# Compact-time continuity along automorphism orbits

Changes of an automorphism can move the support of every finite state used
to test a weight. This lesson proves compact-time strong* continuity of its
weight cocycle by estimating those finite pieces, recovering the omitted
components through unitarity, and then changing the faithful reference.
The algebra, its standard Hilbert space and all directed index sets are
arbitrary.

## Statement and conventions

Let \((M,H,J,P)\) be a standard form of an arbitrary von Neumann algebra.
Let \(\varphi\) be a normal semifinite faithful weight, and let
\(\alpha_i\to\alpha\) in the u-topology on \(\operatorname{Aut}(M)\):
for every \(f\in M_*\), one has
\(\|f\circ\alpha_i-f\circ\alpha\|\to0\).
The index set is any directed set. Define

\[
 C_\varphi(\beta,t)=[D(\varphi\circ\beta^{-1}):D\varphi]_t.
 \tag{OC.1}
\]

Then, for every \(\eta\in H\) and compact \(K\subset\mathbb R\),

\[
 \sup_{t\in K}\|(C_\varphi(\alpha_i,t)
                      -C_\varphi(\alpha,t))\eta\|\longrightarrow0.
 \tag{OC.2}
\]

The same assertion holds for adjoints. Every compact \(K\) is contained
in \([-T,T]\) for some finite \(T\ge0\), so it is enough to prove
the estimates below on those intervals. There is no assumption of
separability, sigma-finiteness, a countable state family, ordered weights,
bounded densities, or strongly convergent inner lifts.

Inner products are linear in the first variable. For a positive
self-adjoint operator \(A\) with possibly nonzero kernel, write
\(A^{[it]}\) for the bounded Borel function equal to \(\lambda^{it}\)
on \(\lambda>0\) and to zero at \(\lambda=0\), including when \(t=0\).
Its value at zero time is the support projection, not necessarily the
identity. Ordinary imaginary powers of an injective positive operator are unitary.

The case \(M=0\) is immediate. We work with \(M\ne0\) below.

## Exact inputs and source boundary

We use \(\mathfrak n_\theta=\{x:\theta(x^*x)<\infty\}\) and the GNS map
\(\Lambda_\theta\) as in The finite part of an extended-valued weight and The GNS construction for an arbitrary weight.
The notation \(M_\theta\) denotes the fixed algebra of the modular group,
as identified with the centralizer.
For a faithful normal semifinite weight \(\chi\) on \(M'\),
\(D_\chi\) is its bounded-vector domain and
\(R_\chi(\zeta)\Lambda_\chi(y)=y\zeta\);
the exact domain and coefficient construction are in
Conventions and the actual prerequisites.

The following existing written providers supply the other inputs:

- The concrete predual and its intrinsic norm through The sigma-strong seminorms are vector seminorms and WS-06: normal functional supports, separation and corner
  compression; WG-008: finite positive contractions detecting semifiniteness;
  NW-12: arbitrary sums of normal weights.
- Every von Neumann algebra has such a weight and CW-05--07: existence of n.s.f. corner weights, faithful diagonal
  completions and centralizer compression. CZ-09 supplies the projection
  criterion; CZ-05, through (CZ.3), supplies finite centralizer right multiplication.
- Canonical comparison of the positive cones through One Hilbert space for all n.s.f. GNS maps and SE: comparison of standard forms, natural-cone vectors,
  \(\|\xi_\omega-\xi_\nu\|\le\|\omega-\nu\|^{1/2}\), common standard
  GNS maps and the polar factor \(J\) of the relative Tomita map.
- SC and SI-09--12: spatial forms, supported derivatives, the graph-core
  construction for nonfaithful numerators, and identification of the faithful
  relative GNS operator with \(d\psi/d\theta^{\mathrm{opp}}\).
- The balanced-matrix cocycle is independent of the reference, CH-02 and PT-02: the spatial formula for faithful cocycles,
  the faithful chain rule and simultaneous naturality.
- SK: Borel functional calculus and scalar spectral integration.

We shall also use two elementary compactness arguments, recorded here
in full. Suppose that \(\sup_{i,t}\|E_i(t)\|\le b<\infty\) and that
\(\sup_{t\in K}\|E_i(t)\zeta\|\to0\) for every fixed vector \(\zeta\).
For a norm-compact set \(L\subset H\), choose a finite \(\varepsilon\)-net
\(\zeta_1,\ldots,\zeta_n\). Then

\[
 \sup_{\zeta\in L,t\in K}\|E_i(t)\zeta\|
 \le b\varepsilon+\max_{1\le k\le n}\sup_{t\in K}\|E_i(t)\zeta_k\|.
 \tag{OC.22}
\]

The maximum tends to zero: finitely many eventual net indices have a
common upper bound. Taking the limit superior and then
\(\varepsilon\downarrow0\) proves uniformity on \(L\).

Inverse convergence in the automorphism group is also
automatic: for every normal \(f\), composition is isometric and
\(\|f\circ\alpha_i^{-1}-f\circ\alpha^{-1}\|
=\|f-f\circ\alpha^{-1}\alpha_i\|\to0\).

For the second argument, let \(\beta_i\to\beta\) in the u-topology and
let \(V_i,V\) be their canonical standard implementers. SF-12 and SE-11
give \(V_i\to V\) strongly. The identity
\(\|(V_i^*-V^*)\zeta\|=\|\zeta-V_iV^*\zeta\|\)
also gives strong convergence of the adjoints. If \(a_t\in M\) is a
uniformly bounded strongly continuous family, and \(\zeta_t\in H\)
is norm continuous, both on a compact \(K\), then

\[
 (\beta_i(a_t)-\beta(a_t))\zeta_t
 =(V_i-V)a_tV^*\zeta_t
       +V_i a_t(V_i^*-V^*)\zeta_t.
 \tag{OC.23}
\]

Both \(\{\zeta_t:t\in K\}\) and
\(\{a_tV^*\zeta_t:t\in K\}\) are norm compact. For the latter assertion,
bounded strong continuity of \(a_t\), together with norm continuity of
\(\zeta_t\), proves continuity of their product on each vector.
Applying (OC.22) to the two differences of implementers proves that
(OC.23) tends to zero uniformly on \(K\). Neither argument requires
sequences or a countable dense set.

Mathematical antecedents include Takesaki, *Theory of Operator Algebras II*,
VIII.2 (strictly semifinite weights), IX.1 (natural cones), IX.3 (spatial
derivatives), and XII.6, Lemma 6.14(iii), printed 450--451. The latter source
states continuity of the quantized modulus, an application of weight
cocycle continuity. The arbitrary-net proof below establishes the general
weight-theoretic input. The proof here handles nonfaithful state summands
inside a generally infinite faithful weight; it does not assume that every
algebra has a faithful normal state. The relative and spatial ingredients, with their faithful graph identities and
closed-form covariance, are treated in Takesaki II, VIII.3 and IX.3; they underlie the derivation in OC-05; that derivation
does not apply an invertible covariance formula to a projection.

The reader can enter through the natural-cone estimate and standard
implementation, the
spatial derivative construction,
and the faithful chain identity.

## A scalar estimate which retains the kernel

For \(T\ge0\), put \(L_T=4\max\{1,T\}\). For every \(\lambda\ge0\),

\[
 \sup_{|t|\le T}|\lambda^{[it]}-1|
       \le L_T|\sqrt\lambda-1|.
 \tag{OC.3}
\]

Here \(0^{[it]}=0\). At \(\lambda=0\) the left side is one. For
\(0<\sqrt\lambda<1/2\), use the bound two and
\(|\sqrt\lambda-1|>1/2\). For \(x=\sqrt\lambda\ge1/2\),

\[
 |e^{2it\log x}-1|\le2|t|\,|\log x|
       \le4T|x-1|,
\]

because the derivative of \(\log x\) is at most two on the interval
between \(x\) and one. This proves (OC.3), including \(x=1\).

Consequently, for every positive self-adjoint \(A\) and
\(\xi\in D(A^{1/2})\), spectral integration gives

\[
 \sup_{|t|\le T}\|(A^{[it]}-1)\xi\|
       \le L_T\|(A^{1/2}-1)\xi\|.
 \tag{OC.4}
\]

The inequality is integrated separately for each \(t\), with one common
bound; no exchange of a supremum with a signed integral is needed. This
estimate asserts continuity on the particular vector. It does not assert
convergence of support projections on all of \(H\).

## A faithful reference assembled from finite orthogonal pieces

There is a family of nonzero normal states \((\omega_j)_{j\in I}\) whose
support projections \(p_j\) are pairwise orthogonal and satisfy
\(\sum_jp_j=1\) strongly. Indeed use Zorn's lemma on orthogonal families
of supports. If their supremum leaves a nonzero projection \(q\), normal
positive separation gives a positive normal functional nonzero on \(q\).
Its compression to \(qMq\), normalized, has nonzero support at most \(q\),
contradicting maximality. This construction places no cardinality bound on
\(I\).

Define, for \(x\in M_+\),

\[
 \theta(x)=\sum_{j\in I}\omega_j(x)
      :=\sup_{F\subset I\text{ finite}}\sum_{j\in F}\omega_j(x).
 \tag{OC.5}
\]

NW-12 proves normality and the weight axioms. If \(\theta(x)=0\), each
\(p_jxp_j\) is zero by faithfulness of \(\omega_j\) on its support corner.
Thus \(x^{1/2}p_j=0\) for all \(j\), whence \(x=0\). For
\(p_F=\sum_{j\in F}p_j\), one has \(\theta(p_F)=|F|<\infty\), and
\(p_F\uparrow1\). WG-008 proves semifiniteness.

For each \(j\), the unitary \(u_j=2p_j-1\) leaves every summand in
(OC.5) invariant under conjugation: it acts by a scalar sign on that
summand's support. Hence \(\theta(u_jxu_j)=\theta(x)\) on all positive
elements. CZ-09 gives \(p_j\in M_\theta\). Its compression is exactly

\[
 \theta(p_jxp_j)=\omega_j(x).
 \tag{OC.6}
\]

Realize \(\theta\)'s GNS construction in the given standard form, using
SF/SE, and abbreviate its standard GNS map by \(\Lambda_\theta\). Then

\[
 \xi_j:=\Lambda_\theta(p_j)=\xi_{\omega_j}\in P,
 \qquad J\xi_j=\xi_j,
 \qquad \Delta_\theta^{it}\xi_j=\xi_j.
 \tag{OC.7}
\]

To check the identities, finite centralizer right multiplication by
\(p_j\) is an orthogonal projection, while adjunction fixes
\(\Lambda_\theta(p_j)\). The same finite-projection argument as SF-10
gives \(\Delta_\theta\xi_j=\xi_j\) and membership in the natural cone.
Its represented functional is \(x\mapsto\theta(p_jxp_j)\), which is
\(\omega_j\); uniqueness of cone representatives proves (OC.7).
Moreover SF-11 gives

\[
 \overline{M'\xi_j}=p_jH,
 \qquad \overline{\operatorname{span}\{M'\xi_j:j\in I\}}=H.
 \tag{OC.8}
\]

## The square-root identity on a finite centralizer vector

The following lemma is the needed support-sensitive ingredient. Let
\(\theta\) be n.s.f., let \(p\in M_\theta\) be a projection with
\(\theta(p)<\infty\), and put
\(\omega(x)=\theta(pxp)\), \(\xi=\Lambda_\theta(p)=\xi_\omega\).
On \(M'\), let \(\chi=\theta^{\mathrm{opp}}\), that is
\(\chi(y)=\theta(JyJ)\) for \(y\in M'_+\).
For any bounded normal positive functional \(\nu\), put
\(A_\nu=d\nu/d\chi\). Then

\[
 \xi\in D(A_\nu^{1/2}),
 \qquad A_\nu^{1/2}\xi=Jp\xi_\nu,
 \qquad
 \|(A_\nu^{1/2}-1)\xi\|
       \le\|\nu-\omega\|^{1/2}.
 \tag{OC.9}
\]

Here is a derivation from the faithful relative-GNS theorem, rather than
an assumed extension of it. If \(\nu=0\), the derivative and cone vector
are zero, and the assertion follows directly. Otherwise let
\(q=s(\nu)\). Choose an n.s.f. weight \(\tau\) on
\((1-q)M(1-q)\), with the zero-corner convention, and form

\[
 \widehat\nu(x)=\nu(qxq)+\tau((1-q)x(1-q)).
 \tag{OC.10}
\]

CW-06 makes this n.s.f., with \(q\) in its centralizer. Since
\(\widehat\nu(q)=\nu(1)<\infty\), the standard finite-projection
identity just used gives
\(\Lambda_{\widehat\nu}(q)=\xi_\nu\).

Write \(\widehat A=d\widehat\nu/d\chi\). The projection \(q\)
reduces \(\widehat A\): the spatial modular implementation sends it
to \(\sigma_t^{\widehat\nu}(q)=q\), and hence it commutes with the
spectral calculus. Centralizer compression of the numerator gives

\[
 A_\nu=(\widehat A|_{qH})\oplus0_{(1-q)H}.
 \tag{OC.11}
\]

This identification includes square-root domains. For clarity, on the
common bounded-vector domain \(D_\chi\), covariance of its coefficient
map gives

\[
 \nu(R_\chi(\zeta)R_\chi(\zeta)^*)
 =\widehat\nu(qR_\chi(\zeta)R_\chi(\zeta)^*q)
 =q_{\widehat\nu,\chi}[q\zeta].
\]

The spatial form core for \(\widehat A\), compressed by its reducing
projection \(q\), is a core on \(qH\). The vectors
\((1-q)D_\chi\) are dense in the orthogonal complement and have zero
energy. Closing their direct sum proves (OC.11), as in SI-10.

Now \(qp\in\mathfrak n_\theta\), since this is a left ideal and
\(p\) is finite. Also \((qp)^*=pq\in\mathfrak n_{\widehat\nu}\),
because
\(\widehat\nu(qpq)=\nu(p)\le\nu(1)\).
Thus \(qp\) belongs to the exact mixed finite-star domain of SF-13
and SI-12. Their common polar conjugation and derivative identification
give

\[
 \widehat A^{1/2}\Lambda_\theta(qp)
       =J\Lambda_{\widehat\nu}(pq)
       =Jp\Lambda_{\widehat\nu}(q)=Jp\xi_\nu.
 \tag{OC.12}
\]

The vector on the left is \(\widehat A^{1/2}q\xi\). Equation (OC.11)
therefore proves the domain and equality in (OC.9). Finally
\(Jp\xi=\xi\) and the cone-vector estimate give

\[
 \|(A_\nu^{1/2}-1)\xi\|
 =\|Jp(\xi_\nu-\xi_\omega)\|
 \le\|\xi_\nu-\xi_\omega\|
 \le\|\nu-\omega\|^{1/2}.
\]

No faithful state on \(M\) was introduced. The auxiliary completion is
used only to invoke already proved faithful graph formulas; the derivative
on the left of (OC.9) is independent of it.

Combining (OC.4) and (OC.9) yields the quantitative estimate

\[
 \sup_{|t|\le T}\|(A_\nu^{[it]}-1)\xi\|
       \le L_T\|\nu-\omega\|^{1/2}.
 \tag{OC.13}
\]

## Automorphism continuity at the identity on one summand

Assume first \(\alpha_i\to\mathrm{id}\). Use the reference \(\theta\)
from OC-04 and its opposite \(\chi\). Write

\[
 \theta_i=\theta\circ\alpha_i^{-1},\quad
 \nu_{ij}=\omega_j\circ\alpha_i^{-1},\quad
 q_{ij}=\alpha_i(p_j),\quad
 B_i=d\theta_i/d\chi,\quad B=d\theta/d\chi=\Delta_\theta.
 \tag{OC.14}
\]

The equality for \(B\) is SI-12 with equal weights. The projection
\(q_{ij}\) lies in \(M_{\theta_i}\), by weight transport, and
\((\theta_i)_{q_{ij}}=\nu_{ij}\). Consequently the same compressed
spatial-form argument used in (OC.11) gives

\[
 (d\nu_{ij}/d\chi)^{[it]}=q_{ij}B_i^{it}.
 \tag{OC.15}
\]

In particular this formula is valid at \(t=0\); its two sides equal
\(q_{ij}\).

By the u-topology, \(\delta_{ij}:=\|\nu_{ij}-\omega_j\|\to0\) for
each fixed \(j\). Using \(B^{-it}\xi_j=\xi_j\), the faithful cocycle
formula and (OC.13) yield

\[
 \sup_{|t|\le T}
  \|q_{ij} C_\theta(\alpha_i,t)\xi_j-\xi_j\|
       \le e_{ij}:=L_T\delta_{ij}^{1/2}\longrightarrow0.
 \tag{OC.16}
\]

A projection on the first term cannot simply be discarded. Here is the
required estimate. Set \(r_j=\|\xi_j\|\) (equal to one for the states
chosen above), \(v=q_{ij}C_\theta(\alpha_i,t)\xi_j\), and
\(w=(1-q_{ij})C_\theta(\alpha_i,t)\xi_j\). They are orthogonal, and
unitarity gives \(\|v\|^2+\|w\|^2=r_j^2\). Hence

\[
 \|w\|^2=(r_j-\|v\|)(r_j+\|v\|)
       \le2r_j\|v-\xi_j\|\le2r_je_{ij}.
\]

The triangle inequality proves

\[
 \sup_{|t|\le T}\|(C_\theta(\alpha_i,t)-1)\xi_j\|
       \le e_{ij}+\sqrt{2r_je_{ij}}\longrightarrow0.
 \tag{OC.17}
\]

This argument does not require the supports of arbitrary norm-convergent
functionals to converge. It recovers the omitted component from the full
cocycle's unitarity.

## From finite vectors to the arbitrary standard Hilbert space

For \(y'\in M'\), membership of the cocycle in \(M\) gives

\[
 \|(C_\theta(\alpha_i,t)-1)y'\xi_j\|
       \le\|y'\|\,\|(C_\theta(\alpha_i,t)-1)\xi_j\|.
\]

Thus (OC.17) holds on finite linear combinations of these vectors. For
finitely many summands, choose a common eventual net index, using
directedness. OC-04 says their span is dense in \(H\). Since
\(\|C_\theta(\alpha_i,t)-1\|\le2\), approximating an arbitrary vector
by one vector in this span proves

\[
 \sup_{|t|\le T}\|(C_\theta(\alpha_i,t)-1)\eta\|\to0
 \quad(\eta\in H).
 \tag{OC.18}
\]

There is no diagonal choice across the possibly uncountable index set
\(I\): each vector approximation uses only finitely many indices.

## Change the reference, change the limit, and take adjoints

For another n.s.f. reference \(\varphi\), put
\(d_t=[D\varphi:D\theta]_t\). Chain rule and naturality give

\[
 C_\varphi(\beta,t)
       =\beta(d_t)C_\theta(\beta,t)d_t^*.
 \tag{OC.19}
\]

For \(\alpha_i\to\mathrm{id}\), subtract the identity from this formula.
On a vector \(\eta\), the resulting terms are

\[
 \alpha_i(d_t)(C_\theta(\alpha_i,t)-1)d_t^*\eta,
 \qquad (\alpha_i(d_t)-d_t)d_t^*\eta.
\]

For compact time intervals, \(\{d_t^*\eta\}\) is norm compact.
OC-07 and the finite-covering estimate (OC.22) control the first term.
Equation (OC.23) controls the second: canonical standard implementers of
\(\alpha_i\) converge strongly together with their adjoints, and the
bounded family \(d_t\) is strongly continuous. This proves (OC.2) at
the identity for every n.s.f. reference.

For a general limit \(\alpha\), set
\(\beta_i=\alpha^{-1}\alpha_i\). Then \(\beta_i\to\mathrm{id}\), and
the chain rule gives

\[
 C_\varphi(\alpha\beta_i,t)
   =\alpha(C_\varphi(\beta_i,t))C_\varphi(\alpha,t).
 \tag{OC.20}
\]

The fixed implementer of \(\alpha\) and the compact family of vectors
obtained from \(C_\varphi(\alpha,t)\eta\) reduce this to the result
at the identity, again by (OC.22). This proves (OC.2).

Finally, if \(U_i(t)=C_\varphi(\alpha_i,t)\) and
\(U(t)=C_\varphi(\alpha,t)\), then

\[
 \|(U_i(t)^*-U(t)^*)\eta\|
   =\|(U(t)-U_i(t))U(t)^*\eta\|.
 \tag{OC.21}
\]

The set \(\{U(t)^*\eta:t\in K\}\) is compact, so (OC.22) proves the
adjoint assertion.

## Scope checks and a two-dimensional example

On \(M=M_2(\mathbb C)\) in its Hilbert--Schmidt standard form, take
\(\theta=\operatorname{Tr}\), \(p=\mathrm{diag}(1,0)\), and
\(\omega(x)=\operatorname{Tr}(px)\). If
\(\nu_\varepsilon(x)=\operatorname{Tr}(h_\varepsilon x)\) with
\(h_\varepsilon=\mathrm{diag}(1,\varepsilon)\), then
\(\|\nu_\varepsilon-\omega\|=\varepsilon\),
\(\xi_\omega=p\), and \(A_{\nu_\varepsilon}\) is left multiplication
by \(h_\varepsilon\). Its supported imaginary powers fix \(p\) for
every \(\varepsilon\ge0\). Nevertheless, the support of
\(h_\varepsilon\) is the identity for every \(\varepsilon>0\), whereas
the limit support is \(p\). Thus the supported-vector lemma proves
exactly what is needed and no false global support-continuity claim.

The proof diagram depicts (OC.3), (OC.9), and (OC.16)--(OC.20).
Its scalar panel uses \(T=1\), \(x=\sqrt\lambda\), the exact time
supremum for \(x>0\), and the separately marked kernel value one
at \(x=0\). The right panel follows one state summand; the lower
panel explains the passage to the whole Hilbert space and to other
faithful reference weights. All arguments remain in OC-03--08.

![A scalar error plot with its proved bound and a diagram of the finite-state estimates leading to compact-time cocycle continuity.](assets/continuity-mechanism.png)

**Figure OC.1.** Compact-time continuity from finite normal state summands. The scalar panel plots the exact supremum for positive x at T = 1 and separately marks the kernel value. The other panels summarize the identity-limit estimates and the transfer to general limits in OC-03–08; all arguments remain in those sections. Original illustration generated from the accompanying plotting code.

## A square-root calculation with noncommuting supports

**Problem.** In the Hilbert--Schmidt standard form of
\(M_2(\mathbb C)\), set \(\theta=\operatorname{Tr}\),
\(p=\operatorname{diag}(1,0)\), and

\[
 u_s=\begin{pmatrix}\cos s&-\sin s\\ \sin s&\cos s\end{pmatrix},
 \qquad q_s=u_spu_s^*,\qquad
 \nu_s(x)=\operatorname{Tr}(q_sx).
 \tag{OC.24}
\]

For \(0<s<\pi/2\), verify (OC.9) without commuting \(p\) and \(q_s\).
Compute its left-hand error and compare it with the functional-norm bound.

**Solution.** Write \(c=\cos s\) and \(a=\sin s\). The standard conjugation
is \(Jz=z^*\), the cone consists of positive Hilbert--Schmidt matrices,
and \(\xi_{\nu_s}=q_s\), since \(q_s\) is a projection. The opposite trace
is the usual trace for the right action. Thus the spatial derivative
\(A_{\nu_s}\) is left multiplication by \(q_s\); its square root is
the same projection. Hence

\[
 A_{\nu_s}^{1/2}p=q_sp=(pq_s)^*=Jp\xi_{\nu_s}.
 \tag{OC.25}
\]

The two products are explicitly

\[
 q_sp=\begin{pmatrix}c^2&0\\ca&0\end{pmatrix},
 \qquad
 pq_s=\begin{pmatrix}c^2&ca\\0&0\end{pmatrix}.
\]

They differ for the stated range of \(s\), so (OC.25) genuinely uses
the conjugation. Moreover

\[
 \|(A_{\nu_s}^{1/2}-1)p\|_2
 =\sqrt{a^4+c^2a^2}=a.
 \tag{OC.26}
\]

The self-adjoint trace-zero matrix \(q_s-p\) has square \(a^2\,1\).
Its two eigenvalues are \(a\) and \(-a\), so the trace norm is \(2a\).
Finite-dimensional trace duality therefore gives
\(\|\nu_s-\omega\|=2a\), where \(\omega(x)=\operatorname{Tr}(px)\).
The bound in (OC.9) reads \(a\le\sqrt{2a}\), as required. At \(s=0\)
both sides vanish. The calculation keeps the left product, the adjoint
and the Hilbert--Schmidt norm in their separate roles.

## An explicit cocycle and its compact-time error

**Problem.** Retain \(u_s\), put \(d=\operatorname{diag}(4,1)\), and let
\(\varphi(x)=\operatorname{Tr}(dx)\). Compute
\(C_\varphi(\operatorname{Ad}u_s,t)\) and its exact operator-norm distance
from the identity, uniformly for \(|t|\le T\).

**Solution.** The density of \(\varphi\circ\operatorname{Ad}u_s^*\)
is \(u_sdu_s^*\). The matrix specialization of the spatial cocycle
formula (SI-16) gives

\[
 C_\varphi(\operatorname{Ad}u_s,t)
       =u_sd^{it}u_s^*d^{-it}.
\]

Set \(z=e^{it\log4}\), \(c=\cos s\), \(a=\sin s\). Multiplication gives

\[
 C_\varphi(\operatorname{Ad}u_s,t)
 =\begin{pmatrix}
      c^2+a^2z^{-1}&ca(z-1)\\
      ca(1-z^{-1})&a^2z+c^2
   \end{pmatrix}.
 \tag{OC.27}
\]

If \(E\) denotes this matrix minus \(1\), direct multiplication, using
\(\overline z=z^{-1}\), gives \(E^*E=a^2|z-1|^2\,1\): the two diagonal
entries are \(a^2(a^2+c^2)|z-1|^2\) and the off-diagonal entries cancel.
Therefore

\[
 \|C_\varphi(\operatorname{Ad}u_s,t)-1\|
       =2|\sin s|\,|\sin(t\log4/2)|,
\]

and

\[
 \sup_{|t|\le T}\|C_\varphi(\operatorname{Ad}u_s,t)-1\|
 =2|\sin s|\,\sin\!\left(\min\{T\log4/2,\pi/2\}\right).
 \tag{OC.28}
\]

For the last equality, the sine is increasing on \([0,\pi/2]\);
if the time interval reaches \(\pi/2\) in phase, its supremum is one.
In particular the error tends to zero with \(s\), even in operator norm.
The full theorem concerns strong* convergence in arbitrary standard
forms; this matrix computation is a check on orientation and constants.

## Joint limits and the need for nets

**Problem.** First show that if \(\alpha_i\to\alpha\) in the u-topology
and \(t_i\to t\) in \(\mathbb R\), then
\(C_\varphi(\alpha_i,t_i)\to C_\varphi(\alpha,t)\) strongly*.
Next let \(I\) be uncountable and consider \(M=\ell^\infty(I)\) in
its standard form on \(\ell^2(I)\). Explain why the finite-piece
construction in OC-04 cannot be replaced by a sequence of finite
subsets covering \(I\).

**Solution.** Eventually \(t_i\in K=[t-1,t+1]\). For each \(\eta\),

\[
\begin{aligned}
 \|(C_\varphi(\alpha_i,t_i)-C_\varphi(\alpha,t))\eta\|
 &\le
 \sup_{r\in K}\|(C_\varphi(\alpha_i,r)-C_\varphi(\alpha,r))\eta\|\\
 &\quad+\|(C_\varphi(\alpha,t_i)-C_\varphi(\alpha,t))\eta\|.
\end{aligned}
 \tag{OC.29}
\]

The first term tends to zero by OC-01, and the second by the fixed-pair
strong continuity of the cocycle from SI-14. The adjoint version follows
by the same argument using the adjoint statement of OC-01 and the
strong* continuity of a fixed unitary cocycle.

For the second question, let \(\omega_j(x)=x_j\), with support the
one-coordinate projection \(p_j\). The sum
\(\theta(x)=\sum_{j\in I}x_j\) on \(M_+\) is the n.s.f. counting
weight. Its GNS Hilbert space is \(\ell^2(I)\); conjugation is coordinatewise
complex conjugation and the natural cone consists of nonnegative real
vectors. If \(F\subset I\) is finite, \(p_F=\sum_{j\in F}p_j\)
increases strongly to \(1\) as \(F\) ranges over all finite subsets:
for \(\eta\in\ell^2(I)\) and \(\varepsilon>0\), the definition of the
summable square norm gives a finite \(F_0\) with
\(\sum_{j\notin F_0}|\eta_j|^2<\varepsilon^2\), and every
\(F\supseteq F_0\) has \(\|(1-p_F)\eta\|<\varepsilon\).

For any sequence \(F_n\) of finite subsets, the union is countable.
Choose \(j\) outside it. Then \(p_{F_n}e_j=0\) for every \(n\), so
this sequence does not converge strongly to \(1\). Nor can one replace
the counting weight by a faithful normal state: its values on the
orthogonal \(p_j\)'s would be strictly positive numbers with bounded
finite sums. For each positive integer \(n\), only finitely many can
be at least \(1/n\); all the positive values would therefore occur on
a countable set, a contradiction. The proof in OC-07 avoids both
restrictions because each vector approximation uses finitely many
coordinates, with a new finite set permitted for each approximation.

A route for further study is to combine the theorem with
reference changes and the
standard implementation of automorphisms to study continuity of induced
actions. Applications to crossed products require their own
representation and integration arguments; the theorem here supplies
the general weight-cocycle continuity input.