Removing a specified unitary from a discrete trace-scaling action

A classical prerequisite reconciliation by GPT-6 Astra (OpenAI), Ultra, October 2026. Independently written programme exposition: CC0 1.0. The construction follows the recorded OA-FLOW linking-algebra argument. The exact general imports below remain distinct from the discrete argument proved here.

This companion supplies the discrete stability step used in Section 6 of Trace scaling on the hyperfinite semifinite factor. It treats a specified unitary in an arbitrary semifinite algebra with separable predual. The fixed algebra can have a nontrivial center. No factor or scalar-coordinate reduction is made.

Theorem DS. Let \(Q\) be a von Neumann algebra with separable predual, let \(T\) be a faithful normal semifinite trace on \(Q\), and let \(\alpha:Q\to Q\) be a normal *-automorphism such that

For every specified \(c\in\mathcal U(Q)\), there is \(z\in\mathcal U(Q)\) satisfying

Consequently every unitary cocycle for the integer action generated by \(\alpha\) is a coboundary, with its specified values, and

The proof of this discrete theorem is given below using the following precise general foundations. A proof of the whole transitive modular and weight-theory chain is not asserted by this companion.

1. The general imports and their exact scope

The completed bounded-topology foundations, the tracial-normality foundations, and Lemmas 3A.1–3A.3 of the bounded-topology lesson provide the concrete predual, positive-functional decomposition, global normality of fixed multiplication, faithful normal states for separable preduals, bounded spectral calculus, arbitrary projection joins and bounded monotone convergence. We use their full proofs at the recorded revisions.

The additional general weight, unbounded spectral and trace inputs are these.

W1 — Extended positives and spectral transport. The extended positive cone \(\widehat B_+\) of a von Neumann algebra supports increasing positive suprema, positive-functional evaluation, bounded sandwiches, and normal-isomorphism transport. Normal weights extend to it preserving increasing suprema. An element fixed by an automorphism has its spectral projections and infinite part in the fixed algebra. Positive self-adjoint affiliated operators have the full Borel spectral calculus; their logarithms, imaginary powers and spectral cuts are transported by normal automorphisms. Scalar integration against their finite vector spectral measures has the monotone and dominated convergence properties used below. These assertions include a possible infinite part of an extended positive; no trace-measurability assumption is imposed on an affiliated density.

W2 — Tracial density and modular identities. If \(S\) is a faithful normal semifinite trace on \(B\) and \(\Phi\) is a faithful normal semifinite weight, there is a unique positive injective self-adjoint affiliated \(H\) with

Its modular group is \(\sigma_t^\Phi=\operatorname{Ad}H^{it}\). Modular groups fix the center pointwise, and a faithful normal semifinite weight with trivial modular group is a trace. The exact programme density provider is OA-MOD, Recognizing a weight by its fixed density, PT01–PT08; PT08 follows from the invariant-weight theorem and retains the spectral, modular and extended-positive dependencies listed there.

W3 — Derivatives under scalar composition. Let \(E:B_+\to\widehat F_+\) be a faithful normal semifinite operator-valued weight for a unital inclusion \(F\subset B\). For faithful normal semifinite \(\omega_1,\omega_2\) whose scalar composites are faithful normal semifinite,

For a faithful normal semifinite weight \(\Psi\) and a unitary \(u\), with \(\operatorname{Ad}u(x)=uxu^*\),

The first identity is the exact general modular-transfer input stated in OA-FLOW lesson 21, (O9). OA-MOD, Operator-valued weight rigidity, OR03–OR04 currently gives a conditional assembly and names its remaining mixed-analytic-graph, finite-matrix-transfer and analytic-generator-comparison contracts. This companion does not declare those contracts proved. The second identity is derived from the balanced-matrix derivative and modular naturality in lesson 21, (HC-2); those general modular inputs retain their own proof boundaries.

T1 — General finite and semifinite traces. An algebra in which every nonzero projection dominates a nonzero finite projection admits a faithful normal semifinite scalar trace. A finite von Neumann algebra \(D\) has a faithful normal normalized center-valued trace \(\operatorname{Tr}_D:D\to Z(D)\). It is positive, linear, a center-module map and preserves partial-isometry equivalence. For \(x\in D\), \(\operatorname{Tr}_D(x)\) lies in the norm-closed convex hull of its unitary conjugates in \(D\). The exact classical imports are Takesaki I, V.2.15, V.2.6–V.2.9 and V.4.6(i). They are not proved merely by the finite matrix trace calculation or by the existence of the given trace on \(Q\).

In Sections 2–3 we prove the projection consequences needed here, using bounded polar decomposition and the stated trace imports. In Sections 4–7 we construct the particular counting weight and its scalar composites; their semifiniteness is proved directly. Thus no unexamined general composition theorem is needed to assert semifiniteness of those particular composites. W3 remains a substantive modular input.

2. Projection comparison at the scope used here

Write \(a\precsim b\) if a partial isometry has initial projection \(a\) and final projection at most \(b\), and write \(a\sim b\) for equivalence. A projection is finite if it is equivalent to no proper subprojection. In its central-support corner, a properly infinite projection is one with no nonzero finite central compression. The splitting assertion for that definition is proved below.

Polar decomposition and sums. For \(x\in B\), bounded calculus gives \(|x|=(x^*x)^{1/2}\). The rule \(|x|\xi\mapsto x\xi\) is an isometry from its range into the range of \(x\), since both vectors have the same norm. Extend it between the closures and put it equal to zero on \(\ker|x|\). The resulting partial isometry \(v\) satisfies \(x=v|x|\). It commutes with every unitary in \(B'\): this is checked on \(\operatorname{ran}|x|\) from the same rule, and on its orthogonal complement. Hence \(v\in B\) by the bicommutant theorem. If partial isometries have separately orthogonal initial and final projections, their finite sums and their adjoints converge strongly, over finite subsets, because the squared vector norms add. Their strong* sum is a partial isometry in \(B\), with the corresponding summed initial and final projections.

Central support and comparison. For a projection \(a\), \(c(a)=\bigvee_{u\in\mathcal U(B)}uau^*\) is central and is the least central projection dominating \(a\). Centrality follows from invariance under every unitary conjugation; unitaries linearly span \(B\), since a self-adjoint contraction is the real part of the unitary \(h+i(1-h^2)^{1/2}\). If \(aBb=0\), then each \(uau^*\) is orthogonal to \(b\), so \(c(a)c(b)=0\). Conversely orthogonal central supports imply \(aBb=0\). If the supports overlap, polar decomposition of a nonzero element of \(aBb\) gives equivalent nonzero subprojections below \(a,b\).

Choose a maximal family of partial isometries with orthogonal initial projections below \(a\) and orthogonal final projections below \(b\). Let their sum be \(v\), and put \(a_0=a-v^*v\), \(b_0=b-vv^*\). Maximality implies \(b_0Ba_0=0\). With \(s=c(b_0)\), it follows that \(a_0s=0\) and \(b_0(1-s)=0\). Compressing \(v\) by these central projections proves

Mutual subequivalence. If \(a\precsim b\precsim a\), replace \(a\) by an equivalent subprojection of \(b\). There is then \(w\in bBb\) with \(w^*w=b\) and \(ww^*\le a\). Put \(d=b-a\), \(d_n=w^nd(w^*)^n\), and \(D=\sum_{n\ge0}d_n\). Since \(dw=0\), the \(d_n\) are orthogonal and \(wDw^*=D-d\). The partial isometry \(wD+(b-D)\) has initial projection \(b\) and final projection \(a\). Thus \(a\sim b\).

The center of a full corner. Suppose \(c(a)=1\). Choose a maximal orthogonal family \((q_i)\) of projections subequivalent to \(a\), including \(q_0=a\). The bridge just proved shows that it sums to \(1\). Choose \(v_i\) with \(v_i^*v_i\le a\), \(v_iv_i^*=q_i\), and \(v_0=a\). For \(h\in Z(aBa)\), the orthogonal strong sum

is bounded by \(\|h\|\). For \(x\in B\), the element \(v_i^*xv_j\in aBa\) commutes with \(h\); comparison of every \((q_i,q_j)\) corner proves that \(\widetilde h\) commutes with \(x\). Thus it is central and compresses to \(h\). A central element with zero compression to \(a\) is zero by fullness. These two maps preserve products, adjoints and bounded increasing suprema, as the same corner calculation shows. They give inverse normal *-isomorphisms of the centers. In particular every central projection of \(aBa\) has the form \(as\), \(s\in Z(B)\).

Finite and properly infinite parts. Subprojections of finite projections are finite: otherwise a strict equivalence below the subprojection, extended by the identity on its complement, contradicts finiteness of the original projection. Finiteness is preserved by equivalence. In a projection corner choose a maximal orthogonal family of finite central projections. Their sum is finite, since an equivalence with a subprojection compresses to equality on each member and hence on their sum. The complementary central projection has no nonzero finite central compression. Equation (DS.8), applied in the central-support algebra, transports this finite/properly infinite split to central projections of \(B\).

Here is the required splitting of a properly infinite unit \(e\). In any nonzero central piece \(s\le e\), choose \(w\) with \(w^*w=s\) and \(ww^*<s\). For \(d=s-ww^*\ne0\), the projections \(w^nd(w^*)^n\) are orthogonal and equivalent. Restrict to \(c(d)\); they then have full central support in that corner. Extend this infinite homogeneous family to a maximal one \((e_j)_{j\in J}\), and let \(q=\sum_j e_j\), \(r=c(d)-q\). Central comparison gives a central piece \(t\) on which \(rt\precsim e_{j_0}t\), while \(e_{j_0}(c(d)-t)\precsim r(c(d)-t)\). The piece \(t\) is nonzero, since otherwise another copy could be put in the remainder, contrary to maximality. On \(t\), map \(rt\) into the \(j_0\) copy and map the infinite family bijectively into its remaining copies. The orthogonal sum gives \(t\precsim qt\); (DS.7) and mutual subequivalence give \(t\sim qt\). Transporting the family by this equivalence makes it fill \(t\). Split its infinite index set into two sets each of the original cardinality. Matching their equivalent members gives two complementary projections each equivalent to \(t\). A maximal collection of central pieces admitting this construction fills \(e\), because the construction works on every nonzero central remainder. Summing gives

Iteration gives countably many orthogonal copies of \(e\) below \(e\).

The countable comparison used by this companion. Let \(e\) be properly infinite, and let \(fBf\) have a faithful normal state, with \(c(f)\le c(e)\). A maximal family of nonzero orthogonal projections \(f_i\le f\) subequivalent to \(e\) fills \(f\): a nonzero remainder has overlapping central support with \(e\) and therefore contains another bridge projection. The family is countable. Indeed its state values are positive and have sum at most one; those at least \(1/n\) form a finite set, and their union contains the entire family. Put each \(f_i\) into a distinct one of the orthogonal copies of \(e\) supplied by (DS.9). Summing the partial isometries proves \(f\precsim e\). Hence two properly infinite projections of full central support whose corners have faithful normal states are equivalent. This conclusion uses the state hypothesis; it is not asserted for arbitrary full properly infinite projections.

3. A trace distinguishes a strict finite enlargement

Lemma DS-F. Let \(B\) admit a faithful normal semifinite trace, and let \(a\) be a finite projection with \(c(a)=1\). If \(a\precsim b\) but \(a\not\sim b\), then there are a nonzero central projection \(s\) and a faithful normal semifinite trace \(S\) on \(Bs\) for which

Proof. Choose \(u^*u=a\), \(a'=uu^*\le b\), and put \(r=b-a'\ne0\), \(t=c(r)\). Fix a faithful normal semifinite trace \(\rho\) on \(B\). Fullness of \(a\) gives \(at\ne0\); semifiniteness gives \(0\ne x\le at\) with \(\rho(x)<\infty\). Let \(h=\operatorname{Tr}_{aBa}(x)\). T1 makes \(h\ne0\), \(h=ht\), and puts \(h\) in the norm-closed convex hull of the unitary orbit of \(x\) inside \(aBa\). Each corner unitary extends by \(1-a\) to a unitary of \(B\), so each of these convex approximants has trace \(\rho(x)\).

For completeness, \(\rho\) is norm lower semicontinuous on \(B_+\). Its finite-trace projections form an upward-directed family: the kernel of \((1-e)f\) on \(fH\) is \(eH\cap fH\), and its range closes to the part of \((e\vee f)H\) orthogonal to \(eH\). Polar decomposition therefore gives \((e\vee f)-e\sim f-(e\wedge f)\), whence \(\rho(e\vee f)\le\rho(e)+\rho(f)\). Their supremum is one, since semifiniteness supplies a nonzero finite-trace spectral cut in any nonzero complementary projection. For \(y\ge0\), normality and traciality give

Each map on the right extends to a bounded positive functional, of norm \(\rho(e)\); its normality is the positive-functional criterion TP1 in the tracial companion. In particular it is norm continuous. Their supremum is norm lower semicontinuous. Applying this to the convex approximants yields \(\rho(h)\le\rho(x)<\infty\).

Choose \(m\) with \(1_{[1/m,\infty)}(h)\ne0\). The full-corner center identification (DS.8) writes this projection as \(as\) for nonzero \(s\in Z(B)\), with \(s\le t\). Since \(as\le mh\), the restriction \(S=\rho|_{Bs}\) satisfies \(S(as)<\infty\). Also \(rs\ne0\), because \(0\ne s\le c(r)\); faithfulness gives \(S(rs)>0\). Compressing \(u\) by \(s\) identifies \(as\) and \(a's\). Thus

The last strict inequality uses the proved finiteness of \(S(as)\). This proves the lemma. \(\square\)

4. The fixed matrix algebra and the counting weight

The zero algebra has the trivial conclusion. First suppose \(Q\ne0\) and \(0<\lambda<1\). Put

This is a normal *-automorphism of \(A\), and \(T_2\gamma=\lambda T_2\) by inner invariance of a trace. The matrix trace is faithful normal semifinite: traciality follows by summing \(T(x_{ij}^*x_{ij})=T(x_{ij}x_{ij}^*)\) over the four entries, normality and faithfulness follow on the diagonal, and finite-trace corner compressions give its semifinite domain. The fixed unital *-algebra \(F\) is ultraweakly closed by normality of \(\gamma\), hence is a von Neumann algebra. The projections \(p,q\) are fixed. A partial isometry \(w\in pFq\) with \(ww^*=p\), \(w^*w=q\) has a unitary entry \(z\in Q\). Its fixedness reads \(\alpha(z)c^*=z\), which is (DS.2). We must therefore prove \(p\sim q\) inside \(F\).

We first construct a wandering projection for \(\gamma\). In any nonzero invariant projection \(e\), choose a nonzero finite-trace projection \(a\le e\). This is possible by semifiniteness and a nonzero spectral cut of a finite-trace positive element. Define \(f=\bigvee_{n\ge0}\gamma^n(a)\). The join estimate proved in (DS.11) gives

Since \(\gamma(f)\le f\) and \(T_2(\gamma(f))=\lambda T_2(f)\), the projection \(b=f-\gamma(f)\) is nonzero. The projections \(f_n=\gamma^n(f)\) decrease for all integer \(n\), so the differences \(\gamma^n(b)=f_n-f_{n+1}\) are pairwise orthogonal. Their sum is invariant and is dominated by \(e\).

Choose a maximal family of such wandering projections whose invariant saturations are orthogonal. Their saturations fill one, since the same construction applies to any nonzero invariant remainder. Summing the initial projections, over finite subsets if necessary, gives one projection \(b_0\) such that

Normality permits commuting each fixed power of \(\gamma\) with these projection sums. The construction does not need an enumeration of the initial maximal family.

For \(x\in A_+\), define

as the supremum of finite positive sums in W1. Addition, homogeneity and normality follow by positive-functional evaluation and interchanging the two directed suprema, over finite subsets of \(\mathbb Z\) and over any increasing net. Reindexing proves both \(E\gamma=E\) and fixedness of its value. The term at zero proves faithfulness. For \(a\in F\), termwise conjugation gives \(E(a^*xa)=a^*E(x)a\).

Let \(B_m=\sum_{|n|\le m}b_n\). Counting each translate in (DS.14) gives

The bounded strong* approximants \(B_mxB_m\to x\) prove that the bounded-output left ideal of \(E\) is ultraweakly dense. Thus \(E:A_+\to\widehat F_+\) is the actual faithful normal semifinite operator-valued weight needed in W3. Neither a real-line integral nor a continuous interpolation of \(\alpha\) has been used.

5. A translating affiliated coordinate

Separable predual gives a faithful normal state on \(Q\), by Lemma 3A.2 of the bounded-topology lesson. Its normalized diagonal sum is a faithful normal state on \(A\): for a positive matrix, vanishing of both diagonal values makes both diagonal operators zero, and then its square root annihilates both coordinate subspaces. Restriction gives a faithful normal state \(\omega\) on \(F\). In particular all projection corners of \(F\) have faithful normal states, after normalizing restrictions.

Set \(\Phi=\omega\circ E\), evaluating the extended positive value as in W1. It is normal and faithful. It is semifinite directly: (DS.16) makes it finite on every bounded positive element of \(B_mAB_m\), and these corners are ultraweakly dense. Reindexing gives \(\Phi\gamma=\Phi\).

For \(u\in\mathcal U(F)\), bimodularity yields \(\Phi\circ\operatorname{Ad}u=(\omega\circ\operatorname{Ad}u)\circ E\). Both weights on \(F\) here are faithful normal states, so their composites are semifinite by the same corner proof. Apply (DS.5) and then (DS.6) on the two algebras. They give

The second identity follows by multiplying by \(u\) and using the linear span of unitaries. This is the exact point at which the general modular-transfer input W3 enters.

By W2, write \(\Phi=(T_2)_H\). The covariance of this density can be checked on each bounded spectral truncation: for a normal automorphism \(\beta\) with \(S\beta=aS\),

Indeed transport \(\beta(x)^{1/2}\) and \(\min(H,k)\) through \(\beta^{-1}\) in (DS.4), use the trace scaling, and take increasing suprema. Scalar homogeneity gives the last equality on the whole positive cone. Therefore invariance of \(\Phi\) and uniqueness of its density give \(H=\lambda\gamma^{-1}(H)\), or \(\gamma(H)=\lambda H\). Put

These are affiliated-operator identities through spectral transport. The density is nonsingular, so \(\log H\) is self-adjoint; no bounded inverse or trace measurability has been assumed.

6. Fourier series recover the algebra and lift its center

For real \(r\), set

The series for \(d\) converges uniformly on \([0,1]\), since its tails are bounded by a constant times \(\sum_{|n|>m}n^{-2}\). Translation gives one-periodicity. A nearest integer shows \(d(r)\ge4/5\); continuity on a period gives a finite upper bound. Thus \(k\) is continuous, strictly positive, at most one, and vanishes at infinity. Periodicity gives the exact identity

The second identity is a strong positive sum, verified against every finite vector spectral measure using W1. Also \(k(R)\) is injective with dense range.

Fix \(x\in A\), and write

For each positive normal functional \(\rho\), positive-functional Cauchy–Schwarz gives

For example, apply Cauchy–Schwarz to \(a=k(R-n)\) and \(\gamma^n(x)a\), and use \(x^*x\le\|x\|^2\). The positive spanning theorem CP07 proves absolute summability for every normal functional. For vectors in a faithful concrete representation, scalar Cauchy–Schwarz also gives

for every finite \(J\subset\mathbb Z\). Multiplication on the left by \(e^{it(R-n)}\) preserves the same estimates: it commutes with the window, so the second vector norm is unchanged. Compactness of the bounded dual ball and the absolutely convergent scalar tests consequently define a unique operator

Reindexing the absolutely convergent tests and using normality of \(\gamma\) proves \(V_t(x)\in F\). The order of the factors in (DS.24) follows from (DS.19); \(x\) has not been commuted with \(R\).

Let \(C=W^*(F,e^{itR}:t\in\mathbb R)\). This algebra contains the spectral projections of \(R\). To see the spectral-generation point explicitly, its resolvent is the ultraweak integral \((R-i)^{-1}=i\int_0^\infty e^{-t}e^{-itR}\,dt\), as follows from the scalar spectral identity; the Borel calculus for that bounded normal resolvent recovers the spectral projections of \(R\). Equation (DS.24) gives \(\widehat v(t)\in C\).

If \(\ell\in A_*\) annihilates \(C\), then the Fourier series \(\sum_n e^{-itn}\ell(v_n)\) vanishes for every \(t\). It converges absolutely and uniformly. Multiplication by \(e^{itm}\) followed by integration over \([0,2\pi]\) recovers \(2\pi\ell(v_m)\), so all coefficients vanish. In particular \(\ell(k(R)xk(R))=0\). Predual separation of the ultraweakly closed linear space \(C\) gives \(k(R)xk(R)\in C\).

Let \(f_m=1_{[1/m,1]}(k(R))\), and let \(j_m=k(R)^{-1}f_m\) be the bounded spectral inverse on this cut. Then

Since \(f_m\uparrow1\), these bounded strong* approximants converge to \(x\). Hence \(C=A\).

If \(s\in Z(F)\) is a projection, its modular orbit for \(\omega\) is constant by W2. Equation (DS.17) makes it commute with every \(H^{it}\), hence with every \(e^{itR}\). Generation gives \(s\in Z(A)\). Spectral approximation extends this to

In particular \(p,q\) have full central support in \(F\): a central \(s\) with \(sp=0\) also commutes with the matrix units, so \(sq=e_{21}(sp)e_{12}=0\) and \(s=0\); interchange the two corners for the other assertion.

7. Every local trace gives the same two corner sizes

Let \(s\in Z(F)\) be a projection such that \(Fs\) admits a faithful normal semifinite trace \(\rho\). By (DS.26) it is central in \(A\) and fixed by \(\gamma\). Restrict the counting weight to \(E_s:(As)_+\to\widehat{Fs}_+\), and put \(\Phi_\rho=\rho\circ E_s\).

Here is a direct semifiniteness proof even when \(\rho(s)=\infty\). Finite-\(\rho\)-trace projections \(e_i\in Fs\) form an increasing net with supremum \(s\), by the argument of (DS.11). Set \(u_{m,i}=sB_m e_i\). This product net consists of contractions and tends strongly* to \(s\). Bimodularity and (DS.16) give

For each bounded \(y\ge0\) in \(As\), the positive element \(u_{m,i}^*yu_{m,i}\) has \(\Phi_\rho\)-value at most \(\|y\|\Phi_\rho(u_{m,i}^*u_{m,i})\). These elements converge ultraweakly to \(y\). Thus the finite positive cone is ultraweakly dense, proving semifiniteness. Normality and faithfulness follow from W1 and faithfulness of \(E_s,\rho\). The same proof applies to \(\rho\circ\operatorname{Ad}u=\rho\) for \(u\in\mathcal U(Fs)\).

Repeat Sections 5–6 on this central piece with \(\rho\) in place of \(\omega\). The argument there used only the proved semifiniteness of the composites. It supplies a density \(H_\rho\), a coordinate \(R_\rho=L^{-1}\log H_\rho\) with \(\gamma(R_\rho)=R_\rho-s\), and generation of \(As\) by \(Fs\) and its imaginary powers. Since \(\rho\) is a trace, (DS.17) makes those powers commute with \(Fs\). They also commute with one another. Generation therefore makes them central in \(As\), so \(R_\rho\) is affiliated with \(Z(As)\). W2 then says that \(\Phi_\rho\) is a trace.

The half-open spectral projection

For every real scalar coordinate, exactly one translate lies in \([0,1)\), including when it is an integer. W1 turns this exact scalar identity into the projection sum. Replacing this half-open interval by a closed one would double count possible atoms and is not permitted.

The projection \(h\) is central in \(As\). For \(a=se_{12}h\), this gives \(aa^*=sph\), \(a^*a=sqh\). Bimodularity and (DS.28), followed by traciality of \(\Phi_\rho\), yield

This is an equality of extended values for every such central piece and every faithful normal semifinite trace on it. It will be used with a trace finite on a smaller finite projection, rather than by comparing two infinite values under one trace.

8. Equivalence in the fixed algebra and the specified solution

Apply the finite/properly infinite central decompositions of Section 2 to \(p,q\) in \(F\), and take their common refinement. On a piece where both are properly infinite, they are equivalent by the countable comparison in Section 2: their central supports are full by (DS.26), and their corners have faithful normal states by Section 5.

On any remaining piece at least one of the two full projections is finite. This piece of \(F\) is semifinite in the projection sense required by T1. Indeed every nonzero projection has a bridge to that finite full projection, and therefore contains a nonzero subprojection equivalent to a subprojection of a finite projection. T1 supplies a faithful normal semifinite trace there.

Apply central comparison (DS.7) and refine once more. If the finite side is subequivalent to the other side but not equivalent, Lemma DS-F produces a further nonzero central piece and a faithful trace finite on the smaller projection and strictly larger on the other. This contradicts (DS.29), which holds for that precise trace and piece. If the finite side is the larger side, the smaller one is finite too; reverse the roles and apply the same lemma. Full central support persists under every nonzero central compression. Thus \(p\sim q\) on every piece.

Orthogonal central sums of the resulting partial isometries give \(w\in pFq\) with \(ww^*=p\), \(w^*w=q\). Write its sole entry as \(z\in Q\). The support equations say \(zz^*=z^*z=1\), and fixedness says

This proves the theorem for \(0<\lambda<1\), with the actual given unitary \(c\).

For \(\lambda>1\), apply that case to \(\alpha^{-1}\) and \(d=\alpha^{-1}(c^*)\). It gives \(d=z^*\alpha^{-1}(z)\). Applying \(\alpha\) and taking adjoints yields \(c=z^*\alpha(z)\), with the same orientation.

For positive integers, the prescribed cocycle satisfies \(c_n=c\alpha(c)\cdots\alpha^{n-1}(c)\); substituting (DS.30) telescopes to \(z^*\alpha^n(z)\). The identity \(1=c_n\alpha^n(c_{-n})\) gives the same formula for negative integers, and the value at zero is one. Finally \(zc=\alpha(z)\) makes the factors in (DS.3) cancel in their displayed order. This proves the stated inner conjugacy, not merely existence of an unspecified conjugacy. \(\square\)

9. Source roles and the remaining programme boundary

The existing OA-FLOW proof in lesson 28, Removing noncentral cocycles from a trace-scaling action, develops the fixed matrix algebra, translating density, Fourier generation, center lifting and local trace comparison for real actions at arbitrary cardinality. Its current projection subsection proves central comparison, mutual subequivalence, the finite/properly infinite split, proper infiniteness and the full-corner center identification used here. Lesson 19 proves the wandering-projection construction. The current lesson 140, Discrete decomposition of type III lambda factors, now gives the full counting-weight and Fourier-series adaptation, including the half-open cut, at its wider stated scope. The earlier frozen version only said those steps were unchanged; that wording alone was insufficient to provide the discrete details.

This companion makes those discrete steps readable together and uses the faithful normal state available under the requested separable-predual hypothesis for the properly infinite comparison. That step is justified in Section 2. It does not replace OA-FLOW's stronger arbitrary-cardinality theorem: its two-sided Fourier support argument and countable amplification comparison remain in that provider with their original scope. The continuous-action theorem and its continuity/measurability obligations likewise remain separate.

The human antecedent is Masamichi Takesaki, Theory of Operator Algebras II (2003), Theorem XII.1.11, printed pp. 378–379. The theorem there is for a real one-parameter action; its proof uses dual recognition, a translation model and a stabilized type III case. Theory of Operator Algebras III (2003), XVII.3.12, printed p. 283, invokes XII.1.11 when promoting outer conjugacy of trace-scaling automorphisms to conjugacy. The counting construction above supplies the discrete proof needed here, without asserting that the real-action statement literally has integer-action hypotheses. The programme's matrix-comparison route differs from the organization of the source's proof.

The action-specific argument is complete with W1–W3 and T1 as stated. W3's general modular transfer still has the explicit OR04 component contracts described in Section 1. W2's PT proof still uses its general modular, generator, spectral and extended-positive foundations. T1's general center-valued trace, finite averaging and semifinite trace-existence proofs are separate imports. The completed bounded/predual companions do not by themselves prove these unbounded or modular assertions. Consequently this companion supplies the full local discrete proof.