Constructing the natural cone from bounded multiplication

Written by GPT-6.1 Sol (OpenAI), Ultra, October 2026. Original text: public domain (CC0).

The methods used here are credited to H. Araki, U. Haagerup, M. Rieffel and A. Van Daele, and F. Hiai. Free primary accounts are Araki's 1974 paper, Haagerup's 1975 paper, Rieffel–Van Daele's 1977 paper, and Hiai's author arXiv version, 2004.02383v1. Hiai's Lemma 3.3 and Theorem 3.2 give a useful cyclic version of the endpoint and middle-cone arguments. We work with the entire spaces of bounded multiplication vectors, so that the endpoint proof also applies to an infinite weight and an arbitrary Hilbert-space cardinality.

NC00. The existing operator construction and its precise interface

Inner products are linear in the second variable. A first-variable-linear source pairing is translated by $\langle u,v\rangle=(v\mid u)$. The conjugate-linear adjoint convention is

All operator equalities below include their actual domains. In particular, a displayed product of unbounded operators is not an equality of formal expressions alone.

The preceding proofs used here are the following.

• NC01 and NC01b construct a faithful normal semifinite weight on an arbitrary algebra, its normal GNS representation, and the full left Hilbert algebra with its original closed involution. NC01b includes the normal-functional GNS and arbitrary direct-sum proofs. For a different, preselected faithful n.s.f. weight, WG003–010 and WH02 and WH09–11 give the corresponding realization.
• RC–GP–RS–IK–MC–MP prove the modular commutant theorem for that Hilbert algebra and identify the original closed involution and its adjoint. The route constructs the bounded real-subspace coordinates, proves both commutant inclusions, and obtains the polar factors with their domains; it does not replace the original graph by a newly defined modular graph.
• MF06–09 prove modular covariance, the common analytic multiplication algebra, its product identities and bounded-multiplier Gaussian approximation. The approximation of merely bounded multiplication vectors needed below is proved directly in NC01a from the full operator ideals.
• QF03–04 represent a densely defined closed positive form, with the exact operator domain, and prove unitary transport of its representing operator. SK04–09 provide the spectral domains, bounded transforms, powers, inverse domains and pairing identities. The bounded continuous calculus and Hilbert representation are proved in BK01.
• For the compact-metric representation step SS3 in the spectral construction, this route uses RM01–RM05 for the finite regular measure of a positive functional on a compact metric space, including a compact rectangle. SC01–SC02 prove rectangle compactness and completion of an arbitrary measure. Together with SS1–SS2 and SS4, these supply scalar Hilbert completeness, compact-metric approximation and measurable representatives. The scalar complex and integration proofs and their boundary and product-integration continuations supply the contour, convergence and Gaussian formulas used by IK and MA.
• MA03/16 prove Gaussian regularization for operators and arbitrary Hilbert vectors, including the scalar Gaussian Fourier identity. Its use below requires only that particular positive Gaussian kernel, not a general positive-definite-function representation theorem.

Here is the combined concrete interface, which fixes the objects used in the rest of this proof. For a faithful n.s.f. weight $\varphi$ on $N$, its GNS representation identifies $N$ faithfully and normally with $M\subset B(H)$, and

The full left and right multiplication spaces are $B_l,B_r$, with multipliers $L_\xi\in M$, $R_\eta\in M'$. Their exact defining mixed identity and covariance are

The multipliers are injective assignments. Their ranges are left ideals, and

In particular, a self-adjoint left multiplier puts its vector in $\mathcal A$, with $S\xi=\xi$; the corresponding assertion holds on the right. Products satisfy $\xi\zeta=L_\xi\zeta$, $\eta\theta=R_\theta\eta$, and the involutions on the two algebras are $S,F$. Both algebras are dense, and are the graph cores of their involutions.

The modular output is

Here $\Delta$ is injective; its spectrum may accumulate at zero or infinity. The domain of a real power is

The modular group $U_t=\Delta^{it}$ implements automorphisms $\sigma_t$ of $M$. The common analytic algebra $\mathcal A_0$ is invariant under every complex power, and for $\xi,\zeta\in\mathcal A_0$,

The algebra $\mathcal A_0$ is dense in the stated involution graphs; the Gaussian approximants to $\xi\in\mathcal A$ converge to $\xi$ and $S\xi$, with uniformly bounded left multipliers converging strongly. The additional approximation proved in NC01a is: every $\eta\in B_r$ has a net $\eta_i\in\mathcal D$ with

The analogous approximation holds on the left. The algebraic product reversal by $J$, and $FJ=JS$, have the graph domains supplied by (5)–(7).

No cone theorem is an input to this interface. In particular, endpoint duality and the middle cone's self-duality are proved below.

NC00a. The bounded-multiplication contracts from the original modular graphs

We record the proof of (3)–(4), to make their bounded tests and adjoint tests explicit. The real-square subspace in RS4 is

and MC5 proves $D(S)=K+iK$, $S(k+i\ell)=k-i\ell$. Approximating $k$ and $\ell$ by real square spans proves that $\mathcal A^2$ is a graph core for this original $S$. The same conclusion holds for the right algebra and $F$, by MC1/MC3/MC5 and MP's adjoint identification.

For a right-bounded vector $\eta$, the operator determined by $R_\eta a=L_a\eta$ commutes with all $L_b$, by associativity on $\mathcal A$, and hence belongs to $M'$. The assignments are linear and injective, because $L_a\eta=0$ for all $a$ implies $\eta=0$ by nondegeneracy. For $y\in M'$, the defining tests immediately give $R_{y\eta}=yR_\eta$. Thus their range $\mathfrak n_r$ is a left ideal.

If $\eta\in B_r\cap D(F)$, then for $a,b\in\mathcal A$,

Density shows that $F\eta\in B_r$ and $R_{F\eta}=R_\eta^*$. Conversely, if $R_\eta^*=R_\theta$ for another $\theta\in B_r$, the same chain of pairings, read backwards, proves

Extend this over the proved product graph core. It is exactly the adjoint-domain test (1), so $\eta\in D(F)$ and $F\eta=\theta$. Consequently the right algebra defined by the two bounded multiplier tests in GP–MC is precisely $\mathcal D=B_r\cap D(F)$, and

The right algebra is full, dense and nondegenerate by MC1–MC6. Apply the proved argument to its opposite left Hilbert algebra. Its closed involution is $F$, its adjoint is $S$, and its generated algebra is $M'$. This gives all the left assertions of (3)–(4), including covariance, injectivity, the ideal intersection, and equality with the full left completion. In particular, the definition of $B_l$ is exactly boundedness of $\eta\mapsto R_\eta\xi$ on $\mathcal D$, with multiplier $L_\xi$.

The mixed identity initially holds for $\eta\in\mathcal D$ by that definition. Its extension to every $\eta\in B_r$ is proved after the approximating net in NC01a. This derivation of the full bounded tests uses the complete modular graph construction, and supplies the exact product-core density used later in NC03.

NC01. Choosing a weight at arbitrary cardinality

Every von Neumann algebra has a faithful normal semifinite weight. Here is the construction used to choose the weight in (2). Start with a faithful normal concrete presentation. Nonzero positive elements are detected by positive normal vector functionals. The support lemma CG01 constructs the support of a normal positive functional, its faithful restriction, and its compression formula, independently of any natural cone.

Choose a maximal family of normal states $(\omega_i)_{i\in I}$ whose nonzero supports $p_i$ are mutually orthogonal. The union of a chain remains such a family. Its support join is one: otherwise a vector state on $(1-\bigvee_i p_i)H$ provides another nonzero disjoint support. Define

Finite nonnegative subsums prove additivity and positive homogeneity. For $a_j\uparrow a$, normality follows by interchanging the two suprema over $j$ and finite $E$. If $\varphi(a)=0$, every $\omega_i(a)=0$, so $a^{1/2}p_i=0$. The finite sums of the $p_i$ converge strongly to one, and therefore $a=0$. Thus the weight is faithful.

For $p_E=\sum_{i\in E}p_i$, supportedness gives $\varphi(p_E)=|E|$. For every $a\in N_+$,

These uniformly bounded finite positive elements converge ultraweakly as well, by H03's series-vector tail estimate. Each finite positive element $c$ lies in the finite span $\mathfrak m_\varphi$, because $c=c^{1/2}c^{1/2}$ and $\varphi((c^{1/2})^*c^{1/2})=\varphi(c)<\infty$. Every element of $N$ is a complex linear combination of positive elements. Consequently $\mathfrak m_\varphi$ is ultraweakly dense in $N$, which is semifiniteness. This also gives the strong density of the finite ideal, since $xp_E$ has finite square weight for every $x\in N$. No countability of $I$ is asserted. The zero algebra uses the zero weight and the empty family.

The following direct construction supplies the GNS representation and full left Hilbert algebra for this weight. It retains the original algebra through a faithful normal representation.

NC01a. Approximating the entire bounded-vector space by finite-star vectors

Let $\mathfrak n$ be either full multiplication left ideal, in its respective von Neumann algebra $B$, and let $\mathcal C=\mathfrak n\cap\mathfrak n^*$. This is a nondegenerate *-algebra by the operator construction. For a finite subset $E\subset\mathcal C$ and $\varepsilon>0$, put

The positive contraction $e_{E,\varepsilon}$ belongs to $\mathfrak n$ by the left-ideal property and bounded calculus, and to $\mathfrak n^*$ by self-adjointness. Thus it belongs to $\mathcal C$. These contractions converge strongly to one along finite-set enlargement and $\varepsilon\downarrow0$. Here is a direct bound, avoiding a countability assumption or an assertion of monotonicity for squares. If $x\in E$, then $x^*x\le h_E$, and

The last inequality is the scalar bound $\varepsilon^2t/(t+\varepsilon)^2\le\varepsilon/4$. Taking adjoints shows convergence of $e_{E,\varepsilon}$ to one on the range of every $x^*$. Their linear span is dense because $\mathcal C$ is self-adjoint and nondegenerate. The uniform contraction bound extends convergence to all of $H$.

Use this net for $\mathfrak n_r=R(B_r)$. For $\eta\in B_r$, set $\eta_i=e_i\eta$. Covariance gives $R_{\eta_i}=e_iR_\eta\in\mathfrak n_r$. Its adjoint $R_\eta^*e_i$ also lies in $\mathfrak n_r$, since $e_i\in\mathfrak n_r$ and this is a left ideal. The ideal intersection (4) therefore puts $\eta_i$ in $\mathcal D$. Strong convergence and the contraction bound give every assertion in (8). The identical proof on $\mathfrak n_l=L(B_l)$ gives the left approximation.

For $\xi\in B_l$, the defining equality $L_\xi\eta_i=R_{\eta_i}\xi$ now tends to $L_\xi\eta=R_\eta\xi$; both operator limits are justified on these fixed vectors. This proves the entire mixed identity in (3). All these proofs apply to nets at arbitrary cardinality and supply no extraneous involution-domain claim for their original merely bounded vectors.

NC01b. Normal GNS models and the original full diagonal-weight algebra

We use the complete Hilbert and concrete-predual constructions H00–H03, their arbitrary-Hilbert-space identification P00, and the bounded calculus F03–F08. H03 identifies a concrete von Neumann algebra $N$ isometrically with $(N_*)^*$. Its quotient predual is a norm-closed subspace of $N^*$: Hahn–Banach gives the quotient norm as the supremum of its pairings with the unit ball of its dual. The normal coefficients therefore remain normal under functional-norm limits.

Normal positive functionals. For $\omega\in N_*^+$, positivity and the scalar quadratic-polynomial test give

Indeed $\omega((b+zc)^*(b+zc))=A+2\operatorname{Re}(z d)+|z|^2C\ge0$, with $A=\omega(b^*b)$, $C=\omega(c^*c)$, and $d=\omega(b^*c)$. For $C>0$, choose $z=-\overline d/C$; for $C=0$, arbitrary phases and magnitudes of $z$ force $d=0$. This proves both cases. Quotient $N$ by the null space of this form and complete, using $\langle[b],[c]\rangle=\omega(b^*c)$. The null space is a left ideal: bounded order and the positive square root give

Thus $\pi_\omega(a)[c]=[ac]$ defines a contractive unital *-representation on $H_\omega$, with cyclic vector $\Omega_\omega=[1]$ representing $\omega$. Multiplication, adjoints and the identity follow on quotient vectors by taking the displayed pairings, and extend by density.

This representation is normal. Indeed every predual functional is a series $f(a)=\sum_j\langle u_j,av_j\rangle$ with $\sum_j\|u_j\|\|v_j\|<\infty$, by H03 and P00. Fixed multiplication gives

In particular the coefficients on quotient vectors, $\omega(b^*ac)$, belong to $N_*$. For arbitrary vectors, approximate both by quotient vectors. Contractivity gives the functional-norm estimate

The norm-closedness just proved puts each limiting coefficient in $N_*$. An ultraweak functional on the target is itself a norm-summable series of coefficients. Its pullback is a norm-convergent series in $N_*$, so $\pi_\omega$ is ultraweak continuous. The zero functional gives the zero space.

Arbitrary sums. For any set-indexed family of such representations, take their Hilbert direct sum $H=\bigoplus_i H_i$, with the componentwise action $\pi(a)$. Each vector has countable support: for every positive integer $n$, only finitely many component norms can exceed $1/n$. Finite-support vectors are dense, and completeness follows by the coordinate argument in H00. The coefficient series obeys

Its finite sums belong to $N_*$ and its tails converge in functional norm. The same target-predual series argument proves that the direct sum is normal, without a cardinality restriction.

Apply this construction to the supported functionals $\omega_i$ from NC01. In fact the following proof allows any nonzero finite normal positive $\omega_i$ with orthogonal supports $p_i$ joining to one; it therefore also covers a faithful finite functional as a one-block family. Define

Its squared norm is $\varphi(x^*x)$. The same left-ideal estimate defines the contractive action $\pi(a)\Lambda(x)=\Lambda(ax)$.

This GNS space is the entire direct sum. For any finite tuple of quotient vectors $([x_i]_i)_{i\in E}$, use $x=\sum_{i\in E}x_ip_i$. Supportedness means $[yp_i]_i=[y]_i$ and $[yp_i]_j=0$ for $j\ne i$, by the support compression and Cauchy–Schwarz. Hence $\Lambda(x)$ is exactly the chosen finite tuple and $x\in\mathfrak n_\varphi$. Such tuples are dense. The normal direct sum above is therefore the original weight's GNS representation.

Let $R_i$ be the coordinate projection and let $\eta_i=\Lambda(p_i)$. They satisfy

Here $xp_i\in\mathfrak n_\varphi$, since its square weight is $\omega_i(x^*x)$. The represented vector functional of $\eta_i$ is $\omega_i$. If $\pi(a)=0$, (9a) gives $\Lambda(ap_i)=0$; faithfulness of $\varphi$ gives $ap_i=0$ for every $i$, hence $a=0$. The representation is isometric. To see the norm assertion from the stated bounded calculus, if $\|\pi(a)\|<\|a\|$, choose a continuous function on the spectrum of $a^*a$ that vanishes on $[0,\|\pi(a)\|^2]$ but is nonzero at $\|a\|^2$. Its nonzero calculus value is killed by $\pi$, contradicting faithfulness.

The image is a von Neumann algebra. We give the compactness step explicitly. For a Banach space $X$, place its dual unit ball in the product of closed scalar discs $\prod_{x\in X}\{z:|z|\le\|x\|\}$, by evaluation. This product is compact. One direct proof uses choice to extend a filter with the finite-intersection property to an ultrafilter. On each scalar disc, repeated finite subdivision into closed squares chooses a nested square sequence of diameters tending to zero whose intersections with the disc belong to the ultrafilter. Scalar completeness gives a limit point in the disc; every neighbourhood belongs to that coordinate ultrafilter. The coordinate limits therefore give a product limit, because a basic neighbourhood tests only finitely many coordinates. If a family of product-closed sets with the finite-intersection property had empty intersection, its extending ultrafilter would give a limit lying in each of them, a contradiction. This proves compactness. The evaluation families satisfying additivity and complex homogeneity form a closed subset of the product, and the coordinate bound makes them precisely the dual unit ball. Its product topology is the weak* topology. This is Banach–Alaoglu with its scalar and choice inputs displayed.

Since $N=(N_*)^*$, its unit ball is compact ultraweakly. Put $B=\pi(N)$, a unital norm-closed C*-algebra by isometry. The complete self-adjoint contraction-density proof ODF03–05 supplies, for each self-adjoint contraction $T\in B''$, a net of self-adjoint contractions $\pi(a_\lambda)$ converging strongly to $T$, with $\|a_\lambda\|\le1$. Here the represented algebra has an identity, so the orbit closure in ODF03 contains each test tuple directly by using that identity. Compactness supplies a convergent subnet by the complete net construction in COMPACT. Explicitly, the closures of the tails have a common cluster point; index the subnet by triples consisting of a neighbourhood, an original tail index and a chosen later index hitting that neighbourhood. Shrink the neighbourhood and advance both indices in the order. A hit beyond both previously chosen indices gives a common successor, so this is directed, and projection onto the chosen index is order preserving and eventually beyond each original index. It gives the convergent subnet. Write its ultraweak limit as $a\in N$. Normality makes $\pi(a_\lambda)$ converge ultraweakly to $\pi(a)$, while its bounded strong convergence has limit $T$, by H03's series-tail test. Coefficients separate operators, so $T=\pi(a)$. Scaling and taking the real and imaginary self-adjoint parts of an arbitrary $T\in B''$ prove $B''=B$. We now write $M=\pi(N)$.

The inverse identification is normal as well. On each closed norm ball, $\pi$ is a continuous bijection from a compact ultraweak space to a Hausdorff ultraweak space, so its inverse is continuous there: a compact subset of a Hausdorff space is closed, by separating one exterior point from its points and taking a finite subcover. Let $f\in N_*$, and put $g=f\circ\pi^{-1}\in M^*$. Its restriction to the unit ball is ultraweak continuous. For $\varepsilon>0$, continuity at zero gives finitely many $h_1,\ldots,h_n\in M_*$ such that $|g(x)|<\varepsilon$ whenever $\|x\|\le1$ and all the $h_j(x)$ are sufficiently small. In particular $g$ has norm at most $\varepsilon$ on the common kernel $V=\bigcap_j\ker h_j$. Hahn–Banach F01 extends this restriction to $G\in M^*$ with $\|G\|\le\varepsilon$. The functional $g-G$ vanishes on $V$, so it factors through the finite-dimensional map $x\mapsto(h_1(x),\ldots,h_n(x))$. Extending a linear functional on its image to $\mathbb C^n$ proves $g-G\in\operatorname{span}\{h_j\}\subset M_*$. Since $\varepsilon$ is arbitrary and $M_*$ is norm closed, $g\in M_*$. Thus $\pi^{-1}$ pulls every normal functional back to a normal functional. Normal positive functionals can consequently be transported in both directions without a normality-converse theorem.

The initial left Hilbert algebra and its closed graph. Let $\mathcal A=\Lambda(\mathfrak n_\varphi\cap\mathfrak n_\varphi^*)$, with product $\Lambda(x)\Lambda(y)=\Lambda(xy)$ and involution $\Lambda(x)^\sharp=\Lambda(x^*)$. The left-ideal estimate gives bounded left multiplication, and the original sum pairings give $\langle\Lambda(x)\Lambda(y),\Lambda(z)\rangle=\langle\Lambda(y),\Lambda(x^*)\Lambda(z)\rangle$. All these series are absolutely convergent by Cauchy–Schwarz. The finite-star set is a *-algebra: left multiplication preserves $\mathfrak n_\varphi$, and applying the same fact to adjoints proves stability under products.

For $a,b\in\mathfrak n_\varphi$, the inequalities $(a^*b)^*(a^*b)\le\|a\|^2b^*b$ and $(a^*b)(a^*b)^*\le\|b\|^2a^*a$ put $a^*b$ in the finite-star set. Thus $p_Ex$ belongs to that set for $x\in\mathfrak n_\varphi$, and $\Lambda(p_Ex)=\pi(p_E)\Lambda(x)\to\Lambda(x)$, since normality and $p_E\uparrow1$ give $\pi(p_E)\uparrow1$. This proves density of $\mathcal A$. When $x$ is finite-star, these vectors are products $\Lambda(p_E)\Lambda(x)$, so the product span is dense as well. Finally $p_Ex p_F$ is finite-star for every bounded $x\in N$, by the two square bounds with right supports $p_F$ and $p_E$. Their represented operators converge strongly to $\pi(x)$. Consequently the left algebra generates exactly $M$.

The original involution is closable. First the subspace $K=\overline{\operatorname{span}\{b'\eta_i:b'\in M',i\in I\}}$ is all of $H$. It reduces $M'$; its projection belongs to $(M')'=M$, hence equals $\pi(p)$ for a projection $p\in N$. Since $\pi(p)\eta_i=\eta_i$, we have $\omega_i(1-p)=0$, so each support $p_i\le p$. Their join is one, whence $K=H$. For every finite-star $x$, every $b'\in M'$, and all $i,j$, (9a), commutation and adjoints give the exact second-variable-linear identity

If $\Lambda(x_n)\to0$ and $\Lambda(x_n^*)\to\zeta$, (9b) makes $\zeta$ orthogonal to $R_iK$ for every $i$. These spaces together span $H$, so $\zeta=0$. This proves closability of precisely the initial involution. Let its closure be $S$. We have now supplied the left Hilbert algebra to which the complete RC–GP–RS–IK–MC–MP construction applies.

The bounded part of the original closed graph. If $x\in\mathfrak n_\varphi$, $\Lambda(x)\in D(S)$, and $S\Lambda(x)=\eta$, take finite-star $x_n$ converging in this original graph. Identity (9b) passes to the limit. The same bounded adjoint computation with $x$ on the right gives

The projected commutant-orbit spans are dense in $R_iH$, so $R_i\eta=\pi(x^*)\eta_i$. Summing squares yields $\varphi(xx^*)=\sum_i\|\pi(x^*)\eta_i\|^2=\|\eta\|^2<\infty$, and then (9a) gives $\eta=\Lambda(x^*)$. Conversely finite-star vectors belong to the initial graph. Thus its closure has added no extra bounded finite-star vectors.

Fullness. For $\xi\in B_l$, its multiplier belongs to $M$, so $L_\xi=\pi(x)$ for a unique $x\in N$. The vectors $\eta_i$ are right-bounded with $R_{\eta_i}=R_i$, by testing $L_a\eta_i=R_i a$ on $\mathcal A$. Their self-adjoint multipliers and NC00a put them in $\mathcal D$. The defining mixed test gives

Thus $x\in\mathfrak n_\varphi$, and (9a) gives $\xi=\Lambda(x)$. Conversely, for $x\in\mathfrak n_\varphi$, the finite-star vectors $\Lambda(p_Ex)\to\Lambda(x)$ have multipliers $\pi(p_Ex)\to\pi(x)$ strongly with bound $\|x\|$. For every $\eta\in\mathcal D$, pass to the limit in $R_\eta\Lambda(p_Ex)=\pi(p_Ex)\eta$. This proves $\Lambda(x)\in B_l$ and $L_{\Lambda(x)}=\pi(x)$. Therefore $B_l=\Lambda(\mathfrak n_\varphi)$. Combining this with the preceding original-graph domain check gives

To check fullness as an algebra, let $\mathcal A''$ be the double completion from MC6. Every $\xi\in\mathcal A''$ satisfies the bounded left-multiplier tests MC6.2, so $\xi\in B_l$. MC6.5 and MC5 prove that its closed involution has the same original graph $S$, so $\xi\in D(S)$. Therefore $\mathcal A''\subset B_l\cap D(S)=\mathcal A$; the reverse inclusion is also proved by MC6.2. Hence $\mathcal A''=\mathcal A$. This is the full original left Hilbert algebra required in (2)–(4). The proof uses no normality converse for a general weight, no opposite-weight reconstruction, and no separability hypothesis.

NC02. A positive symmetric operator has a covariant positive extension

Let $T_0$ be a densely defined positive symmetric linear operator, with domain $D$. Complete $D$ for

The inclusion into $H$ extends to a contraction $\iota:V\to H$. This map is injective. Indeed, if $v_n$ tends to $v\in V$ and to zero in $H$, then for every $w\in D$,

Consequently $v$ is orthogonal in $V$ to its dense subspace $D$, and $v=0$. Identify $V$ with $\iota V\subset H$. The continuous extension of the second term in (10) is a densely defined closed positive form $q$, since its form norm is exactly the complete $V$-norm.

QF03 constructs its positive self-adjoint representing operator $h$. Its operator-domain criterion says that $v\in D(h)$ precisely when the form $q(w,v)$ is $\langle w,f\rangle$ for some $f\in H$, for every $w\in V$, and then $hv=f$. For $v\in D$, symmetry and form-norm approximation give $q(w,v)=\langle w,T_0v\rangle$, so $h$ extends $T_0$.

If every unitary in a von Neumann algebra $B$ preserves $D$ and commutes with $T_0$ there, it preserves (10), hence $V$ and $q$. QF04's uniqueness and transport prove $uhu^*=h$, including domains, for every such unitary. The spectral calculus then places every bounded transform of $h$ in $B'$. Thus $h$ is affiliated with $B'$. This proves the extension and its covariance, including the form-closure step used in Hiai's endpoint argument.

NC03. Endpoint duality using all right-bounded vectors

Define the two closed square sets

For any subset $C\subset H$, its dual is

We claim

First, commuting multipliers and the adjoint identities in (3)–(4) give

The pair is real, so its order is unchanged by conjugation. This proves one inclusion in each equality.

Now let $v\in C_r^\vee$. Define on the whole dense right-bounded space

For $\eta\in\mathcal D$,

For general $\eta\in B_r$, use (8). Both $\eta_i\to\eta$ and $R_{\eta_i}v\to R_\eta v$, so the same nonnegative diagonal inequality passes to $\eta$. Polarization makes $T_v$ symmetric and positive on its full domain. In particular it is closable: if $\eta_n\to0$ and $T_v\eta_n\to w$, testing symmetry against every $\theta\in B_r$ gives $\langle\theta,w\rangle=0$, and density gives $w=0$.

The full-domain choice in (15) also proves covariance directly. Every $y\in M'$ preserves $B_r$, and (3) gives

NC02 therefore supplies a positive self-adjoint extension $h$ affiliated with $M$. It was not necessary first to put $v$ in an involution domain.

Use the bounded resolvents

For $\eta\in B_r$, affiliation gives commutation with $R_\eta$, and the fact that $h$ extends (15) gives

The operator $b_nh$ is bounded and positive. Equations (3)–(4), or the defining bounded-multiplier test, show that $v_n\in B_l$ and $L_{v_n}=b_nh\ge0$. Also $b_n\to1$ strongly, so $v_n\to v$.

Every left-bounded vector $w$ with $a=L_w\ge0$ is a norm limit of left squares. Here is a proof entirely within bounded multiplication. For $\varepsilon>0$, put

Covariance puts $\xi_\varepsilon$ in $B_l$. Its positive self-adjoint multiplier and (4) put it in $\mathcal A$, with $S\xi_\varepsilon=\xi_\varepsilon$. Therefore

Indeed $f_\varepsilon(a)$ tends strongly to the range support $p$ of $a$, and injectivity of multiplication gives $(1-p)w=0$ from $L_{(1-p)w}=(1-p)a=0$. Thus (20) has the stated vector limit.

Apply this to every $v_n$, then use closedness of the square set: $v\in C_l$. This proves $C_r^\vee\subset C_l$. Run the same proof on the opposite right Hilbert algebra to obtain $C_l^\vee\subset C_r$. This proves (13), without a cyclic vector or a prior convexity assumption on the square sets. Dual sets are closed convex cones, so both endpoint sets are now closed convex cones.

Each left square is fixed by $S$; closedness gives

The right assertion is $Fv=v$ on $C_r$. Product reversal by $J$, including $FJ=JS$, gives

The square spans are dense: complex polarization spans the product algebra by squares, and the product graph-core theorem gives density in $H$. Thus neither cone has a nonzero vector together with its negative. This also records the endpoint graph domains needed for taking the quarter powers.

NC04. Defining the middle cone and identifying both power descriptions

For $\xi\in B_l$, define the always meaningful vector

Set

On the analytic multiplication algebra, (5) and (7) give

Every expression in this equality belongs to its stated domain; $\Delta^{1/4}$ maps $\mathcal A_0$ onto itself.

Analytic left squares are dense in $C_l$. For the approximants from NC00, the estimate

gives this assertion first for each square, then for the whole closure. If $u_n,u\in C_l$, (21) gives $\Delta^{1/2}(u_n-u)=J(u_n-u)$, and the spectral pairing inequality gives

Therefore (24) extends to the closure description

For the second equality use (22) and the spectral domain identity on $C_l$. Both images in (26) are convex cones because the respective powers are linear on their entire cone domains. Their closures are convex too.

Equation (7) gives $JQ(\xi)=Q(\xi)$, so $J$ fixes $P$ pointwise. Covariance and commutation of $J$ with the real group give

No cone order theorem is being used to assert this group invariance.

NC05. Self-duality by a positive Gaussian smoothing

For $u\in C_l$, $v\in C_r$, the spectral pairing identity and (13) give

One obtains the identity first on bounded spectral bands and then by convergence in the two quarter-power domains. The two descriptions in (26) imply $P\subset P^\vee$.

Suppose $w\in P^\vee$. Define

The integral is a Hilbert-vector integral, obtained by truncation and scalar pairing; its absolute norm bound is one. The Gaussian Fourier formula in MA03/16 proves the spectral equality. The scalar multiplier converges to one and is bounded by one, so $w_n\to w$. For every real $\alpha$,

which, together with (6), proves $w_n\in D(\Delta^\alpha)$. Thus every power used next has an actual vector value.

The kernel in (29) is nonnegative. Equation (27) implies that $G_n$ preserves $P$; self-adjointness of $G_n$ implies that it preserves $P^\vee$. For $u\in C_l$,

Endpoint duality gives $\Delta^{1/4}w_n\in C_r$. Its inverse quarter power is defined, and equals $w_n$; (26) therefore puts $w_n\in P$. Closedness and its vector limit give $w\in P$. We have proved

NC06. Preservation by every bounded algebra element

The generating set in (23) can be enlarged to all left-bounded vectors:

First use Gaussian approximation for $\xi\in\mathcal A$. Its vector convergence and strong multiplier convergence with a uniform bound give $Q(\xi_r)\to Q(\xi)$. Next approximate $\xi\in B_l$ by the left version of (8); the same estimate gives $Q(\xi_i)\to Q(\xi)$. Both limits lie in $P$. The reverse containment follows from $\mathcal A_0\subset B_l$.

For $a\in M$, $\xi\in B_l$, covariance says that $a\xi$ is still in $B_l$. Since $JaJ\in M'$,