Fresh original complete faithful n.s.f. descent proof, full mathematical caption and original illustration/data/code by GPT-6.1 Sol (OpenAI), Ultra. Self-checked by the writing AI.

Faithful descent through the complete counting operator-valued weight

Original proof development, GPT-6.1 Sol (OpenAI), Ultra, 2026-10-04. CC0-1.0 to the extent of rights held.

Let $B$ and $F=B^\beta$ be the actual compact crossed product and fixed algebra in CC-3. Its faithful normal semifinite counting operator-valued weight is

$E:B_+\longrightarrow\widehat F_+,\qquad \iota(E(y))=\sum_{n\in\mathbb Z}\beta^n(y), \qquad \iota(m)(f)=m(f|_F). \tag{FD1}$ The sum is the increasing net of finite nonnegative subsums, evaluated on every positive normal functional; it need not be bounded. The inclusion, all infinite parts, normality, bimodularity and bounded-value finite left ideal $N_E$ are proved in CC-4, CC-5 and CC-6 and EP-4, EP-5 and EP-6. Let $\tau$ be the faithful normal semifinite trace actually constructed in CD-4, with

$0<\lambda<1,\qquad \tau\circ\beta=\lambda\tau. \tag{FD2}$ There is no factor, separability, countable-decomposability, finite-total-weight or minimal-period hypothesis.

The theorem. Every faithful normal semifinite weight $\Psi$ on $B$ satisfying $\Psi\circ\beta=\Psi$ has a unique faithful normal semifinite weight $\varphi_F$ on $F$ such that

$\Psi(y)=\widehat{\varphi_F}(E(y))\qquad(y\in B_+). \tag{FD3}$ Conversely every faithful normal semifinite $\varphi_F$ has such a composition. Thus composition is a bijection for these exact two weight classes. Through the actual normal isomorphism $\Pi:M\to F$, the descending weight on $M$ is $\varphi=\varphi_F\circ\Pi$. The zero algebra has its unique zero weight and the assertion is immediate; below assume $B\ne0$.

The proof uses the written all-normal tracial density correspondence TD-2, TD-4, TD-5, TD-6, TD-7, the centralizer finite-domain and perturbation arguments CZ-0, CZ-2, CZ-5, CZ-7, GW-1 and GW-4, and the complete spectral domains and transport SF-1, SB-4 and SB-6. Bounded strong-to-ultraweak convergence is SF-2. Each is an earlier actual programme proof. The freely readable human context for tracial densities is Hiai, Theorem 5.13, Corollary 5.14 and Remark 5.15, printed pp.49–50. This citation supplies context; the mathematical inputs are the complete local proofs just linked.

FD-0. The unique density and its exact covariance

By TD, $\Psi=\tau_H$ for a unique positive self-adjoint operator $H$ affiliated with $B$. Semifiniteness removes the infinite-domain projection and faithfulness removes the kernel. In particular $1_{\{0\}}(H)=0$, but no positive lower bound is asserted. For $a\in B_+$, the notation means

$\tau_H(a) =\sup_{k\ge1}\tau((H\wedge k)^{1/2}a(H\wedge k)^{1/2}) =\widehat\tau(a^{1/2}Ha^{1/2}). \tag{FD4}$ These are the complete extended-form sandwiches in TD-2/TD-6; there is no unverified product of unbounded operators.

For any positive affiliated density $A$, normal spectral transport gives $\beta^{-1}(A\wedge k)=\beta^{-1}(A)\wedge k$, including its full operator and square-root domains. Applying (FD2) to every bounded positive cutoff in (FD4) proves

$\tau_A\circ\beta=\lambda\tau_{\beta^{-1}(A)}. \tag{FD5}$ TD-2 proves positive scalar homogeneity of the density map. Since $\Psi\beta=\Psi$, its full uniqueness in TD-5 now gives $H=\lambda\beta^{-1}(H)$, and hence

$\beta(H)=\lambda H,\qquad \beta^n(H)=\lambda^nH\quad(n\in\mathbb Z). \tag{FD6}$ Equality means equality of the transported self-adjoint operators and their complete spectral domains. The same cutoff argument proves (FD5) for $\beta^n$ with factor $\lambda^n$, including negative integers and infinite values.

FD-1. Restriction commutes with the entire scalar extension

We need a restriction statement before knowing that the restriction is semifinite. Let $\rho$ be any normal weight on $B$, and let $\chi=\rho|_{F_+}$. This is a normal weight: bounded increasing positive nets in the concrete von Neumann subalgebra have the same strong supremum in $F$ and $B$.

For $m\in\widehat F_+$, write its complete spectral pair as $(e,A)$, with infinite part $1-e$. The canonical bounded approximants in EP-2/EP-3 are

$a_k=(A\wedge k)e+k(1-e)\in F_+,\qquad a_k\uparrow m. \tag{FD7}$ Their quadratic forms and their values on all positive normal functionals also increase to those of $\iota(m)$ in $B$: CC-4 proves that this inclusion retains the spectral pair. EP-5 therefore gives

$\widehat\chi(m) =\sup_k\chi(a_k)=\sup_k\rho(a_k) =\widehat\rho(\iota(m)). \tag{FD8}$ Every equality includes infinity. No semifiniteness, normal-weight sum theorem, directedness of the dominated functional family, or surjectivity assertion about that family is needed. We will use this identity with a bounded-density weight $\rho=\tau_d$.

FD-2. One spectral window and its complete integer tiling

Put

$p=1_{[\lambda,1)}(H),\qquad d=Hp,\qquad e_n=1_{[\lambda^{n+1},\lambda^n)}(H),\qquad k_n=He_n. \tag{FD9}$ Here $d$ and every $k_n$ are bounded positive elements of $B$; $0\le d\le p\le1$ and $0\le k_n\le\lambda^n e_n$. The products mean bounded spectral functions on the whole Hilbert space. The half-open intervals partition $(0,\infty)$, including every endpoint exactly once. The full spectral calculus and the zero kernel thus prove

$\sum_n e_n=1\ \text{strongly},\qquad \beta^{-n}(p)=e_n,\qquad \lambda^n\beta^{-n}(d)=k_n. \tag{FD10}$ For example the last equality follows by multiplying $\beta^{-n}(H)=\lambda^{-n}H$ by $1_{[\lambda,1)}(\lambda^{-n}H)=e_n$; the powers and endpoints have not been reindexed ambiguously.

The projections $p_j=\beta^j(p)=e_{-j}$ are orthogonal with strong sum $1$. For a finite $J\subset\mathbb Z$ put $q_J=\sum_{j\in J}p_j$. Reindexing the entire counting sum, and testing the orthogonal strong sum on every positive normal functional, gives

$E(p)=1,\qquad E(q_J)=|J|1,\qquad q_J\uparrow1\ \text{strongly},\qquad q_J\in N_E. \tag{FD11}$ The set $J$ is directed by inclusion. This is the spectral sequence of one density and the given integer action; it imposes no countability condition on $B$.

FD-3. The complete reconstruction on every positive element

Define the normal weight

$\varphi_F(x)=\tau_d(x) =\tau(d^{1/2}xd^{1/2})\qquad(x\in F_+). \tag{FD12}$ TD-2 proves normality of $\tau_d$; FD-1 proves normality of its restriction. Semifiniteness and faithfulness will be established separately below.

For $y\in B_+$, use (FD8), the whole counting sum, and EP-5 normality of the scalar extension:

$\widehat{\varphi_F}(E(y)) =\widehat{\tau_d}\!\left(\sum_n\beta^n(y)\right) =\sum_n\tau_d(\beta^n(y)). \tag{FD13}$ This is a supremum over finite subsets, including all infinite values. No finite partial sum in $B$ is falsely regarded as an element of $F$.

Trace scaling and bounded spectral transport give, for every integer $n$,

$\begin{split} \tau_d(\beta^n(y)) &=\lambda^n\tau_{\beta^{-n}(d)}(y)\\ &=\tau_{k_n}(y). \end{split} \tag{FD14}$ The first equality applies $\tau\beta^n=\lambda^n\tau$ to the bounded positive $\beta^{-n}(d)^{1/2}y\beta^{-n}(d)^{1/2}$. The second uses the exact scalar homogeneity and (FD10); no finite-value cyclic manipulation is required.

For finite $J$, $K_J=\sum_{n\in J}k_n$ is a bounded spectral function. On the original Hilbert space, its increasing quadratic forms have supremum

$\sup_J\langle K_J\xi,\xi\rangle =\int_{(0,\infty)}r\,d\mu_\xi^H(r),\qquad D_{\rm finite}=D(H^{1/2}). \tag{FD15}$ This is scalar monotone convergence on the disjoint spectral intervals, proved in SF and SC. Thus $\sum_n k_n=H$ in the complete extended cone, on every normal positive functional by EP-3. TD-2/TD-7 gives equality of the corresponding weights on every bounded positive element, rather than just on a dense finite algebra. Equations (FD13)–(FD15) prove

$\widehat{\varphi_F}(E(y)) =\sum_n\tau_{k_n}(y)=\tau_H(y)=\Psi(y) \qquad(y\in B_+). \tag{FD16}$ The full operator domain is $D(H)=\{\xi:\int r^2\,d\mu_\xi^H(r)<\infty\}$; (FD15) identifies its square-root form domain, not this generally smaller operator domain.

FD-4. The common finite ideal is dense for a proved reason

For the trace $\tau$, TD-1's full closed involution is the antiunitary $J$, and its modular operator is $1$. Thus its centralizer is all of $B$. The CZ unbounded perturbation weight for $H$ is exactly $\tau_H$: its bounded densities $H(1+\varepsilon H)^{-1}$ increase to $H$ in the extended cone, and TD-2 normality of the density map identifies their whole positive-cone supremum with (FD4). CZ-5 therefore gives

$\sigma_t^\Psi(a)=H^{it}aH^{-it}\qquad(a\in B). \tag{FD17}$ Every spectral projection of $H$, including $p_j$ and $q_J$, belongs to $B_\Psi$.

Let $N_\Psi=\{x\in B:\Psi(x^*x)<\infty\}$. The exact centralizer right-multiplication formula in CZ-0, applied to the given faithful n.s.f. $\Psi$, gives for $x\in N_\Psi$

$xq_J\in N_\Psi,\qquad \Psi((xq_J)^*xq_J) =\|J_\Psi\pi_\Psi(q_J)J_\Psi\Lambda_\Psi(x)\|^2 \le\Psi(x^*x). \tag{FD18}$ Also $xq_J\in N_E$, because $q_J\in N_E$ and $N_E$ is a left ideal. Since $q_J\uparrow1$, $xq_J\to x$ boundedly strongly and hence ultraweakly by SF-2. GW-4 proves that $\mathfrak m_\Psi\subset N_\Psi$ is ultraweakly dense. It follows that

$\mathcal I=N_\Psi\cap N_E\quad \text{is ultraweakly dense in }B. \tag{FD19}$ This conclusion comes from these particular centralizer cutoffs. No assertion about the intersection of arbitrary dense finite ideals has been used.

FD-5. Support detection proves semifiniteness of the restriction

For $z\in\mathcal I$, the element

$a_z=E(z^*z)\in F_+,\qquad \varphi_F(a_z)=\Psi(z^*z)<\infty \tag{FD20}$ is bounded; (FD16) justifies its finite scalar value. Let $s=\bigvee_{z\in\mathcal I}s(a_z)\in F$, and set $r=1-s$. If $r\ne0$, then $ra_zr=0$ for every $z$. Whole-cone bimodularity and faithfulness of $E$ give

$E((zr)^*(zr))=rE(z^*z)r=0 \quad\Longrightarrow\quad zr=0. \tag{FD21}$ The map $z\mapsto zr$ is ultraweakly continuous (test the defining square-summable vector coefficients). Its kernel contains the ultraweakly dense $\mathcal I$, so it contains $1$, giving $r=0$. Thus $s=1$.

For $\epsilon>0$, the spectral threshold projection in $F$

$f_{z,\epsilon}=1_{[\epsilon,\infty)}(a_z) \le\epsilon^{-1}a_z \tag{FD22}$ has finite $\varphi_F$-value, by positivity and heredity. As $\epsilon\downarrow0$ these increase to $s(a_z)$. Their supremum over $z,\epsilon$ is consequently $1$. The complete GW-4 criterion proves semifiniteness. Concretely, its construction takes the entire finite positive cone $\mathcal P_{\varphi_F}$, directed by order, and gives

$u_a=a(1+a)^{-1},\quad 0\le u_a\le1,\quad \varphi_F(u_a)<\infty,\quad u_a\uparrow1. \tag{FD23}$ Monotonicity is the inverse-order calculation in GW-4. Each finite threshold projection is approached by $u_{nf}$; their ranges fill the Hilbert space, so the supremum is $1$. For any $b\in F$, $u_abu_a$ belongs to the finite linear algebra and tends boundedly strongly, hence ultraweakly, to $b$. This spells out the finite-domain density conclusion without assuming an increasing net of finite-weight projections.

FD-6. The same window proves faithfulness and uniqueness

For $x\in F_+$, put $a=x^{1/2}px^{1/2}\in B_+$. By (FD11) and full bimodularity,

$E(a)=x^{1/2}E(p)x^{1/2}=x,\qquad \varphi_F(x)=\Psi(x^{1/2}px^{1/2}). \tag{FD24}$ If $\varphi_F(x)=0$, faithfulness of $\Psi$ gives $a=0$, so $x=E(a)=0$. Hence $\varphi_F$ is faithful. This argument is valid even if $\Psi(p)$ is infinite.

If any normal weight $\chi$ on $F$ satisfies $\widehat\chi E=\Psi$, the same test gives $\chi(x)=\Psi(a)=\varphi_F(x)$ for every bounded positive $x$. Thus uniqueness holds even among all normal weights, without first imposing faithfulness or semifiniteness on $\chi$. Transport through $\Pi$ preserves the whole positive cone, finite ideals and normal increasing suprema, as its inverse is also normal.

For completeness, apply bimodularity to $x^*px$, now for arbitrary $x\in F$. The exact finite domains and their finite extensions, supplied by GW-1/GW-2, are

$\begin{split} N_{\varphi_F} &=\{x\in F:px\in N_\Psi\} =\{x\in F:\tau(d^{1/2}x^*xd^{1/2})<\infty\},\\ A_{\varphi_F}&=N_{\varphi_F}\cap N_{\varphi_F}^*,\qquad \mathfrak m_{\varphi_F}=\operatorname{span}N_{\varphi_F}^*N_{\varphi_F},\\ (\varphi_F)_0(x^*y)&=\Psi_0(x^*py) \qquad(x,y\in N_{\varphi_F}). \end{split} \tag{FD25}$ Indeed $E(x^*px)=x^*x$, and polarization of the resulting finite equality proves the final identity, with all four polarized square terms finite before either extension is used. Thus GW-3's complete GNS has the concrete isometric identification

$V\Lambda_{\varphi_F}(x)=\Lambda_\Psi(px),\qquad \operatorname{Ran}V =\overline{\{\Lambda_\Psi(px):x\in N_{\varphi_F}\}}. \tag{FD26}$ The range here is this closed subspace of $H_\Psi$; it is not asserted to be all of $H_\Psi$, or invariant under the whole representation of $F$. Left multiplication remains the actual GNS representation of GW-3. In particular no commutation of $p$ with $F$, or replacement of $px$ by $xp$ in (FD25), is assumed.

FD-7. The converse uses an explicit finite-domain approximation

Let $\varphi_F$ be faithful n.s.f. and put $\Psi=\widehat{\varphi_F}E$. EP-5/EP-6 and CC-5 prove additivity, positive homogeneity and normality for arbitrary increasing bounded positive nets. Its faithfulness follows from faithfulness of both the entire scalar extension and $E$. Reindexing gives $\Psi\beta=\Psi$.

To verify semifiniteness explicitly, choose the GW-4 finite positive contractions $u_i\in F$ with $u_i\uparrow1$. For $z\in N_E$, bimodularity and the boundedness of $E(z^*z)$ give

$\Psi((zu_i)^*(zu_i)) =\varphi_F(u_iE(z^*z)u_i) \le\|E(z^*z)\|\varphi_F(u_i^2)<\infty. \tag{FD27}$ The vectors $zu_i\to z$ strongly. CC-6 makes $N_E$ weak-operator dense; GW-4 now gives semifiniteness of $\Psi$. Combined with FD-0–6 this proves the stated bijection. It proves no arbitrary-cocycle realization, periodic-weight existence, compact duality, factoriality, type classification or whole homogeneity theorem.

FD-8. An exact scalar example and two full-domain exercises

Use CD-5's scalar model with $\lambda=1/2$:

$B=\ell^\infty(\mathbb Z),\quad F=\mathbb C1,\quad (\beta b)_n=b_{n+1},\quad \tau(b)=\sum_n2^n b_n,\quad E(b)=\left(\sum_n b_n\right)1\quad(b\ge0). \tag{FD28}$ Every sum is the supremum of finite nonnegative subsums. Let $\Psi(b)=\frac34\sum_n b_n$. Its faithful n.s.f. density is $H_n=\frac34\,2^{-n}$. The spectral window $[1/2,1)$ contains only $H_0$, so $p=e_0$, $d=\frac34e_0$, and

$\varphi_F(a1)=\tfrac34a,\qquad \tau_{k_n}(b)=\tfrac34 b_n,\qquad \widehat{\varphi_F}(E(b))=\tfrac34\sum_n b_n=\Psi(b). \tag{FD29}$ Here $e_n$ denotes the coordinate projection, and $k_n=\frac34\,2^{-n}e_n$. The trace mass $2^n$ exactly cancels the density mass $2^{-n}$; numerical truncation is not the reason the entire equality holds.

Exercise 1. Replace the window by $[\lambda^{m+1},\lambda^m)$, for an integer $m$, retaining $d_m=H1_{[\lambda^{m+1},\lambda^m)}(H)$. Does the descending weight change?

Solution. Its translated windows still partition $(0,\infty)$. FD-1 and the calculations (FD14)–(FD15) apply with this $d_m$, after shifting the integer index, and reconstruct the same $\Psi$. The projection has $E(p_m)=1$. The uniqueness argument FD-6 then proves $\tau_{d_m}|F=\varphi_F$, including every infinite value. No finite trace of $p_m$ is assumed.

Exercise 2. Describe the form and operator domains in the scalar example, and decide whether $H$ or $H^{-1}$ is bounded.

Solution. In the trace Hilbert space $\ell^2(\mathbb Z,2^n)$, the underlying norm is $\sum_n2^n|\xi_n|^2$. The full domains are

$\begin{split} D(H^{1/2})&=\{\xi\in\ell^2(\mathbb Z,2^n):\sum_n|\xi_n|^2<\infty\},\\ D(H)&=\{\xi\in\ell^2(\mathbb Z,2^n):\sum_n2^{-n}|\xi_n|^2<\infty\}. \end{split} \tag{FD30}$ These follow by multiplying the trace masses by $H_n$, respectively $H_n^2$, and discarding only positive constant factors. Finite coordinate vectors are graph cores by spectral cutoff. $H_n\to\infty$ as $n\to-\infty$, and $H_n^{-1}\to\infty$ as $n\to+\infty$, so both operators are unbounded. Their kernel is nevertheless zero.

The spectral partition and the finite-domain descent mechanism

Exact half-open spectral windows, trace-density cancellation and the complete finite-ideal argument

This original $2400\times1700$ illustration, complete caption, editable SVG, exact rational data and reproduction source are CC0-1.0 to the extent of rights held. Panels A/B are the commutative scalar example in FD-8; panels C/D/E show the general proof in FD-2–6. No finite scalar plot is presented as a model of a factor or as proof of an infinite sum.

For the actual counting weight, the hypotheses are exactly $0<\lambda<1$, $\tau\beta=\lambda\tau$, and a faithful n.s.f. $\Psi$ with $\Psi\beta=\Psi$. TD supplies its nonsingular affiliated density $H$; the full covariance argument gives $\beta(H)=\lambda H$. The window $p=1_{[\lambda,1)}(H)$ has $E(p)=1$, and $d=Hp$ is bounded. The complete reconstruction is $\widehat{\varphi_F}(E(y)) =\sum_{n\in\mathbb Z}\tau_{\lambda^n\beta^{-n}(d)}(y) =\sum_n\tau_{H1_{[\lambda^{n+1},\lambda^n)}(H)}(y) =\tau_H(y)=\Psi(y)\quad(y\in B_+).$ Each sum means a supremum over finite nonnegative subsums. Intrinsic inclusion of the whole extended-positive cone, rather than a restriction only to bounded images, justifies the first equality. The last equality has the exact finite form domain $D(H^{1/2})$; it does not replace the operator domain $D(H)$ by that larger domain.

For panels A/B, take precisely $B=\ell^\infty(\mathbb Z)$, $F=\mathbb C1$, $(\beta b)_n=b_{n+1}$, $\lambda=1/2$, $\tau(b)=\sum_n2^n b_n$, and $\Psi(b)=\frac34\sum_n b_n$. Then $H_n=\frac34\,2^{-n}$, $p=e_0$, $d=\frac34e_0$, and $\varphi_F(a1)=\frac34a$.

Panel A uses the logarithmic coordinate $s=\log_2r$. The spectral interval $[2^{-n-1},2^{-n})$ is exactly $[-n-1,-n)$ in this coordinate: the left endpoint is filled and the right endpoint is hollow. The diamond is the spectral value $s=\log_2(3/4)-n$, strictly inside that interval, associated with the coordinate projection $e_n$. The five displayed rows $n=-2,\ldots,2$ sample the full integer tiling. The identity $\beta(e_n)=e_{n-1}$ concerns these actual projections; no translation law for a bounded finite matrix spectrum is asserted.

Panel B records all seven exact rational samples $n=-3,\ldots,3$. The blue trace mass is $2^n$, the orange density is $\frac34\,2^{-n}$, and the green paired mass is their exact product $3/4$. The vertical axis is logarithmic with base two. Connecting segments are visual guides only. For every bounded positive $b$, not just a finitely supported sequence, positivity and finite-subsums give $\tau_{H e_n}(b)=\tfrac34 b_n,\qquad \sum_n\tau_{H e_n}(b)=\tfrac34\sum_n b_n=\Psi(b),$ including infinite values. The rational data therefore illustrate an identity already proved for the whole positive cone.

Panels C/D/E record the crucial general semifiniteness argument. With $q_J=\sum_{j\in J}\beta^j(p)$, one has $q_J\in B_\Psi$, $E(q_J)=|J|1$, and $q_J\uparrow1$ strongly. Thus for $x\in N_\Psi$, the vectors $z=xq_J$ lie in both $N_\Psi$ and $N_E$, with $\Psi(z^*z)\le\Psi(x^*x)$. These particular centralizer cutoffs prove that the common ideal is ultraweakly dense; there is no theorem asserting density of arbitrary intersections of finite ideals.

The image $a_z=E(z^*z)$ is bounded positive in $F$, with $\varphi_F(a_z)<\infty$. If a projection $r\in F$ annihilates all their supports, faithfulness and bimodularity give $E((zr)^*(zr))=0$, hence $zr=0$ for every $z$ in the dense common ideal. Ultraweak continuity of right multiplication then gives $r=0$. Consequently $\bigvee_zs(a_z)=1$. Every positive-threshold projection $1_{[\epsilon,\infty)}(a_z)\le\epsilon^{-1}a_z$ has finite weight; GW-4 turns their full support into the finite positive contraction net proving semifiniteness.

The lower strip distinguishes the direction of multiplication: the exact descended finite ideal is $N_{\varphi_F}=\{x\in F:px\in N_\Psi\}$, whereas the common-ideal approximation uses $xq_J$. No commutation of $p$ with $F$ is assumed. The separate equality $E(x^{1/2}px^{1/2})=x$ for $x\in F_+$ proves faithfulness and uniqueness, including weights for which $\Psi(p)=\infty$.

The human trace-density context actually read is Hiai, Theorem 5.13, Corollary 5.14 and Remark 5.15, printed pp.49–50. The diagram follows the complete local FD proof and its actual earlier CC/CD/TD/CZ/EP/GW/SF inputs. It asserts neither periodic-weight existence, arbitrary-cocycle realization, compact duality, factoriality nor classification.