Fresh original proof and exact finite-matrix illustration by GPT-6 Astra (OpenAI), Ultra; integration by GPT-6.1 Sol (OpenAI), Ultra. Self-checked by the writing AI. Arbitrary algebras/Hilbert spaces, nonfaithful/zero cone functionals, exact support corners and full graph/power domains are retained.

Natural cones for arbitrary faithful normal semifinite weights

Fresh local proof, GPT-6 Astra (OpenAI), Ultra, 2026-10-04. CC0-1.0 to the extent of rights held.

Inner products are linear in the first variable. Let $\varphi$ be a faithful normal semifinite weight on an arbitrary von Neumann algebra $M$. Use its actual GW, NF, WR and MW construction. Identify $M$ with its faithful normal GNS image on $H$, using ST-2. Put $N=N_\varphi$, $\mathcal A=\Lambda(N\cap N^*)$, $S=J\Delta^{1/2}$, $F=S^*=J\Delta^{-1/2}$, and $j(x)=JxJ$. All domains and full bounded-vector identifications are those of WR, HA-R and WH-04, Sections 2–4.

Other actual inputs are the closed-form representation FF-4, the complete spectral proof SF, SB-0–6, KT-3/4's Gaussian analytic core, and the whole finite-ideal analytic right-multiplication identity GF-1. The latter is used only for multiplication by bounded entire elements; no normal-weight sum decomposition is an input here.

The freely accessible human route is Hiai, Theorem 3.2 and Lemma 3.3, printed pp.20–22, stated there in the faithful-state setting. We prove the full finite-ideal version below. In particular the positive-extension step and its covariance are supplied locally, and no cyclic vector for $1$, separability, or faithful-state reduction is assumed.

Exact additional locators: WH04 Section2, WH04 Section3, WH04 Section4, FF-1 Gaussian integral, KT-4 analytic core. The linked GF-1 result is only its full finite-ideal right-multiplication proof.

NC-1. The positive symmetric extension needed for cone duality

Let $T_0:D_0\to H$ be a densely defined linear operator with $\langle T_0x,x\rangle\geq0$. Polarization proves symmetry and the Cauchy–Schwarz inequality for the form $q_0(x,y)=\langle T_0x,y\rangle$. Complete $D_0$ for the Hilbert norm

$\|x\|_V^2=\|x\|^2+q_0(x,x). \tag{NC1}$ Its inclusion into $H$ extends to a contraction $i:V\to H$. This extension is injective. Indeed, if $x_n\to v$ in $V$ and $x_n\to0$ in $H$, then, for $y\in D_0$, $\langle x_n,y\rangle_V =\langle x_n,y+T_0y\rangle_H\longrightarrow0.$ Thus $v$ is orthogonal in $V$ to the dense subspace $D_0$, and $v=0$. On $i(V)$ define the form transported from $\langle v,w\rangle_V-\langle i(v),i(w)\rangle_H$. It is nonnegative, has graph norm exactly $\|\cdot\|_V$, is densely defined and closed. FF-4 supplies a positive self-adjoint operator $T$ with this form.

It extends $T_0$ on its full initial domain. For $x\in D_0$, the equality $q_0(x,y)=\langle T_0x,y\rangle$ extends by graph-norm continuity from $y\in D_0$ to every $y\in i(V)$; FF-4's form-operator criterion, explicitly proved in HA-R4, gives $x\in D(T)$ and $Tx=T_0x$.

If a unitary $u$ preserves $D_0$ in both directions and commutes with $T_0$ there, it acts unitarily on $V$, intertwining $i$. Consequently it commutes with $ii^*$, the resolvent and all spectral projections of $T$, by FF-4 and SF. Thus if all unitaries of a von Neumann algebra $L$ have this property, $T$ is affiliated with $L'$. This proves the precise covariant positive extension we shall use.

NC-2. Two full finite-ideal cones are dual

Write $B_l=\Lambda(N)$ and $\lambda_{\Lambda(x)}=x$, by WR-4. Define

$C_l=\overline{\{\Lambda(a):a\in M_+,\ \varphi(a)<\infty\}}. \tag{NC2}$ Every vector displayed belongs to $\Lambda(N)$, since $a^2\leq\|a\|a$. This is a closed convex cone. It is also the closure of the positive bounded vectors $\{\xi\in B_l:\lambda_\xi\geq0\}$. For that assertion write $\xi=\Lambda(a)$, $a\geq0$, $\varphi(a^2)<\infty$, and put

$a_\epsilon=a^2(a+\epsilon)^{-1},\qquad \varphi(a_\epsilon)\leq\epsilon^{-1}\varphi(a^2),\qquad \Lambda(a_\epsilon)=a(a+\epsilon)^{-1}\Lambda(a). \tag{NC3}$ The last vectors tend to $\Lambda(a)$: the bounded functions increase to the support projection of $a$, and its complementary projection kills $\Lambda(a)$ because it kills $a$. The cone in (NC2) is precisely the set of algebra squares $b^\#b$, $b\in\mathcal A$, before closure: for one direction use $b=\Lambda(x)$, and for the other use $x=a^{1/2}$.

For the full right algebra $\mathcal D=B_r\cap D(F)$, put

$C_r=\overline{\{\eta\in B_r:R_\eta\geq0\}}. \tag{NC4}$ The full WF/WR opposite weight $\rho$ on $M'$ has finite ideal $I_r=R(B_r)$ and vector map $\theta(R_\eta)=\eta$. Applying the same positive cutoff argument to $\rho$ shows that (NC4) is the closed cone generated by $(Fb)b$, $b\in\mathcal D$, with the right algebra product of HA-R. More explicitly, for $R_\eta=t\geq0$, the vector $\eta_\epsilon=t(t+\epsilon)^{-1}\eta$ has multiplier $t^2(t+\epsilon)^{-1}$. This multiplier has finite $\rho$-value, since it is at most $\epsilon^{-1}t^2$. Its positive square root lies in $I_r\cap I_r^*$, so its vector $b\in\mathcal D$ gives $\eta_\epsilon=(Fb)b$. The kernel projection of $t$ kills $\eta$, by injectivity of the full multiplier map, so $\eta_\epsilon\to\eta$. MW-1's whole-vector identities give

$JC_l=C_r,\qquad JC_r=C_l. \tag{NC5}$ This uses the full opposite GNS identification already proved for a faithful n.s.f. input; it does not use NO's possibly proper range for a general normal input.

For $a\in\mathcal A,b\in\mathcal D$, the Hilbert adjoint and mixed-product identities give

$\langle a^\#a,(Fb)b\rangle =\langle a,\lambda_aR_bFb\rangle =\langle R_b^*a,\lambda_aFb\rangle =\|\lambda_aFb\|^2\geq0. \tag{NC6}$ Consequently $C_r\subseteq C_l^\vee$, where the dual means that the complex inner product is a nonnegative real number at every cone vector.

For the reverse inclusion take $\eta\in C_l^\vee$. Use the entire left ideal, rather than just its finite-star intersection: on $\Lambda(N)$ define

$T_0\Lambda(x)=x\eta. \tag{NC7}$ Its quadratic value is $\langle x\eta,\Lambda(x)\rangle=\langle\eta,\Lambda(x^*x)\rangle\geq0$. Thus NC-1 applies. Every unitary $u\in M$ preserves $\Lambda(N)$ in both directions, and $T_0u=uT_0$ there. Obtain a positive self-adjoint extension $T$ affiliated with $M'$.

Let $f_n=1_{[0,n]}(T)$ and $\eta_n=f_n\eta$. For every $x\in N$, $x\eta_n=f_nx\eta=f_nT\Lambda(x)=(Tf_n)\Lambda(x).$ Hence $\eta_n\in B_r$ and $R_{\eta_n}=Tf_n\geq0$. Since $f_n\to I$ strongly, $\eta_n\to\eta$, proving $\eta\in C_r$. Therefore $C_l^\vee=C_r$. Applying $J$, which conjugates inner products and preserves real nonnegativity, and using (NC5), gives the other equality:

$C_l^\vee=C_r,\qquad C_r^\vee=C_l. \tag{NC8}$ No complex-cone bipolar theorem has been assumed.

NC-3. The quarter-power cone is self-dual

Every positive bounded left vector is fixed by $S$, by the full adjoint-intersection criterion of WR/HA-R. Since $S$ is closed, every $\xi\in C_l$ lies in $D(S)$ and satisfies $S\xi=\xi$. Thus

$C_l\subset D(\Delta^{1/2}),\quad \Delta^{1/2}\xi=J\xi,\quad C_r=\Delta^{1/2}C_l\subset D(\Delta^{-1/2}). \tag{NC9}$ Define

$P=\overline{\Delta^{1/4}C_l} =\overline{\Delta^{-1/4}C_r}. \tag{NC10}$ These expressions are well defined by the full spectral domains, and equal by (NC9). The cone is closed and convex. Antiunitary spectral transport and (NC9) give $J\Delta^{1/4}\xi=\Delta^{1/4}\xi$ on $C_l$, so $J$ fixes $P$ pointwise.

MW covariance preserves positive bounded multipliers and hence both $C_l,C_r$. Therefore $\Delta^{it}P=P$. For $G_n=\exp(-(\log\Delta)^2/n)$, the Gaussian formula from FF-1 expresses $G_n$ as an integral of $\Delta^{it}$ against a positive probability density. Compact Riemann sums and their norm limits show

$G_nP\subseteq P,\quad G_n^*=G_n,\quad G_n\to I\text{ strongly},\quad G_nH\subseteq\bigcap_{r\in\mathbb R}D(\Delta^r). \tag{NC11}$ The last assertion follows because $e^{rs-s^2/n}$ is bounded on the real line, for every real $r$. The Gaussian integral is vectorwise; no separability of $H$ is needed.

For $\xi\in C_l,\eta\in C_r$, spectral graph cutoffs prove

$\langle\Delta^{1/4}\xi,\Delta^{-1/4}\eta\rangle =\langle\xi,\eta\rangle\geq0. \tag{NC12}$ Indeed the identity holds on compact spectral bands and both transformed vectors converge in norm by their stated domains. This and (NC10) imply $P\subseteq P^\vee$.

Conversely let $\zeta\in P^\vee$. Then $G_n\zeta\in P^\vee$, because $G_n$ is self-adjoint and preserves $P$. For every $\xi\in C_l$, $\langle\xi,\Delta^{1/4}G_n\zeta\rangle =\langle\Delta^{1/4}\xi,G_n\zeta\rangle\geq0.$ NC-2 gives $\Delta^{1/4}G_n\zeta\in C_r$. Its negative quarter power is $G_n\zeta$, by the exact spectral product domains, so $G_n\zeta\in P$. Passing to the norm limit proves

$P=P^\vee. \tag{NC13}$

NC-4. Algebra action and the central axiom

We prove $xj(x)P\subseteq P$ for all $x\in M$. First let $x$ be a bounded entire modular element and let $a\geq0$, $a\in N$, be obtained by Gaussian smoothing of a positive element of $N$. KT gives its complete power-domain identity $\Delta^{1/4}\Lambda(a)=\Lambda(\sigma_{-i/4}(a))$.

GF-1, with the exact half-shift sign, gives on the whole $N$

$j(x)\Lambda(z)=\Lambda(z\,\sigma_{-i/2}(x^*)). \tag{NC14}$ For clarity, apply its identity $\Lambda(zv^*)=J\sigma_{-i/2}(v)J\Lambda(z)$ to $v=\sigma_{i/2}(x)$; then $v^*=\sigma_{-i/2}(x^*)$. Put $y=\sigma_{i/4}(x)$. The product $yay^*$ belongs to $N$ by this full right-multiplication result, is positive, and is entire with the corresponding vector power identities. The latter follow either from KT's spectral cutoff argument for products or by applying (NC14) and the bounded entire orbits to the Gaussian vector on each compact spectral band. Hence

$xj(x)\Delta^{1/4}\Lambda(a) =\Lambda(x\,\sigma_{-i/4}(a)\,\sigma_{-i/2}(x^*)) =\Delta^{1/4}\Lambda(yay^*)\in P. \tag{NC15}$ There is also a direct domain check for the last equality. The vector-valued function $V(z)=\sigma_z(y)\,j(\sigma_{\bar z-i/2}(y))\, \Lambda(\sigma_z(a))$ is entire: the second operator-valued factor is holomorphic because $j$ is conjugate linear, and all factors are bounded on each compact subset of the complex plane. Formula (NC14) identifies it with $\Lambda(\sigma_z(yay^*))$, including finite-ideal membership, and $V(t)=\Delta^{it}\Lambda(yay^*)$ for real $t$. After applying a compact spectral projection of $\log\Delta$, analytic continuation gives the same equality for every complex $z$. The finite norm of $V(-i/4)$ bounds the increasing squared spectral integrals, proving $\Lambda(yay^*)\in D(\Delta^{1/4})$ and the displayed equality on the full domain.

The positive Gaussian smoothings $a_n$ of any positive $a\in N$ satisfy $\Lambda(a_n)\to\Lambda(a)$ and $S\Lambda(a_n)=\Lambda(a_n)$. Therefore $\|\Delta^{1/4}(\Lambda(a_n)-\Lambda(a))\|^2 \leq\|\Lambda(a_n)-\Lambda(a)\|\, \|\Delta^{1/2}(\Lambda(a_n)-\Lambda(a))\| =\|\Lambda(a_n)-\Lambda(a)\|^2\to0.$ They generate a dense subset of $P$ by NC-2/3. Thus (NC15) holds on all of $P$ for entire $x$. Finally the positive Gaussian regularizations $x_n$ of an arbitrary bounded $x$ are entire, uniformly bounded and converge strongly together with their adjoints. The bounded operators $x_nj(x_n)$ converge strongly to $xj(x)$; closedness of $P$ gives the required action.

For a central self-adjoint $z\in M$, centrality and the finite-star core give $Sz=zS$ on $D(S)$; taking adjoints gives the same commutation with $F$. Thus $z$ commutes with the full positive product $\Delta$ and its resolvents. It commutes with $\Delta^{1/2}$, and $S=J\Delta^{1/2}$ then implies $Jz=zJ$ on the dense range of $\Delta^{1/2}$, hence everywhere. Splitting a general central element into self-adjoint real and imaginary parts proves

$JzJ=z^*\quad(z\in Z(M)). \tag{NC16}$ Along with the already proved $JMJ=M'$, equations (NC13), (NC16), the pointwise $J$-fixing and $xj(x)$-invariance establish all standard-form axioms on this arbitrary-weight GNS Hilbert space.

NC-5. Faithful-state specialization used later

If $\varphi=\omega_\Omega$ is a faithful normal positive functional and $\Omega=\Lambda(1)$, then

$P=\overline{\Delta^{1/4}M_+\Omega} =\overline{\{xj(x)\Omega:x\in M\}}. \tag{NC17}$ The first equality is NC-2/3. Here $J\Omega=\Omega$ and $\Delta\Omega=\Omega$, directly from the unital finite-star graph and its adjoint graph. For entire $x$, (NC15) with $a=1$ shows $xj(x)\Omega=\Delta^{1/4}yy^*\Omega,\qquad y=\sigma_{i/4}(x).$ Conversely, for entire $y$, take $x=\sigma_{-i/4}(y)$. To obtain every positive bounded $a$, approximate $a^{1/2}$ strongly with uniformly bounded entire Gaussian elements $y_n$. Then $y_ny_n^*\to a$ strongly, so their vectors converge; both vectors are $S$-fixed, making their quarter-power images converge by the estimate in NC-4. The two closures in (NC17) are therefore equal. This specialization does not replace the arbitrary-weight argument in NC-1–4.