n_\varphi),\qquad \lambda_{\Lambda(x)}=\pi(x)\quad(x\in\mathfrak n_\varphi). \tag{WH.33}$$ Moreover $\mathcal A_\varphi$ is full, and the reconstructed weight satisfies $$\psi(\pi(a))=\varphi(a)\qquad(a\in M_+), \tag{WH.34}$$ including infinite values.

Proof. Equation (WH.27) implies that $\eta_\omega$ is right bounded for $\mathcal A_\varphi$, with $R_{\eta_\omega}=h_\omega^{1/2}$. This operator is self-adjoint. RD-07, applied to the pair $\eta_\omega,\eta_\omega$, gives $$\eta_\omega\in\mathcal A_r,\qquad F\eta_\omega=\eta_\omega. \tag{WH.35}$$

Take $\xi\in\mathcal B_l$. Since $\lambda_\xi\in\pi(M)$, there is a unique $x\in M$ with $\lambda_\xi=\pi(x)$. Using NW-11, (WH.4), (WH.29) and (WH.35), $$\begin{aligned} \varphi(x^*x) &=\sup_{\omega\in\Phi_\varphi}\|\pi(x)\eta_\omega\|^2\\ &=\sup_{\omega\in\Phi_\varphi}\|R_{\eta_\omega}\xi\|^2 =\|\xi\|^2<\infty. \end{aligned} \tag{WH.36}$$ Thus $x\in\mathfrak n_\varphi$. For each $\omega$, (WH.27) and (WH.4) give $$h_\omega^{1/2}\Lambda(x) =\pi(x)\eta_\omega =h_\omega^{1/2}\xi.$$ Their common kernel is zero, so $\Lambda(x)=\xi$.

Conversely, let $x\in\mathfrak n_\varphi$ and write $x=u|x|$. The self-adjoint element $|x|$ lies in $\mathfrak a_\varphi$, since $\varphi(|x|^2)=\varphi(x^*x)<\infty$. Therefore $\Lambda(|x|)\in\mathcal A_\varphi\subseteq\mathcal B_l$. Covariance under $\pi(u)$ gives $$\Lambda(x)=\pi(u)\Lambda(|x|)\in\mathcal B_l,\qquad \lambda_{\Lambda(x)}=\pi(u)\pi(|x|)=\pi(x).$$ This proves (WH.33).

The ideal intersection (WH.9) now reads $$\lambda(\mathcal A_l) =\pi(\mathfrak n_\varphi)\cap\pi(\mathfrak n_\varphi)^* =\pi(\mathfrak a_\varphi) =\lambda(\mathcal A_\varphi).$$ Injectivity gives $\mathcal A_l=\mathcal A_\varphi$, proving fullness.

Finally, $\psi(\pi(a))<\infty$ if and only if $\pi(a^{1/2})\in\lambda(\mathcal B_l)=\pi(\mathfrak n_\varphi)$. Faithfulness of $\pi$ makes this equivalent to $a^{1/2}\in\mathfrak n_\varphi$, or $\varphi(a)<\infty$. In that case the square-root vector is $\Lambda(a^{1/2})$, whose squared norm is $\varphi(a)$. Otherwise both values are infinite. $\square$

Thus the passage from a full left Hilbert algebra to its weight and back is inverse, up to the specified canonical unitary, to the passage from a faithful normal semifinite weight to its finite-star algebra.

OA-MOD-WH-12 — The right weight and the variational formula

Return to an arbitrary left Hilbert algebra $\mathcal A$, with full completion $\mathcal A_l$, right algebra $\mathcal A_r$, and reconstructed weight $\psi$ on $M$. The symmetric construction on $\mathcal A_r^{\mathrm{op}}$ defines a faithful normal semifinite weight $\chi$ on $M'$: $$\chi(b)= \begin{cases} \|\eta\|^2,&b^{1/2}=R_\eta,\ \eta\in\mathcal B_r,\\ +\infty,&\text{otherwise}, \end{cases} \qquad b\in(M')_+. \tag{WH.37}$$ Its finite ideal is $\mathfrak n_r=R(\mathcal B_r)$, and $$\widetilde\chi(R_\eta^*R_\zeta)=\langle\zeta,\eta\rangle. \tag{WH.38}$$ These are WH-05–08 applied to the opposite right algebra. That application has the same right-bounded space on the other side: $\mathcal A\subseteq\mathcal A_l$, while (WH.10) proves boundedness on $\mathcal A_l$ for every original $\eta\in\mathcal B_r$; restriction gives the converse.

Theorem. Under the GNS unitary of WH-08, $\chi$ is the opposite weight $\psi^{\mathrm{opp}}$ of OW. Furthermore, for every $a\in M_+$, $$\psi(a)= \sup_{\substack{\eta\in\mathcal A_r\\\|R_\eta\|\leq1}} \langle a\eta,\eta\rangle = \sup_{\substack{\eta\in\mathcal A_r\\\|R_\eta\|<1}} \langle a\eta,\eta\rangle. \tag{WH.39}$$ The analogous formula for $\chi$ uses $\xi\in\mathcal A_l$ with $\|\lambda_\xi\|\leq1$ or $<1$.

Proof of the opposite identification. Identify the GNS space of $\psi$ with $H$, so $\Lambda_\psi(\lambda_\xi)=\xi$ and $\pi_\psi(x)=x$. For $\eta\in\mathcal B_r$, the vector functional $\omega_\eta(x)=\langle x\eta,\eta\rangle$ is normal and positive. If $x=\lambda_\xi\in\mathfrak n_\psi$, then (WH.10) gives $$\omega_\eta(x^*x)=\|R_\eta\xi\|^2 \leq\|R_\eta\|^2\psi(x^*x). \tag{WH.40}$$ Testing finite positive elements with their square roots, and using any strictly larger positive constant for infinite values if necessary, proves domination by a finite multiple of $\psi$. Its comparison operator is $$h_{\omega_\eta}=R_\eta^*R_\eta, \tag{WH.41}$$ by polarization on all $\xi=\Lambda_\psi(x)$.

If $\chi(b)<\infty$, take $b^{1/2}=R_\eta$. Equation (WH.41) gives $h_{\omega_\eta}=b$, and $$\psi^{\mathrm{opp}}(b)=\|\omega_\eta\|=\|\eta\|^2=\chi(b).$$ Conversely, suppose $b=h_\omega$ for a normal positive functional dominated by a finite multiple of $\psi$. OW-03 supplies $\eta_\omega\in H$ with $$b^{1/2}\Lambda_\psi(x)=x\eta_\omega \quad(x\in\mathfrak n_\psi),\qquad \|\eta_\omega\|^2=\|\omega\|.$$ Taking $x=L_a$ with $a\in\mathcal A$ gives $b^{1/2}a=L_a\eta_\omega$. Thus $\eta_\omega\in\mathcal B_r$, $R_{\eta_\omega}=b^{1/2}$, and $\chi(b)=\|\omega\|=\psi^{\mathrm{opp}}(b)$. The finite cones and their values agree, so the infinite values agree as well.

Proof of the variational formula. Equation (WH.40) shows that every $\omega_\eta$ with $\|R_\eta\|\leq1$ is at most $\psi$ on $M_+$. Conversely, if $0\leq\omega\leq\psi$, OW-02 gives $h_\omega\leq I$, and the implementing vector in the preceding paragraph has $R_{\eta_\omega}=h_\omega^{1/2}$. Because this operator is self-adjoint, RD-07 gives $\eta_\omega\in\mathcal A_r$. Thus every dominated normal positive functional is one of the displayed vector functionals with right norm at most one. NW-11, applied to the already proved normal weight $\psi$, gives the first equality in (WH.39).

For $0<t<1$, the vector $t\eta$ has right norm $<1$ whenever $\|R_\eta\|\leq1$, and its functional is $t^2\omega_\eta$. Letting $t\uparrow1$ proves equality with the strict-norm supremum, including the case of an infinite supremum. Applying the same reasoning to the opposite right algebra proves the formula for $\chi$. $\square$

The same suprema in (WH.39) result if $\mathcal A_r$ is replaced by $\mathcal B_r$: (WH.40) bounds all such vector functionals by $\psi$, and $\mathcal A_r\subseteq\mathcal B_r$ already attains the stated supremum. This also applies to the strict norm bound, and symmetrically to the left side.

This proof also identifies the two relevant functional sets exactly: $$\{\omega_\eta:\eta\in\mathcal A_r,\ \|R_\eta\|\leq1\} =\{\omega\in M_*^+:\omega\leq\psi\}, \tag{WH.42}$$ $$\{\omega_\eta:\eta\in\mathcal A_r,\ \|R_\eta\|<1\} =\bigcup_{0\leq c<1}\{\omega\in M_*^+:\omega\leq c\psi\}. \tag{WH.43}$$ For the reverse inclusion in (WH.43), OW-02 gives $\|R_{\eta_\omega}\|\leq\sqrt c<1$; the case $c=0$ gives the zero vector. Both sets are hereditary and convex: domination is preserved when a positive functional is decreased, and for a convex combination in (WH.43) a common constant $c<1$ is the maximum of the finitely many constants. This proves the functional-set conclusion of source Corollary VII.2.3. WH-17 supplies the separate completely positive maps of source Lemma VII.2.2.

OA-MOD-WH-13 — Every von Neumann algebra has such a weight

Theorem. Every von Neumann algebra admits a faithful normal semifinite weight. No sigma-finiteness assumption is required.

Proof. In a faithful concrete representation, normal positive functionals separate nonzero positive elements: if $a\ne0$, choose $\xi$ with $\langle a\xi,\xi\rangle>0$. The vector functional is normal.

For a nonzero normal positive functional $\omega$, denote by $s(\omega)$ its support projection, using WS-04–06 in the bounded case. In particular, $$\omega(x^*x)=0\quad\Longleftrightarrow\quad xs(\omega)=0. \tag{WH.44}$$ Choose, by Zorn's lemma, a maximal family of normal states $(\omega_i)_{i\in I}$ whose nonzero supports $p_i=s(\omega_i)$ are pairwise orthogonal. For a chain of such families, its union is again such a family, so the maximality argument applies.

The strong sum $p=\sum_i p_i$ equals $1$. Otherwise choose a unit vector in $(1-p)H$. Its vector state has a nonzero support below $1-p$, since it vanishes on $p$, contradicting maximality.

Define $$\varphi(a)=\sum_{i\in I}\omega_i(a) =\sup_{F\subseteq I\text{ finite}}\sum_{i\in F}\omega_i(a), \qquad a\in M_+. \tag{WH.45}$$ Suprema of finite nonnegative subsums commute with addition and nonnegative scalar multiplication; for addition, combine the two finite sets used to approximate the two separate suprema. Thus (WH.45) is a weight.

If $a_j\uparrow a$, normality of each finite sum and commutation of the two suprema give $$\varphi(a)=\sup_F\sup_j\sum_{i\in F}\omega_i(a_j) =\sup_j\varphi(a_j).$$ This proves arbitrary-net normality.

If $\varphi(a)=0$, then every $\omega_i(a)=0$. Equation (WH.44) gives $a^{1/2}p_i=0$ for every $i$. Their finite sums converge strongly to $1$, so $a^{1/2}=0$; the weight is faithful.

For a finite $F\subseteq I$, let $p_F=\sum_{i\in F}p_i$. Supportedness and orthogonality give $\omega_i(p_F)=1$ for $i\in F$ and zero otherwise. Hence $\varphi(p_F)=|F|<\infty$. The projections $p_F\uparrow1$, so WG-008 proves semifiniteness. For the zero algebra, the zero weight has all three properties and the empty family suffices. $\square$

The projections in this proof form a specially chosen orthogonal family. It does not assert that the set of all finite projections of an arbitrary weight is directed.

Editable source · Proof dependencies and component terms