# Approximate innerness and absorption descent

Crossing by an approximately inner automorphism can introduce Fourier degrees, but those degrees can be removed without spoiling a sufficiently good modular estimate. The correction is a unitary of the opposite degree. We prove the estimate in standard form, including nonfaithful positive functionals, and then deduce tensor absorption.

## 1. The descent theorem and its exact inputs

Fix \(0<\lambda<1\), and let \(R_\lambda\) be the constant two-by-two product defined in LA. A type III factor with separable predual is called strongly \(\lambda\)-stable here when it is normally isomorphic to its tensor product with this particular \(R_\lambda\).

**Descent theorem.** Let \(M\) be a type III factor with separable predual, and let \(\theta\) lie in the closure of \(\operatorname{Int}(M)\) for pointwise norm convergence on the predual. Set

\[
N=M\rtimes_\theta\mathbb Z.
\]

If

\[
N\cong N\bar\otimes R_\lambda
\tag{DAS.1}
\]

by a normal isomorphism with normal inverse, then

\[
M\cong M\bar\otimes R_\lambda.
\tag{DAS.2}
\]

The hypothesis (DAS.1) makes sense even when \(N\) is not a factor. No aperiodicity of \(\theta\), invariant state for \(\theta\), or approximation property of \(M\) is required. In particular this proves the usual descent assertion when \(\theta\) is aperiodic and the crossed product is strongly \(\lambda\)-stable.

Here are the earlier complete arguments used in the proof:

- FS02–04 gives the faithful normal GNS representation and the closed Tomita operators.
- SF04–07 and SF10–11 constructs the natural cone, its positive representatives and their norm estimate. SE06 identifies the cone associated with a cyclic separating positive vector.
- AI03–05 compares bounded functional intertwiners with their positive-vector intertwiners, including nonfaithful functionals and algebras which are not factors.
- Circle Fourier analysis, CC0 supplies probability Haar measure, its character integrals and the onto scalar Fourier map.
- LA05 proves the halving property in every absorbed product. LA06, with LA01–04, proves that the aggregate vector criterion implies (DAS.2).

We use the right action \(\eta a=Ja^*J\eta\), and the functional actions

\[
(\varphi a)(b)=\varphi(ab),\qquad
(a\varphi)(b)=\varphi(ba).
\]

For positive functionals \(\varphi_1,\ldots,\varphi_m\), put \(c=\sqrt\lambda\) and define

\[
\begin{aligned}
D_M(y)&=\sum_{j=1}^m\|y\xi_{\varphi_j}\|^2
       =\sum_{j=1}^m\varphi_j(y^*y),\\
Q_M(y)&=\sum_{j=1}^m
 \|\xi_{\varphi_j}y-cy\xi_{\varphi_j}\|^2.
\end{aligned}
\tag{DAS.3}
\]

The criterion in LA06 asks for a nonzero \(y\in M\) with \(Q_M(y)\le\varepsilon D_M(y)\), for every finite positive family and every \(\varepsilon>0\). It permits \(D_M(y)=0\); we will handle that case explicitly.

## 2. The regular crossing and its faithful expectation

We first work with any von Neumann algebra \(M\) having a faithful normal state \(\rho\), represented on its GNS space \(H_\rho\). On \(K=\ell^2(\mathbb Z;H_\rho)\), define

\[
(\pi(a)\eta)_n=\theta^{-n}(a)\eta_n,\qquad
(U\eta)_n=\eta_{n-1}.
\tag{DAS.4}
\]

Thus \(U\pi(a)U^*=\pi(\theta(a))\). The representation \(\pi\) is faithful and normal: faithfulness is seen in coordinate zero, and its vector coefficients are absolutely summable sums of normal coefficients, by Cauchy–Schwarz and CP04–06. Define \(N\) as the von Neumann algebra generated by \(\pi(M)\) and \(U\), and identify \(M\) with \(\pi(M)\).

The algebraic Fourier polynomials \(\sum_n a_nU^n\) are ultraweakly dense in \(N\), by the ultraweak bicommutant argument in FS02. Compression to coordinate zero defines a normal unital completely positive map \(E:N\to M\). To verify its range and faithfulness directly, write \(X_{kl}\) for an operator matrix entry. For a Fourier polynomial,

\[
E(X)=X_{00}=a_0,\qquad
X_{nn}=\theta^{-n}(X_{00}).
\tag{DAS.5}
\]

The entries of \(X\), and the displayed applications of normal \(\theta^{-n}\), are normal maps. The identities and \(X_{00}\in M\) therefore extend to all \(X\in N\). Compression also gives

\[
E(aXb)=aE(X)b,\qquad a,b\in M.
\tag{DAS.6}
\]

If \(X\ge0\) and \(E(X)=0\), every diagonal block of \(X\) is zero. For a vector \(\eta\) supported in one coordinate,
\(\|X^{1/2}\eta\|^2=\langle X\eta,\eta\rangle=0\).
Finite-coordinate vectors are dense, so \(X=0\). This proves faithfulness.

Use the circle convention

\[
\gamma_z(a)=a,\qquad \gamma_z(U)=zU,\qquad z\in\mathbb T.
\tag{DAS.7}
\]

It is implemented on \(K\) by \((W_z\eta)_n=z^n\eta_n\). These unitaries are strongly continuous: truncate a vector to finitely many coordinates, check continuity there and use the unitary bound on the tail. Hence the action is point-strong* continuous. Its fixed algebra is exactly \(M\). Indeed a fixed operator has only diagonal matrix entries, and (DAS.5) identifies it with \(\pi(X_{00})\).

With probability Haar measure on \(\mathbb T\), define

\[
X_q=\int_{\mathbb T}z^{-q}\gamma_z(X)\,dz.
\tag{DAS.8}
\]

These bounded integrals may be evaluated on vectors; their normal coefficient integrals agree with the resulting operators. They belong to \(N\), by weak closure, and have norm at most \(\|X\|\). Invariance of Haar measure gives \(\gamma_z(X_q)=z^qX_q\). Conversely an operator of this degree satisfies \(X_qU^{-q}\in N^\gamma=M\). Thus the degree-\(q\) space is

\[
N_q=MU^q.
\tag{DAS.9}
\]

In particular \(X_0=E(X)\), by the diagonal calculation. Our positive-character convention is fixed by (DAS.7); using the opposite convention reverses all degree labels together.

## 3. Comparing the two standard forms

Put \(\psi=\rho E\) and \(\Omega=\delta_0\xi_\rho\). This vector represents \(\psi\), and is cyclic for \(N\): the vectors \(aU^q\Omega\), with \(a\in M\), are dense in each coordinate. Faithfulness of \(E\) and \(\rho\) makes \(\psi\) faithful, so \(\Omega\) is also separating. FS supplies the closed Tomita operator \(S_N\), its polar decomposition and the standard form of \(N\) on this same \(K\).

Let \(K_0=\overline{M\Omega}\), and let \(e\) be the projection onto it. This is coordinate zero, naturally identified with \(H_\rho\). For \(X\in N\),

\[
eX\Omega=E(X)\Omega,\qquad
eXe=E(X)e.
\tag{DAS.10}
\]

For the second equality, test against \(a\Omega,b\Omega\), \(a,b\in M\), and use
\(\psi(b^*Xa)=\rho(b^*E(X)a)\).
The subspace \(K_0\) reduces \(M\).

There is also reduction of the closed Tomita graph, not just Hilbert-space invariance. On the core \(N\Omega\),

\[
S_N eX\Omega=E(X)^*\Omega=eS_NX\Omega.
\tag{DAS.11}
\]

Boundedness of \(e\) and graph approximation extend this equality to \(D(S_N)\). The same holds for \(1-e\); the adjoint-domain definition then gives reduction of \(S_N^*\) as well. Moreover, projecting a core approximation of a vector in \(D(S_N)\cap K_0\) gives approximants in \(M\Omega\). Consequently the restriction to \(K_0\) is exactly \(S_M\), not a larger closed extension. Polar decomposition and spectral calculus now give

\[
J_N|_{K_0}=J_M,\qquad
\Delta_N|_{K_0}=\Delta_M
\tag{DAS.12}
\]

with their full restricted domains.

For a finite faithful GNS functional, SF04 and SE06 describe the natural cone as the closure of \(aJaJ\Omega\). Applying (DAS.12) for \(a\in M\) shows that the natural cone \(P_M\subset K_0\) is contained in \(P_N\). If \(\varphi\in M_*^+\) and \(\eta=\xi_\varphi\in P_M\), then (DAS.10) gives, for every \(X\in N\),

\[
\langle X\eta,\eta\rangle
 =\langle E(X)\eta,\eta\rangle
 =\varphi(E(X)).
\]

Existence and uniqueness of positive representatives in SF07 and SF10–11 proves

\[
\xi_{\varphi E}=\xi_\varphi\in K_0.
\tag{DAS.13}
\]

This includes zero and nonfaithful functionals. Both bounded module actions by \(M\) agree under this identification, by (DAS.12). Therefore every residual and mass in (DAS.3), evaluated at \(y\in M\), has exactly the same norm whether computed in the standard form of \(M\) or of \(N\).

## 4. Parseval for both sides of a modular residual

The unitaries \(W_z\) in Section 2 fix \(\Omega\), implement \(\gamma_z\), and conjugate \(S_N\) to itself on its complete graph. Polar uniqueness gives \(W_zJ_N=J_NW_z\). The cone description \(P_N=\overline{\{aJ_NaJ_N\Omega:a\in N\}}\) then shows that \(W_zP_N=P_N\). For \(\psi_j=\varphi_jE\), invariance under \(\gamma\) and uniqueness of the positive representative imply

\[
W_z\xi_{\psi_j}=\xi_{\psi_j}.
\tag{DAS.14}
\]


The operators

\[
P_q=\int_{\mathbb T}z^{-q}W_z\,dz
\]

are mutually orthogonal projections. Indeed the unitary representation law and character integration give \(P_qP_r=0\) for \(q\ne r\), \(P_q^2=P_q=P_q^*\), and \(W_zP_q=z^qP_q\). Their ranges span \(K\). Here is the completeness argument without a pointwise Fourier-convergence assumption. For \(z=e^{it}\), the Fejer kernel

\[
F_L(t)=\frac1{L+1}\left|\sum_{k=0}^Le^{ikt}\right|^2
 =\sum_{|q|\le L}\left(1-\frac{|q|}{L+1}\right)e^{iqt}
\tag{DAS.15}
\]

is nonnegative and has integral one for \(dt/(2\pi)\). Outside any fixed neighborhood of zero modulo \(2\pi\), the geometric-sum formula bounds it by a constant divided by \(L+1\). Strong continuity of \(W\), with this bound, gives

\[
\int F_L(t)W_{e^{it}}\eta\,\frac{dt}{2\pi}\longrightarrow\eta.
\]

Each integral is a finite linear combination of the \(P_q\eta\). Orthogonality and completeness prove Parseval's identity for every \(\eta\in K\).

Bounded integration, (DAS.14) and covariance of both module actions yield

\[
\begin{aligned}
P_q(X\xi_{\psi_j})&=X_q\xi_{\psi_j},\\
P_q(\xi_{\psi_j}X)&=\xi_{\psi_j}X_q.
\end{aligned}
\tag{DAS.16}
\]

For the right action, the exact covariance is
\(W_z(\eta X)=(W_z\eta)\gamma_z(X)\); it follows from commutation with \(J_N\). The right-hand vector has the same degree as the left-hand vector.

Define \(D_N,Q_N\) by (DAS.3), using \(\psi_j\) in place of \(\varphi_j\). Applying Parseval first to the left vectors and then to the differences gives

\[
D_N(X)=\sum_{q\in\mathbb Z}D_N(X_q),\qquad
Q_N(X)=\sum_{q\in\mathbb Z}Q_N(X_q).
\tag{DAS.17}
\]

All terms are nonnegative; the equalities include the whole bounded operator \(X\), not merely Fourier polynomials.

Suppose that \(\sum_j\psi_j\) is faithful and \(X\ne0\). Then \(D_N(X)>0\). If \(Q_N(X)\le dD_N(X)\), at least one nonzero coefficient \(x=X_q\) satisfies

\[
Q_N(x)\le dD_N(x),\qquad D_N(x)>0.
\tag{DAS.18}
\]

Otherwise every nonzero coefficient would contribute a strictly positive term to
\(\sum_q[Q_N(X_q)-dD_N(X_q)]\), contradicting (DAS.17). Terms with zero mass are zero operators by faithfulness. Thus the selection also proves \(x\ne0\).

## 5. Absorption supplies witnesses with positive mass

Assume (DAS.1), and let \(\psi_1,\ldots,\psi_m\in N_*^+\) have nonzero total mass
\(C=\sum_j\psi_j(1)>0\).
LA05, together with normal-isomorphism transport from LA02, gives a sequence of halving partial isometries \(h_k\in N\) satisfying

\[
h_k^*h_k+h_kh_k^*=1,\qquad
\|\psi_jh_k-\lambda h_k\psi_j\|\longrightarrow0
\quad(1\le j\le m).
\tag{DAS.19}
\]

For example one may choose error less than \(1/k\) simultaneously for this finite family. No factor hypothesis on \(N\) is used by LA05.

The operators are contractions. AI04–05, applied separately to the positive pair \(\psi_j,\lambda\psi_j\), therefore gives

\[
Q_N(h_k)\longrightarrow0.
\tag{DAS.20}
\]

The denominator cannot disappear. Evaluation of each functional defect in (DAS.19) at \(h_k^*\) shows

\[
|\psi_j(h_kh_k^*)-\lambda\psi_j(h_k^*h_k)|
 \le\|\psi_jh_k-\lambda h_k\psi_j\|.
\]

Adding the identities and using the halving relation yields

\[
\left|C-(1+\lambda)D_N(h_k)\right|
 \le\sum_j\|\psi_jh_k-\lambda h_k\psi_j\|
 \longrightarrow0.
\tag{DAS.21}
\]

Thus \(D_N(h_k)\to C/(1+\lambda)>0\). Given \(d>0\), a sufficiently large \(k\) satisfies

\[
h_k\ne0,\qquad Q_N(h_k)\le dD_N(h_k).
\tag{DAS.22}
\]

This is the precise relative-error witness required in Section 4. Neither (DAS.20) alone nor an assertion of small absolute error would suffice.

## 6. Removing a degree by approximate innerness

For an automorphism \(\alpha\), write \(\alpha_*\varphi=\varphi\circ\alpha^{-1}\). This is an isometry of \(M_*\). Composition and inversion are continuous for pointwise predual norm convergence. Explicitly,

\[
\begin{aligned}
\|(\alpha_i\beta_i)_*\varphi-(\alpha\beta)_*\varphi\|
&\le\|(\beta_i)_*\varphi-\beta_*\varphi\|
 +\|(\alpha_i)_*(\beta_*\varphi)-\alpha_*(\beta_*\varphi)\|,\\
\|(\alpha_i^{-1})_*\varphi-(\alpha^{-1})_*\varphi\|
&=\|\varphi-(\alpha_i)_*((\alpha^{-1})_*\varphi)\|.
\end{aligned}
\tag{DAS.23}
\]

Consequently every integer power of an approximately inner \(\theta\) is approximately inner.

Fix \(q\in\mathbb Z\) and \(x\in N_q\). Choose unitaries \(v_i\in M\) with
\(\operatorname{Ad}(v_i)\to\theta^q\) on the predual, and set

\[
u_i=v_iU^{-q},\qquad
\beta_i=\operatorname{Ad}(v_i)\circ\theta^{-q}.
\tag{DAS.24}
\]

Then \(u_i\) is a unitary of degree \(-q\), \(\operatorname{Ad}(u_i)|_M=\beta_i\), and \(\beta_i\to\mathrm{id}_M\). Thus

\[
y_i=u_ix\in N_0=M,\qquad y_i^*y_i=x^*x.
\tag{DAS.25}
\]


The expectation is equivariant under this normalizer:

\[
E(u_iXu_i^*)=\beta_i(E(X)).
\tag{DAS.26}
\]

One may check (DAS.26) on Fourier polynomials and extend by normality. Equivalently, \(\gamma_z(u_i)=z^{-q}u_i\) makes the scalar phases cancel inside the averaging formula for \(E\).

The lift \(M_*\to N_*\), \(\varphi\mapsto\varphi E\), is isometric. Contractivity gives one inequality; restriction to the unit ball of \(M\), where \(E\) is the identity, gives the other. Hence

\[
\|u_i\psi_ju_i^*-\psi_j\|
 =\|(\beta_i)_*\varphi_j-\varphi_j\|
 \longrightarrow0.
\tag{DAS.27}
\]

Here \(u_i\psi_ju_i^*\) means \(X\mapsto\psi_j(u_i^*Xu_i)\).
The positive-vector estimate in SF07, applied to the cone vector
\(u_i\xi_{\psi_j}u_i^*\), gives

\[
\|\xi_{\psi_j}u_i-u_i\xi_{\psi_j}\|^2
 \le\|u_i\psi_ju_i^*-\psi_j\|.
\tag{DAS.28}
\]

Right multiplication by a unitary preserves the Hilbert norm, which explains the left side of this inequality.

Finally, associativity of the two commuting module actions gives the exact identity

\[
\begin{aligned}
\xi_{\psi_j}u_ix-cu_ix\xi_{\psi_j}
={}&(\xi_{\psi_j}u_i-u_i\xi_{\psi_j})x\\
 &+u_i(\xi_{\psi_j}x-cx\xi_{\psi_j}).
\end{aligned}
\tag{DAS.29}
\]

Take the norm in the finite Hilbert direct sum over \(j\), and use the operator-norm bound on right multiplication by \(x\). If

\[
b_i^2=\sum_j\|(\beta_i)_*\varphi_j-\varphi_j\|,
\]

then (DAS.25), (DAS.28) and (DAS.29) imply

\[
D_N(y_i)=D_N(x),\qquad
\sqrt{Q_N(y_i)}
 \le\sqrt{Q_N(x)}+\|x\|b_i,\qquad b_i\longrightarrow0.
\tag{DAS.30}
\]

This estimate is uniform for the fixed finite family. Its tolerance may depend on the chosen coefficient \(x\), as the proof permits.

**Proposition 6.1 (equality of the optimal local errors).** Let \(M\) be any nonzero von Neumann algebra, let \(\theta\) be approximately inner, and let the finite positive family \((\varphi_j)\) have faithful sum. For \(N=M\rtimes_\theta\mathbb Z\), lift this family by the coefficient expectation and define \(Q,D\) as in (DAS.3). Then

\[
 \inf_{0\ne y\in M}\frac{Q_M(y)}{D_M(y)}
   =\inf_{0\ne X\in N}\frac{Q_N(X)}{D_N(X)}.
 \tag{DAS.30a}
\]

All denominators are strictly positive. This assertion requires neither a type III hypothesis, separable predual, nor tensor absorption.

**Proof.** Normalize the sum of the given functionals to obtain the faithful normal state used in Sections 2–4. Its lift is faithful, so every nonzero operator on either side has positive mass. The inclusion \(M\subset N\) and (DAS.13) show that the right-hand infimum is at most the left-hand one.

For the reverse inequality, fix \(X\ne0\) and put \(d=Q_N(X)/D_N(X)\). The Parseval selection in (DAS.18) gives a nonzero coefficient \(x=X_q\) with \(Q_N(x)/D_N(x)\le d\). The elements \(y_i=u_ix\) from (DAS.25) are nonzero, belong to \(M\), and have the same positive mass as \(x\). Dividing (DAS.30) by \(\sqrt{D_N(x)}\), then taking the upper limit, gives

\[
 \limsup_i\frac{Q_M(y_i)}{D_M(y_i)}
   \le\frac{Q_N(x)}{D_N(x)}\le d.
 \tag{DAS.30b}
\]

The standard-form comparison justifies the subscript \(M\) here. Hence the left-hand infimum is at most the ratio of every nonzero \(X\in N\), proving (DAS.30a). The approximately inner implementers may form a net; selecting one sufficiently accurate index for each fixed tolerance is all the argument needs. \(\square\)


## 7. The full absorption conclusion

Let \(\varphi_1,\ldots,\varphi_m\in M_*^+\) and \(\varepsilon>0\) be arbitrary. Empty families impose no condition. For a nonempty family put

\[
s=\bigvee_j s(\varphi_j).
\]

If \(s\ne1\), choose \(y=1-s\ne0\). SF06's support description, with \(J\xi_{\varphi_j}=\xi_{\varphi_j}\), gives
\(y\xi_{\varphi_j}=\xi_{\varphi_j}y=0\).
Thus \(Q_M(y)=D_M(y)=0\), satisfying the required inequality. This also handles the family consisting only of zero functionals.

Now suppose \(s=1\). Then \(\varphi=\sum_j\varphi_j\) is faithful: if \(\varphi(a)=0\) with \(a\ge0\), then \(a^{1/2}\xi_{\varphi_j}=0\) for every \(j\). Commutation with \(M'\) makes \(a^{1/2}\) vanish on \(\overline{M'\xi_{\varphi_j}}=s(\varphi_j)H\). Their joined support is one, so \(a=0\). Set \(\rho=\varphi/\varphi(1)\), and use Sections 2–4 with this faithful normal state. The lifted family \(\psi_j=\varphi_jE\) has faithful sum, and its positive representatives agree with those of the original family by (DAS.13).

By Section 5 choose \(X\ne0\) with

\[
Q_N(X)\le\frac{\varepsilon}{4}D_N(X).
\]

Section 4 supplies a coefficient \(x=X_q\ne0\) satisfying the same inequality and \(D_N(x)>0\). Choose \(i\) in Section 6 sufficiently far out that

\[
\|x\|b_i<\frac{\sqrt\varepsilon}{2}\sqrt{D_N(x)}.
\]

Equations (DAS.25) and (DAS.30) give a nonzero \(y=u_ix\in M\) with

\[
Q_N(y)<\varepsilon D_N(y).
\tag{DAS.31}
\]

The identification of standard forms in Section 3 changes neither side, so (DAS.31) is exactly the desired inequality \(Q_M(y)<\varepsilon D_M(y)\).

Every finite positive family has now been treated, including nonfaithful members and a common null corner. LA06's aggregate vector criterion, followed by LA01–04's construction of commuting weighted matrix blocks and their normal tensor splitting, proves (DAS.2). This completes the descent theorem.

The separable-predual and type III assumptions enter through that final absorption criterion for \(M\). The Fourier decomposition, opposite-degree correction and functional-to-vector estimates required neither assumption on \(N\). The proof also explains why the classical additional aperiodicity assumption is unnecessary for the stronger formulation (DAS.1): no factoriality of the crossing was used.

## 8. Exact checks and the range of the result

**An inner action which does not preserve the test state.** Let

\[
M=M_2(\mathbb C),\quad
v=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad
\theta=\operatorname{Ad}(v),\quad
h=\begin{pmatrix}4/5&0\\0&1/5\end{pmatrix}.
\]

The state \(\varphi(a)=\operatorname{Tr}(ha)\) is not \(\theta\)-invariant. In the regular crossing put \(Z=v^*U\). It commutes with \(M\). Finite Fourier polynomials identify the crossing normally with
\(M_2\bar\otimes L^\infty(\mathbb T)\), with \(U=v\otimes z\), \(Z=1\otimes z\), and \(E=\mathrm{id}\otimes\int_{\mathbb T}\).
For completeness, the representation comparison is the coordinate unitary
\((T\eta)_n=v^n\eta_n\), which sends \(\pi(a)\) to the constant coefficient \(a\) and \(U\) to \(v\) times the bilateral shift. The onto scalar Fourier unitary of CC0 sends that shift to multiplication by \(z\). These unitaries give the normal inverse as well.

Take \(q=1\), \(x=e_{22}U\), and \(u=vU^{-1}\). Then

\[
x=e_{21}\otimes z,\qquad
u=1\otimes z^{-1},\qquad
y=ux=e_{21}\otimes1.
\tag{DAS.32}
\]

In the matrix standard form, \(\xi_\varphi=h^{1/2}\). For \(\lambda=1/4\),

\[
h^{1/2}e_{21}=\tfrac12e_{21}h^{1/2},\qquad
D_M(e_{21})=4/5,\qquad Q_M(e_{21})=0.
\tag{DAS.33}
\]

Thus the correction preserves a genuine positive-mass estimate even though the original state is not invariant. Using a degree-\(+1\) correction would instead leave degree \(2\), and would not produce an element of \(M\).

**Why all positive tests matter.** This one exact matrix relation does not imply tensor absorption. For the normalized trace \(\tau\) on \(M_2\),

\[
\xi_\tau=2^{-1/2}I,\qquad
Q_M(y)=(1-\sqrt\lambda)^2D_M(y)
\quad(y\in M_2).
\tag{DAS.34}
\]

The mass is positive for every nonzero \(y\). Therefore arbitrarily small relative error is impossible for this test when \(0<\lambda<1\). This finite-dimensional example checks the estimate and its quantifiers; it is outside the type III hypothesis of the theorem.

**A zero denominator.** For \(\varphi(a)=a_{11}\), the positive representative is \(e_{11}\), and \(y=e_{22}\) gives both mass and residual zero. It is a valid witness in the common null corner. Dividing by its mass would be invalid, which is why Section 7 separates this case before Fourier selection.

**The endpoint.** At \(\lambda=1\), the vector criterion is satisfied by \(y=1\) in every algebra. It therefore cannot be used to infer absorption of a tracial infinite product. The range \(0<\lambda<1\) is retained in the descent theorem.

## Source

Masamichi Takesaki, *Theory of Operator Algebras III*, Chapter XVIII, Section 4, Lemma 4.20, printed pp.391–393. The proof above makes the conditional-expectation standard-form comparison, the finite-family Parseval identity and the positive denominator explicit. With the circle convention (DAS.7), the correcting unitary \(v_iU^{-q}\) has degree \(-q\). The formulation by the specified product \(R_\lambda\) also proves descent when the crossing itself is not a factor. Proposition 6.1 isolates the exact local-error equality behind that descent.