# Homogeneous operators in the continuous core

**Self-checked by the writing AI.**

The continuous core turns a nontracial weight into a homogeneous positive
operator in a semifinite algebra. The ordinary trace of that operator is usually
infinite, so the integral cannot be obtained by silently applying the tracial
\(L^p\) construction. This lesson builds the graded integral, proves the
weight--operator dictionary needed by the source exercises, and proves the
Banach, Hölder, cyclicity and onto duality conclusions in their exact ranges.

Two exact inputs remain visible. The OA-FLOW core contract supplies the
functorial continuous core, its dual action, scaling trace, fixed points and
normalized localizers. The OA-MOD density contract identifies normal semifinite
weights with positive grade-one affiliated operators and gives the inverse
localizer pairing. The arguments below prove the analytic, Banach, product and
duality conclusions from those inputs and the named measurable-operator, trace,
complex-analysis and Hahn--Banach providers. The state-dependent interpolation
identification in HL-08 remains a separate supplementary proof target.

Throughout, \(M\) is a factor with separable predual. Write
\((\widetilde M,\theta,\tau)\) for its fixed continuous core. Thus

\[
 M=\widetilde M^\theta,
 \qquad
 \tau\mathbin\circ\theta_s=e^{-s}\tau .
\tag{HL.1}
\]

The measurable algebra for \(\tau\) is denoted by
\(\mathscr S_\tau(\widetilde M)\). Products of measurable operators mean the
closed measurable products constructed in
Adjoint, addition and multiplication keep the domains.

## Equivariant copies of the translation algebra are weights

Put

\[
 \begin{aligned}
 A&=L^\infty(\mathbb R),\\
 (\rho_t f)(r)&=f(r+t),
 \end{aligned}
\tag{HL.2}
\]

and define

\[
 \begin{aligned}
 V(t)(r)&=e^{-irt},\\
 H(r)&=e^{-r}.
 \end{aligned}
\tag{HL.3}
\]

The characters \(V(t)\) generate \(A\), \(V(t)=H^{it}\), and
\(\rho_s(V(t))=e^{-ist}V(t)\).

Let \(\varphi\) be a nonzero normal semifinite weight on \(M\), with support
\(e=s(\varphi)\). The density contract gives a positive self-adjoint operator
\(h_\varphi\), affiliated with \(e\widetilde M e\), such that

\[
 \begin{aligned}
 \theta_s(h_\varphi)&=e^{-s}h_\varphi,\\
 h_\varphi^{it}&=\varphi^{it},\\
 s(h_\varphi)&=e.
 \end{aligned}
\tag{HL.4}
\]

Here \(\varphi^{it}\) is the canonical core unitary symbol; no bounded density
inside \(M\) is being asserted. Spectral calculus defines

\[
 \begin{aligned}
 \pi^\varphi(f)&=f(-\log h_\varphi),\\
 f&\in L^\infty(\mathbb R).
 \end{aligned}
\tag{HL.5}
\]

The spectral measure class of \(-\log h_\varphi\) is translation invariant.
The core contract says it is the Lebesgue class on its support, so (HL.5) is a
faithful normal star isomorphism onto its range. Equations (HL.3--4) give

\[
 \begin{aligned}
 \pi^\varphi(V(t))&=h_\varphi^{it}=\varphi^{it},\\
 \theta_s\pi^\varphi(f)&=\pi^\varphi(\rho_s f).
 \end{aligned}
\tag{HL.6}
\]

This proves existence. It also proves uniqueness: an equivariant normal
homomorphism with the prescribed values on every \(V(t)\) agrees with
\(\pi^\varphi\) on the von Neumann algebra generated by those characters.

Conversely, let

\[
 \pi:(A,\rho)\longrightarrow(e\widetilde M e,\theta)
\tag{HL.7}
\]

be an equivariant normal star isomorphism onto its range. Set \(h=\pi(H)\), using affiliated
spectral calculus. Then

\[
 \theta_s(h)=e^{-s}h,
 \qquad s(h)=e.
\tag{HL.8}
\]

The positive grade-one density theorem supplies a unique normal semifinite
weight \(\varphi_\pi\) with \(h=h_{\varphi_\pi}\). Since \(V(t)=H^{it}\),

\[
 \pi(V(t))=h^{it}=\varphi_\pi^{it}.
\tag{HL.9}
\]

The first uniqueness argument now gives \(\pi=\pi^{\varphi_\pi}\). Hence
nonzero normal semifinite weights, including their nonzero supports, classify
precisely the equivariant copies of the translation algebra in core corners.
The zero weight has zero support and cannot yield a faithful copy of the
nonzero algebra \(A\).

## Localizers recover the weight and scaling measures

For \(a\in\widetilde M_+\), let

\[
 I_\theta(a)=\int_{\mathbb R}\theta_t(a)\,dt
\tag{HL.10}
\]

be the positive extended orbit integral. A **normalized localizer** satisfies
\(I_\theta(a)=1\). Such elements exist by the core contract. If
\(h=h_\varphi\) and \(x\in M_+\), write
\(b_x=x^{1/2}a x^{1/2}\) and \(c_x=x^{1/2}h x^{1/2}\). The inverse density
pairing is

\[
 \begin{aligned}
 \varphi(x)
 &=\tau\!\left(
     h^{1/2}b_xh^{1/2}
   \right)\\
 &=\tau\!\left(
     a^{1/2}c_xa^{1/2}
   \right).
 \end{aligned}
\tag{HL.11}
\]

Both lines mean the value of the corresponding closed positive form and may be
infinite. The second line follows first for bounded spectral truncations by
tracial cyclicity and then for the full form by the extended-positive
monotone limit from
Radon–Nikodym densities relative to a semifinite trace. Since the left side does
not mention \(a\), (HL.11) also proves independence of the normalized
localizer.

The source uses the following affiliated regularizer \(r_\varepsilon\).
It is unbounded: its scalar multiplier grows like
\(\varepsilon^{-1/2}\lambda^{1/4}\) as \(\lambda\to\infty\).
A bounded alternative is \(s_\varepsilon\). Put

\[
 \begin{aligned}
 r_\varepsilon&=h^{1/2}(1+\varepsilon h^{1/2})^{-1/2},\\
 s_\varepsilon&=h^{1/2}(1+\varepsilon h)^{-1/2},\\
 \|s_\varepsilon\|&\leq\varepsilon^{-1/2}.
 \end{aligned}
\tag{HL.12}
\]

For the bounded positive operator \(b_x\), the normal weight
\(\Phi_{b_x}(c)=\tau(b_x^{1/2}c b_x^{1/2})\) extends normally to the
extended positive cone. Spectral calculus gives
\(r_\varepsilon^2=h/(1+\varepsilon h^{1/2})\uparrow h\) and
\(s_\varepsilon^2=h/(1+\varepsilon h)\uparrow h\) as
\(\varepsilon\downarrow0\). On bounded spectral bands, tracial cyclicity
identifies \(\Phi_{b_x}(r_\varepsilon^2)\) with the closed-form trace
\(\tau(r_\varepsilon b_xr_\varepsilon)\). Normal extension gives the same
identity for the full affiliated operator, including infinite values.
Consequently normality and (HL.11) prove

\[
 \varphi(x)
 =\lim_{\varepsilon\downarrow0}
   \tau\!\left(
     r_\varepsilon b_xr_\varepsilon
   \right).
\tag{HL.13}
\]

The same limit holds with \(s_\varepsilon\) in place of
\(r_\varepsilon\). The trace values increase because the squared spectral
multipliers increase; the noncommuting sandwiches themselves are not asserted
to increase in operator order.

There is a normalization defect in the printed scaling-measure exercise. Let
\(\mu\) be a normal weight on \(A=L^\infty(\mathbb R)\) such that

\[
 \mu\mathbin\circ\rho_t=e^{-t}\mu
\tag{HL.14}
\]

and suppose \(0<\mu(f_0)<\infty\) for some \(f_0\in A_+\). Normality represents
\(\mu\) by an extended measurable density \(g\):

\[
 \mu(f)=\int_{\mathbb R}g(r)f(r)\,dr.
\tag{HL.15}
\]

Changing variables in (HL.14) yields

\[
 \begin{aligned}
 g(r-t)&=e^{-t}g(r)\\
 &\text{for almost every }r.
 \end{aligned}
\tag{HL.16}
\]

Thus \(q(r)=e^{-r}g(r)\) is translation invariant modulo null sets. A
measurable function with this property is almost everywhere constant. One
direct proof convolves each finite truncation of \(q\) with a compactly
supported approximate identity: every convolution is continuous and invariant
under the dense subgroup of rational translations, hence constant; monotone
recovery of the truncations gives the assertion. The finite nonzero value of
\(f_0\) rules out both constants zero and infinity. Consequently

\[
 \boxed{
 \begin{aligned}
 \mu(f)&=c\int_{\mathbb R}e^r f(r)\,dr,\\
 c&>0 .
 \end{aligned}}
\tag{HL.17}
\]

This weight is faithful and semifinite. The source's coefficient \(1\) follows
only after an additional normalization fixing \(c=1\). Every \(c>0\) satisfies
the printed hypotheses, so omitting \(c\) changes the statement.

Without the finite nonzero test, the same invariant-density argument gives
an extended constant \(c\in[0,\infty]\). A nonzero scaling weight has
\(c>0\), and \(c=\infty\) means that every nonzero positive function
has infinite weight. This extended case is needed for infinite weights below.

## Finite weights are exactly measurable core densities

Let \(h=h_\varphi\) and
\(\mathcal A^\varphi=\pi^\varphi(A)=W^*(h^{it}:t\in\mathbb R)\). For
\(\lambda>0\), write

\[
 d_h(\lambda)=\tau(1_{(\lambda,\infty)}(h)).
\tag{HL.18}
\]

Grade one and trace scaling give the exact distribution law

\[
 d_h(e^s\lambda)=e^{-s}d_h(\lambda).
\tag{HL.19}
\]

Indeed
\(\theta_s(1_{(\lambda,\infty)}(h))\)
is \(1_{(e^s\lambda,\infty)}(h)\), and (HL.1) applies. More explicitly, the
restriction \(\mu_\varphi=\tau\mathbin\circ\pi^\varphi\) is a normal scaling
weight on \(A\), so (HL.17) gives
\(\mu_\varphi(f)=c_\varphi\int e^r f(r)\,dr\). Choose
\(g\in L^1(\mathbb R)_+\cap L^\infty(\mathbb R)\) with
\(\int g(r)\,dr=1\). Then \(a=\pi^\varphi(g)\) is a normalized localizer on
the support corner; adjoining a normalized complementary-corner localizer does
not change the following trace. The inverse-density formula (HL.11), at
\(x=1\), gives

\[
 c_\varphi
 =\mu_\varphi(Hg)
 =\tau(a^{1/2}ha^{1/2})
 =\varphi(1).
\]

Since \(1_{(\lambda,\infty)}(H)=1_{(-\infty,-\log\lambda)}\), the scaling
law therefore has the exact extended constant

\[
 d_h(\lambda)=\frac{\varphi(1)}{\lambda}
\tag{HL.20}
\]

including infinite values. For \(\varphi(1)=\infty\) the coefficient is
the extended value just described, rather than a finite instance of (HL.17).

If \(\varphi(1)<\infty\), (HL.20) makes every high spectral tail finite and
tending to zero. The spectral-tail criterion
Spectral tails characterize the resulting operators shows that \(h\) is
\(\tau\)-measurable. Conversely, if \(h\) is measurable, some nonzero high
spectral tail has finite trace unless \(h\) is bounded. In the bounded case,
grade one and preservation of the operator norm already force \(h=0\), which
corresponds to the zero finite weight. In the nonzero case, (HL.19) gives a
finite constant in (HL.20), so \(\varphi(1)<\infty\).

The intersection criterion is now exact. If \(\varphi\) is finite and nonzero,
then for \(0<a<b<\infty\), put \(p_{a,b}=1_{[a,b]}(h)\). We have

\[
 \begin{aligned}
 p_{a,b}&\in\mathcal A^\varphi\cap\mathfrak m_\tau,\\
 \tau(p_{a,b})&=\varphi(1)(a^{-1}-b^{-1}).
 \end{aligned}
\tag{HL.21}
\]

whenever the band is nonzero. Conversely, suppose
\(0\neq z\in\mathcal A^\varphi\cap\mathfrak m_\tau\). A nonzero spectral
projection of \(|z|\) has finite trace and still belongs to
\(\mathcal A^\varphi\). Thus the restriction of \(\tau\) to this copy of
\(L^\infty(\mathbb R)\) has a nonzero finite value. The scaling-measure lemma
(HL.17) applies, so its coefficient is finite; (HL.20) then gives
\(\varphi(1)<\infty\). We have proved

\[
 \boxed{
 \begin{gathered}
 \varphi(1)<\infty\\
 \Updownarrow\\
 \mathcal A^\varphi\cap\mathfrak m_\tau\neq\{0\}\\
 \Updownarrow\\
 h_\varphi\in\mathscr S_\tau(\widetilde M).
 \end{gathered}}
\tag{HL.22}
\]

For the zero weight, the first and third conditions hold while the displayed
nonzero-intersection condition does not. The source works with nonzero weights;
that hypothesis is necessary for the three-way formulation.

## Distribution scaling forbids negative real grade

For a closed densely defined operator \(T\) affiliated with \(\widetilde M\),
say that \(T\) has **grade** \(\alpha\in\mathbb C\) when

\[
 \theta_s(T)=e^{-\alpha s}T
 \qquad(s\in\mathbb R).
\tag{HL.23}
\]

Let

\[
 \begin{gathered}
 \mathscr H(\alpha)\\
 =\{T:\\
 \operatorname{grad}(T)=\alpha,\\
 T\in\mathscr S_\tau(\widetilde M)\}.
 \end{gathered}
\tag{HL.24}
\]

If \(T=u|T|\) and \(\alpha=p+iq\), uniqueness of polar decomposition gives

\[
 \begin{aligned}
 \theta_s(|T|)&=e^{-ps}|T|,\\
 \theta_s(u)&=e^{-iqs}u.
 \end{aligned}
\tag{HL.25}
\]

Writing \(d_T(\lambda)=\tau(1_{(\lambda,\infty)}(|T|))\), the same calculation
as in (HL.19) yields

\[
 d_T(e^{ps}\lambda)=e^{-s}d_T(\lambda).
\tag{HL.26}
\]

Suppose \(p<0\) and \(T\neq0\). If \(T\) is bounded, preservation of operator
norm under \(\theta_s\) contradicts
\(\|T\|=e^{-ps}\|T\|\). If \(T\) is unbounded, measurability supplies a
\(\lambda\) with \(0<d_T(\lambda)<\infty\). For \(s>0\), the threshold
\(e^{ps}\lambda\) is smaller than \(\lambda\), so monotonicity gives

\[
 d_T(e^{ps}\lambda)\ge d_T(\lambda),
\]

whereas (HL.26) gives the strictly smaller value
\(e^{-s}d_T(\lambda)\). This contradiction proves

\[
 \begin{aligned}
 \mathscr H(\alpha)&=\{0\}\\
 &\text{when }\Re\alpha<0.
 \end{aligned}
\tag{HL.27}
\]

Now let \(\alpha=0\). Every spectral projection of a grade-zero measurable
operator is fixed by \(\theta\). If such an operator were unbounded,
measurability would supply a nonzero finite-trace high spectral projection
\(e\). But

\[
 \tau(e)=\tau(\theta_s(e))=e^{-s}\tau(e)
\]

is impossible. Hence the operator is bounded, and the fixed-point identity in
(HL.1) places it in \(M\). Conversely, every \(x\in M\) is bounded,
\(\tau\)-measurable and fixed. Therefore

\[
 \boxed{\mathscr H(0)=M.}
\tag{HL.28}
\]

Purely imaginary nonzero grades are different: core unitaries can have such a
grade. Statement (HL.28) concerns the exact grade zero.

## Every positive real grade has a functional density

Let \(T\in\mathscr H(\alpha)\), with \(\alpha=p+iq\) and \(p>0\). Functional
calculus, (HL.25), and the measurable-power theorem give

\[
 \theta_s\bigl(|T|^{1/p}\bigr)
 =e^{-s}|T|^{1/p}.
\tag{HL.29}
\]

Thus \(|T|^{1/p}\) is a positive measurable grade-one operator. The density
contract and (HL.22) produce a unique normal positive functional
\(\omega_T\in M_*^+\) such that

\[
 |T|^{1/p}=h_{\omega_T}.
\tag{HL.30}
\]

In particular

\[
 |T|=h_{\omega_T}^{\,p}.
\tag{HL.31}
\]

For a real grade \(p\), (HL.25) puts the polar partial isometry \(u\) in \(M\),
so \(T=u h_{\omega_T}^{p}\). For a complex grade, choose once and for all a
faithful normal state \(\chi\) and put \(d=h_\chi\). The unitary \(d^{iq}\) has
grade \(iq\). Hence \(u d^{-iq}\) has grade zero and lies in \(M\), giving the
normal form

\[
 \begin{aligned}
 T&=v d^{iq}h_{\omega_T}^{p},\\
 v&=u d^{-iq}\in M.
 \end{aligned}
\tag{HL.32}
\]

The fixed choice of \(\chi\) is only a coordinate for the imaginary grade; the
operator \(T\), its modulus and the functional \(\omega_T\) do not depend on it.
Also, if a nonzero operator has two grades, then
\(e^{-\alpha s}T=e^{-\beta s}T\) for every \(s\), forcing
\(\alpha=\beta\). Distinct homogeneous spaces therefore have zero
intersection.

## The canonical integral and the grade-one predual

Let \(T\in\mathscr H(1)\), with polar decomposition

\[
 \begin{aligned}
 T&=u h_\omega,\\
 u&\in M,\\
 \omega&\in M_*^+.
 \end{aligned}
\tag{HL.33}
\]

We first establish linearity and the module action of the density map; neither
follows from polar decomposition alone. For positive normal functionals write
\(D(\omega)=h_\omega\). A sum of positive measurable grade-one operators
is again positive, measurable and of grade one. Evaluate its inverse density
by the second line of (HL.11). Linearity of the extended trace on positive
forms gives
\(D(\omega_1+\omega_2)=h_{\omega_1}+h_{\omega_2}\), and positive scalar
multiplication is preserved as well. The Jordan decomposition therefore
extends \(D\) to a well-defined complex linear map on \(M_*\). Its
injectivity follows from injectivity on positive functionals: a self-adjoint
element of its kernel has equal positive and negative density parts, and the
real and imaginary parts handle the general kernel.

For \(x\in M\), put \(\omega_x(y)=\omega(x^*yx)\). We claim
\(D(\omega_x)=xh_\omega x^*\). To check this using the inverse formula,
fix \(y\in M_+\) and write \(y^{1/2}x=v|y^{1/2}x|\), with
\(p=v^*v\). If \(a\) is a normalized localizer, then
\(a'=v^*av+(1-p)a(1-p)\) is another: \(v,p\in M\) are fixed by
\(\theta\), so its orbit integral is \(p+(1-p)=1\). The complement term
vanishes between the two factors \(|y^{1/2}x|\). Tracial closed-form
pairings thus give

\[
 \begin{aligned}
 \tau\bigl(a^{1/2}y^{1/2}xh_\omega x^*y^{1/2}a^{1/2}\bigr)
 &=\tau\bigl(h_\omega^{1/2}|y^{1/2}x|a'|y^{1/2}x|h_\omega^{1/2}\bigr)\\
 &=\omega(x^*yx).
 \end{aligned}
\]

The first equality is understood through the normal positive trace pairings
of HL-02, so it does not assume boundedness of \(h_\omega\).
Uniqueness in the density contract proves the claim. Polarizing this
congruence identity gives
\(D[y\mapsto\omega(b^*yc)]=c h_\omega b^*\) for \(b,c\in M\).
In particular the normal functional \(y\mapsto\omega(yu)\) has density
\(u h_\omega\). Its polar support is \(u^*u=s(\omega)\), as in (HL.33).
Conversely HL-05 puts every grade-one operator in this form. Hence \(D\)
is onto \(\mathscr H(1)\).

Define \(\int T=D^{-1}(T)(1)\). This linear functional agrees with the
following localizer formula:

\[
 \int T
 =\tau(a^{1/2}Ta^{1/2}).
\tag{HL.34}
\]

Indeed, decompose \(D^{-1}(T)\) into a complex linear combination of
positive functionals. Each localized positive density in (HL.11), at
\(x=1\), has finite trace. Their linear combination is trace integrable,
and the linearity just proved identifies it with the localized measurable
product in (HL.34). Thus (HL.34) is finite and independent of \(a\), and

\[
 \int u h_\omega=\omega(u).
\tag{HL.35}
\]

The polarized congruence identity fixes the orientation: the functional
associated with \(T\) is

\[
 \begin{aligned}
 F_T(x)&=\int xT,\\
 x&\in M.
 \end{aligned}
\tag{HL.36}
\]

Every normal functional has a polar decomposition of the form
\(x\mapsto\omega(xu)\), and its density is \(u h_\omega\). Thus

\[
 T\longmapsto F_T
\tag{HL.37}
\]

is a linear bijection from \(\mathscr H(1)\) onto \(M_*\). Moreover

\[
 \begin{aligned}
 \|T\|_1&:=\int|T|\\
 &=\omega(1)=\|F_T\|.
 \end{aligned}
\tag{HL.38}
\]

Since the predual is complete, (HL.37--38) prove that
\(\mathscr H(1)\) is a Banach space. They also show why the ordinary core trace
cannot replace the integral: in the scalar model \(M=\mathbb C\), the density
\(ce^{-r}\) has infinite ordinary \(\tau\)-integral, while its canonical
integral is \(c\).

### Spectral size and an initial product bound

For a measurable operator \(T\) of grade \(a+ib\), \(a>0\), put
\(g(T)=(\int|T|^{1/a})^a\). The definition applies even when \(a>1\);
it is not yet asserted to be a norm. The distribution formula in HL-03 gives

\[
 d_T(\lambda)=g(T)^{1/a}\lambda^{-1/a},
 \qquad \mu_t(T)=g(T)t^{-a}.
\]

Here \(\mu_t(T)\) is the infimum of \(\|Te\|\) over projections with
\(\tau(1-e)\leq t\). The spectral-cut characterization follows from the
projection comparison of MT-02: a cut with norm below \(\lambda\) must
discard at least the dimension of \(1_{(\lambda,\infty)}(|T|)\).
At positive real grade, therefore, convergence in measure within one fixed
homogeneous space is exactly convergence of the gauge of the difference.
In particular, on grade one it is predual norm convergence.

We will use two estimates for arbitrary measurable operators:
\(\mu_{s+t}(A+B)\leq\mu_s(A)+\mu_t(B)\) and
\(\mu_{s+t}(AB)\leq\mu_s(A)\mu_t(B)\). For the first, intersect the two
bounded-domain cuts. For the second, take cuts \(e,f\) for \(A,B\).
The right support \(r\) of \((1-e)B\) has trace at most \(\tau(1-e)\),
because its polar range lies in \(1-e\). On \(q=f\wedge(1-r)\),
\(Bq=eBq\), and the ordinary product is defined and bounded by
\(\|Ae\|\|Bf\|\). Projection subadditivity bounds \(\tau(1-q)\) by
\(s+t\). Infima, with arbitrarily small slack in the norm bounds, prove the
estimate for the closed measurable product. The same argument gives
\(\mu_t(CAD)\leq\|C\|\mu_t(A)\|D\|\) for bounded core operators \(C,D\).

If \(A,B\) have grades \(a+ib\) and \(1-a-ib\), \(0<a<1\), their product
has grade one. Set \(s=a\), \(t=1-a\). Then

\[
 \|AB\|_1
 \leq a^{-a}(1-a)^{-(1-a)}g(A)g(B).
\]

This rough estimate uses actual bounded-domain projections and makes no
claim that the cut operators remain homogeneous. Its constant is at most
two and tends to one at both endpoints. At real grade zero, HL-04's
spectral-tail argument gives boundedness, and the bounded module estimate
supplies the corresponding endpoint bound.

## Complex powers interpolate positive densities

Let \(h,k\in\mathscr H(1)_+\) and \(0\leq\Re z\leq1\). Positive real
powers are measurable. At real exponent zero use the supported imaginary
powers, which are bounded partial unitaries. Closed measurable multiplication
and grade addition give

\[
 h^z k^{1-z}\in\mathscr H(1).
\tag{HL.39}
\]

We establish holomorphy of the actual product

\[
 F(z)=h^z k^{1-z}
\tag{HL.40}
\]

in the open strip. For any positive measurable \(c\), \(z\mapsto c^z\)
is differentiable in measure on \(\Re z>0\), with derivative the spectral
function \(c^z\log c\), defined as zero on the kernel. On \(0\leq\lambda\leq R\),
the difference quotients of \(\lambda^z\) converge uniformly near each
\(z_0\) with positive real part: powers times logarithms tend to zero at
\(\lambda=0\), and their second derivatives are uniformly bounded on a small
closed disk about \(z_0\). Functional calculus proves the bounded-cut assertion.
The omitted high spectral projection has arbitrarily small trace as
\(R\to\infty\); this proves differentiability in measure.

The product rule in MT-09 now gives

\[
 F'(z)=h^z\log h\,k^{1-z}-h^z k^{1-z}\log k
\]

as a limit in measure. This formula denotes the two measurable spectral
products, not an assertion that \(\log h\) or \(\log k\) alone is measurable.
The difference quotients have grade one, and continuity of the dual action
in measure keeps their limit in that grade. The last lemma of HL-06 turns
this convergence into predual norm convergence. Thus \(F\) is norm holomorphic.
The spectral cuts prove differentiability in the measure algebra; they are
never asserted to approximate \(F\) inside a homogeneous analytic space.

On the two boundary lines the direct module formulas are

\[
 F(it)=(h^{it}k^{-it})k,
\tag{HL.41}
\]

\[
 F(1+it)=h(h^{it}k^{-it}).
\tag{HL.42}
\]

The parenthesized operator has grade zero and is a contraction in \(M\).
Consequently

\[
 \begin{aligned}
 \|F(it)\|_1&\leq\|k\|_1,\\
 \|F(1+it)\|_1&\leq\|h\|_1.
 \end{aligned}
\tag{HL.43}
\]

For the sharp interior estimate, first take \(\|h\|_1=\|k\|_1=1\).
The initial product bound gives
\(\|F(u+it)\|_1\leq B(u)=u^{-u}(1-u)^{-(1-u)}\).
Fix \(0<\delta<1/2\). The function is norm holomorphic and bounded on the
closed inner strip \(\delta\leq\Re z\leq1-\delta\), and both boundary
bounds are \(B(\delta)\). Apply CI-02 there. For each fixed interior point
let \(\delta\downarrow0\); since \(B(\delta)\to1\), its norm is at most one.
This use of inner strips requires no unproved norm continuity across an
outer boundary. Rescaling \(h,k\) and using (HL.43) proves

\[
 \boxed{\begin{aligned}
 \|h^z k^{1-z}\|_1
 &\leq\|h\|_1^{\Re z}
        \|k\|_1^{1-\Re z}.
 \end{aligned}}
\tag{HL.44}
\]

Zero inputs give a zero product; endpoint geometric means use \(c^0=1\).
All operator powers remain supported. The mathematical source comparison is
Hiai, section 11, Lemmas 11.14 and 11.17--11.21; the derivative and inner-strip
arguments above include their actual topology and support conditions.

## The corrected Banach range for a fixed grade

For \(\alpha=p+iq\), \(0<p\leq1\), define

\[
 \|T\|_\alpha=\left(\int|T|^{1/p}\right)^p.
\tag{HL.45}
\]

HL-05 makes this finite. We first prove the complementary product bound.
If \(0<p<1\), write \(S=u h^p\) and \(T=k^{1-p}v\), using the left polar
decomposition for \(S\) and right polar decomposition for \(T\).
Here \(h,k\) are positive grade-one densities, while \(u,v\) are bounded
core partial isometries whose imaginary grades cancel. Their spectral
supports give \(\|S\|_\alpha=(\int h)^p\) and
\(\|T\|_{1-\alpha}=(\int k)^{1-p}\). The bounded module estimate for \(\mu_1\)
and (HL.44) prove
\(\|ST\|_1\leq\|S\|_\alpha\|T\|_{1-\alpha}\).
This does not assume \(u,v\in M\). At the zero/one real-grade endpoints use
bounded multiplication and the same \(\mu_1\) estimate.

The pairing is norming without an onto duality assumption. For nonzero
\(S=u h^p\), set \(T_0=h^{1-p}u^*\). Then
\(ST_0=u h u^*\), and its nonzero spectral tails are equivalent to those
of \(h\). Thus \(\int ST_0=\int h\) and
\(\|T_0\|_{1-\alpha}=(\int h)^{1-p}\).
Normalize \(T_0\) to have gauge one. Together with the product bound this gives

\[
 \|S\|_\alpha
 =\sup_{\|T\|_{1-\alpha}\leq1}\left|\int ST\right|.
\]

Linearity of the grade-one integral proves the triangle inequality from this
supremum. Homogeneity and separation follow from the distribution law.
For \(p=1\), the imaginary-grade isometry below reduces to HL-06.

Completeness retains the actual measurable limit. The exact size formula
\(\mu_t(T)=\|T\|_\alpha t^{-p}\) turns every norm-Cauchy net into a
measure-Cauchy net. MT-05--09 give a measurable limit, and trace scaling makes
\(\theta_s\) continuous in measure, preserving the grade. For fixed \(n\)
(or a fixed index in the original directed net), apply
\(\mu_{s+t}(S+T)\leq\mu_s(S)+\mu_t(T)\) to the differences with the limit.
Let the moving index tend to its measure limit and then let \(t\downarrow0\).
It gives \(\|T_n-T\|_\alpha\leq\liminf_m\|T_n-T_m\|_\alpha\).
The Cauchy condition proves norm convergence. Hence the homogeneous space
is Banach.

**A separate interpolation target.** To specify a meaningful compatible couple,
fix a faithful state \(\chi\), put \(d=h_\chi\), and embed
\(X_0=J_\chi(M)\) in \(X_1=M_*\), where \(J_\chi(x)(y)=\chi(yx)\) and
\(X_0\) has the transported operator norm. Faithfulness proves injectivity
by testing at \(y=x^*\). The supplementary identification target is

\[
 [X_0,X_1]_p.
\tag{HL.46}
\]

The proposed map is \(T\mapsto F_{T d^{1-p}}\). Its onto isometry and
grade-preserving strip construction remain an additional open theorem, recorded
by HL-DEP-GRADED-STRIP. The Banach proof above has its own complete argument.
The raw inclusions of grades zero and one into the measurable algebra would
give a zero intermediate space: the bounded endpoint projections induced by
\((e^{-s}\mathrm{Id}-\theta_s)/(e^{-s}-1)\), \(s\ne0\), annihilate every
interpolation element by CI-05.

For the imaginary part, the core unitary \(d^{-iq}\) gives a linear bijection

\[
 R_{d^{-iq}}:\mathscr H(p+iq)\longrightarrow\mathscr H(p).
\tag{HL.47}
\]

Indeed \(|Td^{-iq}|=d^{iq}|T|d^{-iq}\). Traciality preserves every spectral-tail
trace, so the distribution constant and gauge are unchanged:

\[
 \|Td^{-iq}\|_p=\|T\|_{p+iq}.
\tag{HL.48}
\]

At real grade zero the spectral-tail argument of HL-04 gives boundedness;
the same unitary coordinate identifies an imaginary grade with \(M\)
isometrically.

For the counterexample beyond the Banach range, take orthogonal positive
grade-one densities \(h_1,h_2\) of integral one, and let \(p>1\).
Set \(T_j=h_j^p\). Spectral calculus on the two supports gives

\[
 (T_1+T_2)^{1/p}=h_1+h_2,
\tag{HL.49}
\]

\[
 \|T_1\|_p=\|T_2\|_p=1,\qquad
 \|T_1+T_2\|_p=2^p>2.
\tag{HL.50}
\]

The displayed gauge is then a quasi-norm. The corrected positive Banach range is

\[
 0<\Re\alpha\leq1.
\tag{HL.51}
\]

A separable factor other than the scalar factor supplies the two orthogonal
nonzero supports used in the counterexample; the source's universal assertion
is refuted already by a matrix factor. In the scalar factor each homogeneous
space is one-dimensional, so its gauge is a norm even outside this universal
Banach range.

## Products, Hölder bounds and the cyclic integral

Let \(T_j\in\mathscr H(\alpha_j)\), \(1\leq j\leq n\), with

\[
 \alpha_1+\cdots+\alpha_n=1.
\tag{HL.52}
\]

For a nonzero product every factor is nonzero, so HL-04 forces
\(p_j=\Re\alpha_j\geq0\). These real grades sum to one and are at most one.
Grade-zero real parts give bounded operators. Zero-factor products are
trivial; on a zero space of negative real grade set the gauge of its sole
element to zero. The desired estimate is

\[
 \|T_1\cdots T_n\|_1\leq
 \prod_{j=1}^n\|T_j\|_{\alpha_j}.
\tag{HL.53}
\]

The product belongs to the measurable algebra by MT-09, and grade addition
already proves

\[
 T_1\cdots T_n\in\mathscr H(1).
\tag{HL.54}
\]

We prove the full finite-product bound without presuming a multilinear
homogeneous strip representation. For \(p_j>0\) put
\(r_j=1/p_j\), \(C_j=\|T_j\|_{\alpha_j}^{r_j}\) and
\(A_j=u_j\min(|T_j|,1)\). These bounded caps are auxiliary core operators,
not homogeneous operators. The exact spectral distribution gives, for
\(\varepsilon>0\),

\[
 \tau(|A_j|^{r_j(1+\varepsilon)})
 =C_j(1+\varepsilon^{-1}).
\]

For \(p_j=0\) put \(A_j=T_j\). Apply the tracial product theorem TI-10
iteratively, with output exponent \(1+\varepsilon\) and input exponents
\(r_j(1+\varepsilon)\). All intermediate exponents are at least one,
since the partial reciprocal sums are at most \(1/(1+\varepsilon)\).

Write \(X=T_1\cdots T_n\), \(A=u_X\min(|X|,1)\) and
\(Q=A_1\cdots A_n\). Each difference \(T_j-A_j\), for \(p_j>0\), has
finite left and right trace support. Measurable multiplication preserves
finite trace rank: the right support of \(BY\) lies under that of \(Y\),
and polar equivalence bounds its left trace rank by that of \(Y\);
the right-multiplication argument is analogous. Projection subadditivity
handles sums. Expanding \(X-Q\) and \(X-A\) therefore shows that \(Q-A\)
has finite trace rank. It is also bounded, so its tracial
\(L^{1+\varepsilon}\) norm is uniformly bounded as \(\varepsilon\downarrow0\).

The cap \(A\) has distribution constant \(\|X\|_1\), hence
\(\varepsilon^{1/(1+\varepsilon)}
 \|A\|_{L^{1+\varepsilon}(\tau)}\to\|X\|_1\).
Tracial Minkowski makes the same limit hold for \(Q\). After multiplication
by \(\varepsilon^{1/(1+\varepsilon)}\), the tracial Hölder bound tends to
\(\prod_j\|T_j\|_{\alpha_j}\), because \(\sum_jp_j=1\).
This proves (HL.53), including imaginary grades and bounded endpoints.

For cyclicity we prove the two-factor identity first. If \(S,T\) have the
same grade \(1/2+ib\), polar equivalence gives
\(\int S^*S=\int SS^*\). Polarize this identity using \(S+i^kT\),
\(k=0,1,2,3\). It gives \(\int ST^*=\int T^*S\).
This argument applies to the common complex half grade, not just the real one.

Next let \(A,B\) have complementary real grades \(p,1-p\), \(0<p<1\).
The real and imaginary parts decompose each into a complex linear combination
of positive elements of its same real grade. For positive elements write
\(A=h^p\), \(B=k^{1-p}\). By HL-07 the scalar functions
\(\int h^z k^{1-z}\) and \(\int k^{1-z}h^z\) are holomorphic in the
open strip. On \(z=1/2+it\), the common-half-grade identity applies to
\(S=h^z\) and \(T=k^z\): both have grade \(1/2+it\), and
\(T^*=(k^z)^*=k^{\bar z}=k^{1-z}\) on this line. Thus
\(\int ST^*=\int T^*S\) says precisely that the two functions agree there.
The identity theorem gives equality throughout the strip, in particular
at \(z=p\). Linearity gives \(\int AB=\int BA\) for the original elements.
The real zero/one endpoints are the module identity of HL-06.

For complementary complex grades \(p+ib,1-p-ib\), put \(w=d^{ib}\).
Then \(A=Sw^*\) and \(B=wT\) have complementary real grades,
\(ST=AB\), and \(TS=w^*BAw\). Conjugation by this core unitary preserves
the integral of a positive grade-one operator by equality of spectral tails;
linearity extends it to all of \(\mathscr H(1)\). Thus
\(\int ST=\int TS\), including the bounded endpoints.
Group the first \(n-1\) factors in (HL.52) and apply this two-factor result:

\[
 \int T_1T_2\cdots T_n
 =\int T_nT_1T_2\cdots T_{n-1}.
\tag{HL.55}
\]

No localizer has been commuted past a homogeneous factor.

## Complementary grades are exact dual partners

Let \(\alpha=p+iq\), \(0<p<1\). The pairing is

\[
 \langle S,T\rangle=\int ST,\qquad
 S\in\mathscr H(\alpha),\ T\in\mathscr H(1-\alpha).
\tag{HL.56}
\]

HL-08--09 give

\[
 |\langle S,T\rangle|
 \leq\|S\|_\alpha\|T\|_{1-\alpha}.
\tag{HL.57}
\]

For \(S=u h^p\), the exact polar equality test is

\[
 T_0=h^{1-p}u^*.
\tag{HL.58}
\]

Its spectral tails give

\[
 \begin{aligned}
 \int ST_0&=\int h,\\
 \|S\|_\alpha&=(\int h)^p,\\
 \|T_0\|_{1-\alpha}&=(\int h)^{1-p}.
 \end{aligned}
\tag{HL.59}
\]

It proves norming in each variable, using cyclicity for the reversed orientation.
We now prove onto representation rather than assume it in a strip contract.

**Uniform convexity for \(0<p\leq1/2\).** Put \(r=1/p\geq2\).
For the real-grade space the required Clarkson bound is

\[
 \|A+B\|_p^r+\|A-B\|_p^r
 \leq2^{r-1}(\|A\|_p^r+\|B\|_p^r).
\]

Here is a direct strip proof. Normalize the right-hand parenthesis to one
and write \(A=uH^p\), \(B=vK^p\), with
\(\int H+\int K=1\). To test the norm of the pair \((A+B,A-B)\),
take \(C=P^{1-p}w\), \(D=Q^{1-p}x\) with
\(\|C\|_{1-p}^{r'}+\|D\|_{1-p}^{r'}=1\), \(r'=1/(1-p)\).
Then \(\int P+\int Q=1\). The norming supremum for the two-component
space is its scalar \(\ell^r\) norm: apply (HL.59) to each component and
choose the scalar conjugate-exponent weights.

Use the holomorphic scalar function

\[
 G(z)=\int (uH^z+vK^z)P^{1-z}w
          +(uH^z-vK^z)Q^{1-z}x.
\]

Holomorphy follows term by term from HL-07 and bounded \(M\)-module
multiplication. On each line \(\Re z=\delta>0\), Hölder bounds its modulus
by

\[
 ((\int H)^\delta+(\int K)^\delta)
 ((\int P)^{1-\delta}+(\int Q)^{1-\delta})\leq2.
\]

On \(\Re z=1/2\), Cauchy--Schwarz in the common-half-grade Hilbert
space, applied to the two components, gives \(|G(z)|\leq\sqrt2\).
That Hilbert structure is obtained by polarizing
\(\int S^*S\); positivity, (HL.45), and the common-half-grade cyclic
identity of HL-09 identify its norm, while HL-08 supplies completeness.
The parallelogram identity makes the squared norm of
\((uH^z+vK^z,uH^z-vK^z)\) equal to two, and the dual test pair has norm one.
For \(p<1/2\), apply CI-02 on \(\delta\leq\Re z\leq1/2\) and let
\(\delta\downarrow0\). It gives \(|G(p)|\leq2^{1-p}\).
At \(p=1/2\) use the Hilbert estimate itself. Take the two-component
norming supremum and undo normalization to obtain Clarkson's bound.
Zero densities remain supported throughout the argument.

For unit vectors with \(\|A-B\|_p\geq\eta\), the bound implies
\(\|(A+B)/2\|_p^r\leq1-(\eta/2)^r\). This is uniform convexity.
The unitary isometry (HL.47--48) gives it for imaginary translates too.

**Why uniform convexity gives reflexivity here.** For any Banach space \(E\),
the canonical image of its unit ball is weak-star dense in the bidual unit
ball. Indeed, if its image under finitely many functionals failed to
approximate the corresponding bidual coordinates, finite-dimensional real
separation would produce \(f\in E^*\) and \(x^{**}\) of norm at most one
with \(\Re x^{**}(f)>\sup_{\|x\|\leq1}\Re f(x)=\|f\|\), a contradiction.
This is the finite-coordinate Hahn--Banach argument.

Let now \(E\) be uniformly convex and \(\|x^{**}\|=1\).
Take a net \(x_i\) in its unit ball converging weak-star to \(x^{**}\).
For any prescribed \(\eta>0\), choose the convexity deficit \(\delta>0\)
and a unit \(f\in E^*\) with \(\Re x^{**}(f)>1-\delta/2\).
Eventually \(\Re f(x_i)>1-\delta\). Every midpoint in that tail has norm
greater than \(1-\delta\), so uniform convexity gives
\(\|x_i-x_j\|<\eta\). The net is norm Cauchy and has a limit \(x\in E\);
its canonical image is \(x^{**}\). Scaling handles every bidual element.
Thus \(E\) is reflexive. Its dual is reflexive as well: any element of
\(E^{***}\) restricts on the surjective canonical copy of \(E\) to an
element of \(E^*\), which represents it on all of \(E^{**}\).

**Onto dual representation.** The map
\(\Phi:T\mapsto[S\mapsto\int ST]\) is a linear isometry by (HL.59).
Its range is norm closed because the complementary space is Banach, and a
closed linear subspace is weakly closed by Hahn--Banach.
It is weak-star dense: a proper weak-star closure would, by separation on
finitely many evaluations, give a nonzero \(S\) annihilating the range,
contradicting the polar norming test.

If \(p\leq1/2\), \(\mathscr H(\alpha)\) is reflexive by the preceding proof.
Weak and weak-star topologies on its dual coincide, so the closed and dense
range is the whole dual. If \(p>1/2\), apply that case to \(1-\alpha\).
It identifies \(\mathscr H(\alpha)\) with the dual of its reflexive
complementary space. Taking the bidual and using cyclicity identifies the
dual of \(\mathscr H(\alpha)\) with \(\mathscr H(1-\alpha)\) in the
orientation (HL.56). Hence

\[
 \boxed{\mathscr H(1-\alpha)\cong\mathscr H(\alpha)^*,\qquad
 T\longmapsto[S\mapsto\int ST]}
\tag{HL.60}
\]

is an onto isometry. At the endpoints, \(\mathscr H(1)\cong M_*\) and
\(M=\mathscr H(0)\cong M_*^*\). The full dual of \(M\) need not consist of
normal functionals. The convexity/reflexivity source comparison is Hiai,
section 11, Propositions 11.24--11.26 and Theorem 11.28; the proof here
supplies the convexity and finite-coordinate reflexivity steps.

## Core commutants recover centralizers

Fix a faithful normal semifinite weight \(\varphi\), and put

\[
 \begin{aligned}
 \mathcal A^\varphi&=\pi^\varphi(A),\\
 \mathcal D^\varphi&=Z(\widetilde M)\vee\mathcal A^\varphi.
 \end{aligned}
\tag{HL.61}
\]

The algebra \(\mathcal A^\varphi\) is generated by
\(h_\varphi^{it}=\varphi^{it}\). The modular implementation supplied by the
core density contract is

\[
 \begin{aligned}
 \sigma_t^\varphi(x)&=h_\varphi^{it}x h_\varphi^{-it},\\
 x&\in M.
 \end{aligned}
\tag{HL.62}
\]

Hence

\[
 \begin{gathered}
 x\in(\mathcal A^\varphi)'\cap M\\
 \Updownarrow\\
 xh_\varphi^{it}=h_\varphi^{it}x\quad(t\in\mathbb R)\\
 \Updownarrow\\
 \sigma_t^\varphi(x)=x\quad(t\in\mathbb R)\\
 \Updownarrow\\
 x\in M_\varphi .
 \end{gathered}
\tag{HL.63}
\]

This proves

\[
 (\mathcal A^\varphi)'\cap M=M_\varphi.
\tag{HL.64}
\]

The two algebras joined in (HL.61) commute, so
\(\mathcal D^\varphi\) is abelian. If
\(x\in\mathcal D^\varphi\cap M\), then \(x\) commutes with
\(\mathcal A^\varphi\), and (HL.64) places \(x\) in \(M_\varphi\). Every
\(y\in M_\varphi\) commutes with \(\mathcal A^\varphi\) by (HL.64), and it
commutes with \(Z(\widetilde M)\) by definition. It therefore commutes with all
of \(\mathcal D^\varphi\), in particular with \(x\). Thus

\[
 \boxed{
 \mathcal D^\varphi\cap M\subseteq Z(M_\varphi).}
\tag{HL.65}
\]

The source states an inclusion in (HL.65), and the proof preserves that exact
strength. Equality needs additional information about the core center and is
not inferred here.

HL-01--11 cover the eleven XII.6 callback groups. Equations (HL.17) and
(HL.51) retain the source corrections. The author arguments include the
actual measure-to-predual holomorphy bridge, inner-strip bound, norming
triangle inequality, finite-product trace-cap limit, graded cyclicity,
Clarkson convexity and onto dual representation. Their core and density
inputs, the prerequisites behind them and the supplementary state-dependent
interpolation identification are not proved here.