# Modular restriction and cocycle preservation for operator-valued weights

*CC0-1.0 to the extent of rights held.*

Let \(N\subset M\) be a unital inclusion of arbitrary von Neumann algebras, and let \(T:M_+\to\widehat N_+\) be a faithful normal semifinite operator-valued weight. Thus \(T\) is additive, positively homogeneous, preserves increasing positive suprema, and satisfies \(T(b^*xb)=b^*T(x)b\) for \(b\in N\). Faithfulness means \(T(x)=0\Rightarrow x=0\). Semifiniteness means that
\[
 N_T=\{x\in M:T(x^*x)\text{ is a bounded element of }N_+\}
\]
is weak-operator dense in \(M\). Let \(\varphi,\psi\) be faithful normal semifinite weights on \(N\), and use the actual EP extension to set
\[
 \widetilde\varphi=\widehat\varphi\circ T,\qquad
 \widetilde\psi=\widehat\psi\circ T.
\]
Then, for every real \(t\),

<a id="equation-ot1"></a>

\[
 \sigma_t^{\widetilde\varphi}(x)=\sigma_t^\varphi(x)\quad(x\in N),
 \qquad
 (D\widetilde\psi:D\widetilde\varphi)_t=(D\psi:D\varphi)_t\in N.    \tag{OT1}
\]
The cocycles on each side are the balanced-matrix cocycles constructed in BC. There is no separability, finiteness or countable-decomposability hypothesis.

Exact earlier written proofs are [EP2](OA-FLOW-EP.md#oa-flow.ep.2), [EP3](OA-FLOW-EP.md#oa-flow.ep.3), [EP4](OA-FLOW-EP.md#oa-flow.ep.4), [EP5](OA-FLOW-EP.md#oa-flow.ep.5), [EP6](OA-FLOW-EP.md#oa-flow.ep.6); [BC1](OA-FLOW-BC.md#oa-flow.bc.1), [BC2](OA-FLOW-BC.md#oa-flow.bc.2), [BC3](OA-FLOW-BC.md#oa-flow.bc.3), [BC4](OA-FLOW-BC.md#oa-flow.bc.4); [GW1](OA-FLOW-GW.md#oa-flow.gw.1), [GW2](OA-FLOW-GW.md#oa-flow.gw.2); [SF0](OA-FLOW-SF.md#oa-flow.sf.sf0), [SB0](OA-FLOW-SF.md#oa-flow.sf.sb0), [SB1](OA-FLOW-SF.md#oa-flow.sf.sb1), [SB2](OA-FLOW-SF.md#oa-flow.sf.sb2), [SB3](OA-FLOW-SF.md#oa-flow.sf.sb3), [SB4](OA-FLOW-SF.md#oa-flow.sf.sb4), [SB5](OA-FLOW-SF.md#oa-flow.sf.sb5), [SB6](OA-FLOW-SF.md#oa-flow.sf.sb6); [MA1](OA-FLOW-MA.md#oa-flow.ma.1), [MA2](OA-FLOW-MA.md#oa-flow.ma.2), [MA3](OA-FLOW-MA.md#oa-flow.ma.3), [MA4](OA-FLOW-MA.md#oa-flow.ma.4); [AS1](OA-FLOW-AS.md#oa-flow.as.1), [AS2](OA-FLOW-AS.md#oa-flow.as.2), [AS3](OA-FLOW-AS.md#oa-flow.as.3), [AS4](OA-FLOW-AS.md#oa-flow.as.4); and [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP3](OA-FLOW-CP.md#oa-flow.cp.3), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP5](OA-FLOW-CP.md#oa-flow.cp.5), [CP6](OA-FLOW-CP.md#oa-flow.cp.6). EP binds positive-functional Cauchy–Schwarz to [GNS Lemma2.1](OA-FLOW-GNS.md#gns-lemma-2-1) and [Lemma2.2](OA-FLOW-GNS.md#gns-lemma-2-2), the norm identity to [GNS Theorem4.1](OA-FLOW-GNS.md#gns-theorem-4-1), and the Hilbert-sum construction to [GNS Lemma7.1](OA-FLOW-GNS.md#gns-lemma-7-1). The finite algebra \(m_T=\operatorname{span}N_T^*N_T\) and its \(N\)-valued bimodule linear extension \(T_0:m_T\to N\) on this finite algebra are precisely [EP6](OA-FLOW-EP.md#oa-flow.ep.6). No uniform norm bound is asserted.

The free development source is [F. Hiai's author manuscript, Theorem 8.7 and Lemmas 8.13–8.16, printed pp.71–80](https://arxiv.org/pdf/2004.02383v1#page=71). The proof below obtains matrix positivity and mixed finite products directly from square-root columns. Resolvent contractions then approximate the whole GNS finite domains. The closed-operator strip construction in MA and the upper-half-plane comparison in AS identify the two groups. EP supplies the extension by normal minorants.

<a id="oa-flow.ot.1"></a><a id="ot-1"></a>

## OT1. Positivity of the coefficient-algebra-valued finite matrix extension

For a positive integer \(k\), define
\[
 R:M_k(m_T)\longrightarrow M_k(N),\qquad
 R([x_{ij}])=[T_0(x_{ij})].
\]
The bimodule identity is a finite entrywise calculation:

<a id="equation-ot2"></a>

\[
 R(CXD)=CR(X)D\qquad(C,D\in M_k(N),\ X\in M_k(m_T)).              \tag{OT2}
\]
The space \(M_k(m_T)\) is spanned by its positive elements. Indeed, a matrix with one entry \(y^*x\), \(x,y\in N_T\), is \((yE_{ri})^*(xE_{rj})\) for the required entry \((i,j)\). The polarization formula for two such matrices expresses each product as a linear combination of matrices \(Z^*Z\) with entries in \(m_T\). This proves the span assertion.

We claim

<a id="equation-ot3"></a>

\[
 X\in M_k(m_T),\quad X\geq0\quad\Longrightarrow\quad R(X)\geq0.    \tag{OT3}
\]
Take the positive square root \(Y=X^{1/2}=[y_{\ell j}]\) in \(M_k(M)\), using [SB1–SB6](OA-FLOW-SF.md#oa-flow.sf.sb1). Its columns satisfy
\[
 x_{jj}=\sum_{\ell=1}^k y_{\ell j}^*y_{\ell j}.
\]
Each \(x_{jj}\) belongs to the positive part of \(m_T\), whose \(T\)-value is bounded by [EP6](OA-FLOW-EP.md#oa-flow.ep.6). Since
\(0\leq y_{\ell j}^*y_{\ell j}\leq x_{jj}\), heredity of that bounded-value cone gives \(y_{\ell j}\in N_T\).

For a column \(c=(c_j)_{j=1}^k\in N^k\), put
\(w_\ell=\sum_j y_{\ell j}c_j\).
The right \(N\)-module property and linearity of \(N_T\), both proved in EP6, give \(w_\ell\in N_T\). Expanding the finite sums and using bimodularity of \(T_0\) yields

<a id="equation-ot4"></a>

\[
 c^*R(X)c
 =\sum_{i,j,\ell}c_i^*T_0(y_{\ell i}^*y_{\ell j})c_j
 =\sum_{\ell=1}^k T(w_\ell^*w_\ell)\geq0.                        \tag{OT4}
\]
All terms are bounded elements of \(N\). Thus positivity has been tested on every coefficient column, without composing \(T\) with an auxiliary scalar weight.

For completeness these column tests imply matrix positivity. The matrix \(H=R(X)\) is self-adjoint because \(T_0\) preserves adjoints. If \(H\) were not positive, its spectral calculus would give a nonzero projection
\(P=1_{(-\infty,-\varepsilon]}(H)\) for some \(\varepsilon>0\). Let \(\iota_j\) be the \(j\)-th coordinate column with entry \(1_N\), and take \(c=P\iota_j\). Since \(PHP\leq-\varepsilon P\), OT4 gives

<a id="equation-ot5"></a>

\[
 0\leq c^*Hc\leq-\varepsilon c^*c.                              \tag{OT5}
\]
Hence \(c^*c=0\), so \(P\iota_j=0\), for every \(j\). All columns of \(P\) vanish, contradicting \(P\ne0\). This proves OT3 for every matrix size.

<a id="oa-flow.ot.2"></a><a id="ot-2"></a>

## OT2. Transfer of a finite mixed product

[EP6](OA-FLOW-EP.md#oa-flow.ep.6) proves that \(\widetilde\varphi,\widetilde\psi\) are faithful normal semifinite. Write \(N_\omega=\mathfrak n_\omega\) for each scalar weight \(\omega\), and
\[
 Z_{\varphi,\psi}=\operatorname{span}(N_\varphi^*N_\psi).
\]
If \(y\in N_{\widetilde\varphi}\cap N_T\) and \(z\in N_{\widetilde\psi}\cap N_T\), then

<a id="equation-ot6"></a>

\[
 T_0(y^*z)\in Z_{\varphi,\psi}.                                  \tag{OT6}
\]
Indeed the Gram matrix
\[
 X=\begin{pmatrix}y^*y&y^*z\\z^*y&z^*z\end{pmatrix}
   =\begin{pmatrix}y&z\\0&0\end{pmatrix}^*
     \begin{pmatrix}y&z\\0&0\end{pmatrix}
\]
is positive in \(M_2(m_T)\). [OT1](OA-FLOW-OT.md#oa-flow.ot.1) makes \(R(X)\) positive in \(M_2(N)\). Write its positive square root by columns:
\[
 R(X)^{1/2}=
 \begin{pmatrix}r_1&s_1\\r_2&s_2\end{pmatrix}.
\]
Multiplication gives
\[
 T(y^*y)=\sum_{\ell=1}^2r_\ell^*r_\ell,\qquad
 T(z^*z)=\sum_{\ell=1}^2s_\ell^*s_\ell,\qquad
 T_0(y^*z)=\sum_{\ell=1}^2r_\ell^*s_\ell.
\]
The first two sums have finite \(\varphi\)- and \(\psi\)-values respectively:
their values are \(\widetilde\varphi(y^*y)\) and
\(\widetilde\psi(z^*z)\). Positivity bounds each summand by its sum, so
\(r_\ell\in N_\varphi\) and \(s_\ell\in N_\psi\). The last displayed equality is therefore an explicit expression of \(T_0(y^*z)\) in \(Z_{\varphi,\psi}\), proving OT6.

We will also use the exact compatibility of finite extensions. If
\(v,w\in N_{\widetilde\omega}\cap N_T\), where \(\widetilde\omega=\widehat\omega T\), then

<a id="equation-ot7"></a>

\[
 \widetilde\omega_0(v^*w)=\omega_0(T_0(v^*w)).                    \tag{OT7}
\]
Polarize \(v^*w\) into the squares of \(v+i^j w\), \(j=0,1,2,3\). This intersection is a linear space, and every square has bounded \(T\)-value and finite \(\widetilde\omega\)-value. On such squares the equation is exactly the definition of composition. Their \(T\)-values belong to \(m_\omega\); linearity of both finite extensions proves ([OT7](OA-FLOW-OT.md#equation-ot7)) and the domain of its right side.

<a id="oa-flow.ot.3"></a><a id="ot-3"></a>

## OT3. Entire base elements give bounded mixed right multipliers upstairs

Let \(a\in N\) be norm entire for \(\beta^{\psi,\varphi}\), and put
\[
 b=\beta_{-i}^{\psi,\varphi}(a),\qquad
 c=\beta_{-i/2}^{\psi,\varphi}(a),\qquad k=\max(1,\|c\|).
\]
[MA11](OA-FLOW-MA.md#equation-ma11)–[MA13](OA-FLOW-MA.md#equation-ma13) on \(N\) give

<a id="equation-ot8"></a>

\[
 \psi(aha^*)\leq k^2\varphi(h),\qquad
 \varphi(b^*hb)\leq k^2\psi(h)\quad(h\in N_+),                    \tag{OT8}
\]
and \(\psi_0(az)=\varphi_0(zb)\) for every \(z\in Z_{\varphi,\psi}\).

The two inequalities extend to all \(m\in\widehat N_+\). To see this, take [EP2](OA-FLOW-EP.md#oa-flow.ep.2)'s increasing bounded approximants \(h_n\uparrow m\) in the extended cone. Sandwiching preserves their increasing pointwise supremum: for each normal positive \(f\),
\[
 (a h_n a^*)(f)=h_n(f_{a^*})\uparrow m(f_{a^*})=(a m a^*)(f).
\]
[EP5](OA-FLOW-EP.md#oa-flow.ep.5)'s normal extension now yields \(\widehat\psi(a m a^*)\leq k^2\widehat\varphi(m)\), and the same proof gives the second inequality. Composing with \(T\) and using its bimodule property gives

<a id="equation-ot9"></a>

\[
 \widetilde\psi(a h a^*)\leq k^2\widetilde\varphi(h),\qquad
 \widetilde\varphi(b^*hb)\leq k^2\widetilde\psi(h)
 \quad(h\in M_+).                                               \tag{OT9}
\]
In particular

<a id="equation-ot10"></a>

\[
 N_{\widetilde\varphi}a^*\subset N_{\widetilde\psi},\qquad
 N_{\widetilde\psi}b\subset N_{\widetilde\varphi},                 \tag{OT10}
\]

<a id="equation-ot11"></a>

\[
 \|\Lambda_{\widetilde\psi}(ya^*)\|
       \leq k\|\Lambda_{\widetilde\varphi}(y)\|,\qquad
 \|\Lambda_{\widetilde\varphi}(zb)\|
       \leq k\|\Lambda_{\widetilde\psi}(z)\|.                     \tag{OT11}
\]
These hold on the entire indicated finite ideals, including elements with unbounded \(T(y^*y)\).

For \(y_0\in N_{\widetilde\varphi}\cap N_T\), \(z_0\in N_{\widetilde\psi}\cap N_T\), put \(x_0=y_0^*z_0\). By ([OT6](OA-FLOW-OT.md#equation-ot6)), \(T_0(x_0)\in Z_{\varphi,\psi}\), so
\[
 \psi_0(aT_0(x_0))=\varphi_0(T_0(x_0)b).
\]
The elements \(y_0a^*,z_0b\) still belong to \(N_T\), since [EP6](OA-FLOW-EP.md#oa-flow.ep.6) makes \(N_T\) a right \(N\)-module. They belong to the appropriate scalar finite ideals by ([OT10](OA-FLOW-OT.md#equation-ot10)). Thus ([OT7](OA-FLOW-OT.md#equation-ot7)) and the \(N\)-bimodule identity for \(T_0\) identify both sides, proving

<a id="equation-ot12"></a>

\[
 \widetilde\psi_0(a x_0)=\widetilde\varphi_0(x_0 b).               \tag{OT12}
\]
This equality is between finite values of actual elements of the respective finite algebras.

<a id="oa-flow.ot.4"></a><a id="ot-4"></a>

## OT4. Resolvent cutoffs cover the whole mixed finite domain

Let \(y\in N_{\widetilde\varphi}\), and put \(m=T(y^*y)\in\widehat N_+\). Then \(\widehat\varphi(m)<\infty\). In [EP2](OA-FLOW-EP.md#oa-flow.ep.2)'s representation of \(m\), its infinite-value projection \(p\) must vanish: \(m\geq n p\) for every positive integer \(n\), whereas \(\widehat\varphi(m)<\infty\). Faithfulness of \(\varphi\) therefore forces \(p=0\). Thus \(m\) is represented by a densely defined positive self-adjoint operator \(A_m\) affiliated with \(N\).

For each positive integer \(n\), define the bounded resolvent contraction
\[
 c_n=(1+A_m/n)^{-1}\in N.
\]
The Borel spectral calculus in [SB4–SB6](OA-FLOW-SF.md#oa-flow.sf.sb4) gives \(0\leq c_n\uparrow1\) strongly. Introduce the following extended positive elements, all defined by nonnegative scalar functions of \(A_m\):

<a id="equation-ot13"></a>

\[
 \begin{gathered}
 m_n=c_nmc_n,\qquad s_n=(1-c_n^2)m,\qquad
 r_n=(1-c_n)m(1-c_n),\\
 m=m_n+s_n,\qquad 0\leq r_n\leq s_n.                            \tag{OT13}
 \end{gathered}
\]
Here \(s_n\) denotes spectral multiplication by
\(\lambda[1-(1+\lambda/n)^{-2}]\); no difference of unbounded operators is being defined. The equalities and inequality follow from scalar identities, including
\((1-c)^2\leq1-c^2\) for \(0\leq c\leq1\), by testing every normal positive functional as in [EP3](OA-FLOW-EP.md#oa-flow.ep.3).

The function \(\lambda(1+\lambda/n)^{-2}\) is bounded by \(n/4\): after setting \(u=\lambda/n\), the estimate is \(u/(1+u)^2\leq1/4\), equivalent to \((u-1)^2\geq0\). These functions increase pointwise to \(\lambda\). Consequently \(m_n\in N_+\), \(m_n\uparrow m\) in the extended cone, and [EP5](OA-FLOW-EP.md#oa-flow.ep.5)'s normality and additivity give

<a id="equation-ot14"></a>

\[
 \widehat\varphi(m_n)\uparrow\widehat\varphi(m)<\infty,\qquad
 0\leq\widehat\varphi(r_n)\leq\widehat\varphi(s_n)
 =\widehat\varphi(m)-\widehat\varphi(m_n)\longrightarrow0.        \tag{OT14}
\]
The difference here is between finite real numbers, justified by \(m=m_n+s_n\).

Bimodularity of \(T\) now gives

<a id="equation-ot15"></a>

\[
 \begin{gathered}
 T((yc_n)^*(yc_n))=m_n,\qquad
 yc_n\in N_T\cap N_{\widetilde\varphi},\\
 \|\Lambda_{\widetilde\varphi}(y-yc_n)\|^2
 =\widehat\varphi(r_n)\longrightarrow0.                          \tag{OT15}
 \end{gathered}
\]
The second line applies \(T\) to the positive element
\((1-c_n)y^*y(1-c_n)\).
For \(z\in N_{\widetilde\psi}\), the same construction from \(T(z^*z)\) produces positive contractions \(d_n\in N\) with
\(zd_n\in N_T\cap N_{\widetilde\psi}\) and
\(\Lambda_{\widetilde\psi}(zd_n)\to\Lambda_{\widetilde\psi}(z)\).

Apply [OT12](OA-FLOW-OT.md#equation-ot12) to
\(x_n=(yc_n)^*(zd_n)\). The bounded right-multiplier estimates in [OT11](OA-FLOW-OT.md#equation-ot11), together with OT15, give convergence of all four vectors in
\[
 \begin{split}
 \widetilde\psi_0(a x_n)
 &=\langle\Lambda_{\widetilde\psi}(zd_n),
             \Lambda_{\widetilde\psi}(yc_n a^*)\rangle,\\
 \widetilde\varphi_0(x_n b)
 &=\langle\Lambda_{\widetilde\varphi}(zd_n b),
             \Lambda_{\widetilde\varphi}(yc_n)\rangle .
 \end{split}
\]
Taking the limits proves

<a id="equation-ot16"></a>

\[
 \widetilde\psi_0(a y^*z)=\widetilde\varphi_0(y^*zb)
 \quad(y\in N_{\widetilde\varphi},\ z\in N_{\widetilde\psi}).       \tag{OT16}
\]
Both products belong to their asserted finite algebras by OT10; these are actual finite values. Linearity extends the equality to the entire space
\(\operatorname{span}(N_{\widetilde\varphi}^*N_{\widetilde\psi})\).

The sharp cutoffs used in the illustration also converge in this norm. Indeed, for \(e_s=1_{[0,s]}(A_m)\), the spectral decomposition gives \(m=e_sme_s+(1-e_s)m(1-e_s)\), with \(e_sme_s\leq s1\) increasing to \(m\). Normality and additivity of \(\widehat\varphi\) therefore give
\[
 \|\Lambda_{\widetilde\varphi}(y-ye_s)\|^2
 =\widehat\varphi(m)-\widehat\varphi(e_sme_s)\longrightarrow0.
\]
This is a second approximation of the same whole finite domain; the preceding proof of OT16 uses the resolvent contractions.

<a id="oa-flow.ot.5"></a><a id="ot-5"></a>

## OT5. Comparison and both conclusions

For every entire \(\beta^{\psi,\varphi}\)-element \(a\in N\), ([OT10](OA-FLOW-OT.md#equation-ot10)) and ([OT16](OA-FLOW-OT.md#equation-ot16)) are exactly [MA12](OA-FLOW-MA.md#equation-ma12)–[MA13](OA-FLOW-MA.md#equation-ma13) for the upstairs weights, with \(b=\beta_{-i}^{\psi,\varphi}(a)\). [MA4](OA-FLOW-MA.md#oa-flow.ma.4) therefore supplies a bounded ultraweak strip whose edges are

<a id="equation-ot17"></a>

\[
 B_a(t)=\beta_t^{\widetilde\psi,\widetilde\varphi}(a),\qquad
 B_a(t-i)=
 \beta_t^{\widetilde\psi,\widetilde\varphi}
       (\beta_{-i}^{\psi,\varphi}(a)).                           \tag{OT17}
\]
[BC3](OA-FLOW-BC.md#oa-flow.bc.3) proves that both real groups in ([OT17](OA-FLOW-OT.md#equation-ot17)) are groups of weak-star continuous surjective linear isometries, with pointwise ultraweakly continuous orbits. The inclusion \(N\subset M\) carries the inherited ultraweak topology: in a concrete representation each vector-series functional on \(N\) extends to \(M\) by the same series, as proved in CP01–06. Thus [AS4](OA-FLOW-AS.md#oa-flow.as.4) applies with \(Y=N\), \(X=M\), and gives

<a id="equation-ot18"></a>

\[
 \beta_t^{\widetilde\psi,\widetilde\varphi}(x)
       =\beta_t^{\psi,\varphi}(x)
 \qquad(x\in N,\ t\in\mathbb R).                                \tag{OT18}
\]

Take \(\psi=\varphi\). [BC4](OA-FLOW-BC.md#oa-flow.bc.4) for two identical weights says their off-diagonal group is \(\sigma^\varphi\), and similarly upstairs. Hence ([OT18](OA-FLOW-OT.md#equation-ot18)) proves the modular restriction in ([OT1](OA-FLOW-OT.md#equation-ot1)). For general \(\varphi,\psi\), put \(x=1\) in ([OT18](OA-FLOW-OT.md#equation-ot18)). [BC18](OA-FLOW-BC.md#oa-flow.bc.4) defines the two cocycles as precisely these values of \(\beta_t\), proving the second assertion in ([OT1](OA-FLOW-OT.md#equation-ot1)).

The proof establishes both assertions for every faithful normal semifinite \(T,\varphi,\psi\) with their full extended positive values and their whole scalar finite ideals. It does not construct \(T\), classify operator-valued weights, realize arbitrary abstract cocycles, establish a sum decomposition for general normal weights, or settle standard-form and spatial-derivative theorems.

<a id="oa-flow.ot.6"></a><a id="ot-6"></a>

## OT6. Three explicit mechanisms in operator-valued weight transport

![Operator-valued weight transport, complete spectral cutoffs, and analytic uniqueness](../assets/ovw-modular/assets/ovw-modular.png)

The figure gives exact examples for [OT1–OT5](OA-FLOW-OT.md#oa-flow.ot.1), the finite-domain strip construction in [MA2–MA4](OA-FLOW-MA.md#oa-flow.ma.2), and the zero argument in [AS1](OA-FLOW-AS.md#oa-flow.as.1). The matrices and tail formulas below are proved, not inferred from plotted samples. Source context for the general theorem is [Hiai's free author manuscript, printed pp.71–80](https://arxiv.org/pdf/2004.02383v1#page=71). These particular examples and the finite-product illustration are local constructions.

### The common tensor density cancels in the cocycle

Let \(M=M_2(\mathbb C)\otimes M_2(\mathbb C)\), \(N=M_2(\mathbb C)\otimes I\), and identify \(N\) with \(M_2(\mathbb C)\). Put

<a id="equation-of1"></a>

\[
 r=\operatorname{diag}(1,3),\qquad
 T(x\otimes y)=x\,\operatorname{Tr}(ry),\qquad
 d_\varphi=\tfrac15\operatorname{diag}(1,4),\qquad
 d_\psi=R d_\varphi R^*,\quad
 R=2^{-1/2}\begin{pmatrix}1&-1\\1&1\end{pmatrix}.                 \tag{OF1}
\]
Thus \(d_\psi=\frac12I-\frac3{10}(E_{12}+E_{21})\). Define \(\varphi(x)=\operatorname{Tr}(d_\varphi x)\) and \(\psi(x)=\operatorname{Tr}(d_\psi x)\). The map \(T\) is the sum of the two positive coordinate compression maps, with coefficients \(1\) and \(3\), so it is positive. For positive \(X\), \(T(X)=0\) forces both diagonal compressions of \(X\) to vanish; equivalently \(X^{1/2}\) kills every coordinate vector, so \(X=0\). It is faithful, normal in finite dimension, \(N\)-bimodular, and finite everywhere, hence semifinite. Its value at the identity is \(4I\); it is not normalized to a conditional expectation.

Matrix units and the finite trace show that \(\widetilde\varphi=\varphi T\), \(\widetilde\psi=\psi T\) have densities \(d_\varphi\otimes r\), \(d_\psi\otimes r\). For any positive invertible matrix \(d\), the modular group of \(\operatorname{Tr}(d\,\cdot)\) is

<a id="equation-of2"></a>

\[
 \sigma_t(x)=d^{it}x d^{-it}.                                    \tag{OF2}
\]
For a direct verification, this group preserves the weight by cyclicity of the finite trace, and
\(G_{a,b}(z)=\operatorname{Tr}(d\,d^{iz}a d^{-iz}b)\) is entire and bounded on each horizontal band. Its upper unit-strip edges satisfy
\(G_{a,b}(t)=\varphi(\sigma_t(a)b)\) and
\(G_{a,b}(t+i)=\varphi(b\sigma_t(a))\), by multiplying the powers of \(d\) and cyclically moving the last factor. Every element is in the finite-star algebra, so [KU](OA-FLOW-KU.md#oa-flow.ku.3) identifies ([OF2](OA-FLOW-OT.md#equation-of2)) with the recovered modular group. Applying ([OF2](OA-FLOW-OT.md#equation-of2)) to the balanced block density \(\operatorname{diag}(d_\varphi,d_\psi)\) gives the BC cocycle
\[
 u_t=d_\psi^{it}d_\varphi^{-it}.
\]
Diagonalizing the positive matrices proves \((d\otimes r)^{it}=d^{it}\otimes r^{it}\) entry by entry. Hence, exactly,

<a id="equation-of3"></a>

\[
 \begin{split}
 \sigma_t^{\widetilde\varphi}(x\otimes I)
   &=\sigma_t^\varphi(x)\otimes I,\\
 (D\widetilde\psi:D\widetilde\varphi)_t
   &=(d_\psi^{it}d_\varphi^{-it})\otimes(r^{it}r^{-it})
     =u_t\otimes I.                                             \tag{OF3}
 \end{split}
\]
At \(t_*=\pi/\log4\), the common scalar factor \(5^{-it_*}\) cancels, while \(4^{it_*}=-1\). Thus
\[
 u_{t_*}=R\operatorname{diag}(1,-1)R^*\operatorname{diag}(1,-1)
   =\begin{pmatrix}0&-1\\1&0\end{pmatrix}.
\]
Panel B shows the exact \(4\times4\) matrix \(u_{t_*}\otimes I\), in the ordered basis \((e_1\otimes e_1,e_1\otimes e_2,e_2\otimes e_1,e_2\otimes e_2)\). Shading records entry modulus only. Panel A displays the two density routes and the exact restriction identity, not a claim that \(T\) itself intertwines every real modular action on every element of \(M\).

### A bounded element with unbounded operator-valued weight

Let \(M=\prod_{n\geq1}M_2(\mathbb C)\), the algebra of bounded matrix sequences, and \(N=\ell^\infty(\mathbb N)I\), its central scalar subalgebra. For \(X=(X_n)\in M_+\) put

<a id="equation-of4"></a>

\[
 T(X)_n=n\operatorname{Tr}(X_n),\qquad
 \varphi(a)=\sum_{n\geq1}2^{-n}a_n.                              \tag{OF4}
\]
The positive normal functionals on \(N\) are exactly the \(\ell^1_+\) sequences: normality on the increasing finite coordinate projections gives the sum representation, and nonnegative summation proves normality conversely. The sequence on the right defining \(T\) may be unbounded. It defines an extended positive element through
\(T(X)(f)=\sum_n f_n n\operatorname{Tr}(X_n)\) for \(f\in\ell^1_+\).
Finite partial sums are continuous positive linear functionals on \(\ell^1_+\); their supremum is lower semicontinuous and additive by nonnegative series, so this is precisely EP's extended cone. Monotone convergence of scalar series proves normality of \(T\), its faithfulness follows coordinatewise from the faithful finite trace, and central scalar multiplication gives bimodularity. Finite coordinate truncations of any bounded matrix sequence are in \(N_T\) and converge strongly in the direct-sum representation. Hence \(T\) is semifinite. The state \(\varphi\) is faithful and normal since every coefficient \(2^{-n}\) is positive and sums to \(1\).

Take \(y=I\). Then \(m=T(y^*y)\) is the unbounded sequence \(m_n=2n\), so \(y\notin N_T\), but
\[
 \widetilde\varphi(y^*y)=\sum_{n\geq1}2n\,2^{-n}=4<\infty.
\]
The finite sum identity
\(\sum_{n=1}^k n2^{-n}=2-(k+2)2^{-k}\)
follows by induction, starting at \(k=0\); letting \(k\to\infty\) proves the preceding total and the tail identity below. The resolvent contraction in [OT4](OA-FLOW-OT.md#oa-flow.ot.4), with parameter \(q\), is
\[
 c_q(n)=\frac{q}{q+2n},\qquad
 \|\Lambda_{\widetilde\varphi}(y-yc_q)\|^2
 =\sum_{n\geq1}2n\,2^{-n}\left(\frac{2n}{q+2n}\right)^2.
\]
This tends to zero: on a fixed finite set every summand tends to zero, and the remaining sum is bounded by the summable tail of \(2n2^{-n}\), independently of \(q\).

For a second explicit approximation, use the coordinate projection \(e_{2k}=1_{[0,2k]}(A_m)\). Its error has the closed formula

<a id="equation-of5"></a>

\[
 e_{2k}(n)=1_{\{n\leq k\}},\qquad
 \|\Lambda_{\widetilde\varphi}(y-ye_{2k})\|^2
   =2(k+2)2^{-k}.                                               \tag{OF5}
\]
Panel C plots this squared GNS error for \(k=0,\ldots,14\), on a logarithmic vertical scale. At \(k=0\) the cutoff is zero. The exact formula tends to zero, although the untruncated \(T(y^*y)\) has no bounded norm. This explicitly displays why the proof must pass beyond \(N_T\).

### The finite-product obstruction to a nonzero difference

[AS1](OA-FLOW-AS.md#oa-flow.as.1) considers a bounded analytic function \(g\) on the upper half-plane with \(g(in)=0\) for every positive integer. If \(g(i/2)\neq0\), the disk transform
\[
 H(w)=g\!\left(\tfrac i2\,\frac{1+w}{1-w}\right)
\]
has zeros \(a_n=(2n-1)/(2n+1)\). Dividing by a finite set of its Blaschke factors, with the removable singularities and the disk maximum principle proved in [AS1](OA-FLOW-AS.md#oa-flow.as.1), gives

<a id="equation-of6"></a>

\[
 \frac{|H(0)|}{\|g\|_\infty}
 \leq\prod_{n=1}^N\frac{2n-1}{2n+1}
 =\frac1{2N+1}\longrightarrow0.                                \tag{OF6}
\]
The product telescopes exactly. Panel D plots this upper bound for \(N=1,\ldots,60\); it does not plot values of an assumed nonzero \(g\), whose existence the bound rules out. In the general [AS1](OA-FLOW-AS.md#oa-flow.as.1) proof the point \(iy\) is chosen wherever \(g(iy)\neq0\); the divergence of \(\sum_n2y/(n+y)\) replaces the special telescoping identity. [AS4](OA-FLOW-AS.md#oa-flow.as.4) applies this obstruction to the bounded, exponentially damped difference between the upper-half-plane strip construction and the entire base orbit.

[Editable original SVG](../assets/ovw-modular/assets/ovw-modular.svg), [figure data](../assets/ovw-modular/ovw-modular-numerics.json), and [reproduction source](../assets/ovw-modular/render_ovw_modular.py).