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

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

and define

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

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

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

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

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

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

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

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

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

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

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

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

Changing variables in (HL.14) yields

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

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

Grade one and trace scaling give the exact distribution law

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

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

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

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

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

Let

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

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

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

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

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

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

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

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

In particular

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

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

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

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:

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

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

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

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

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

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

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

We establish holomorphy of the actual product

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 \(\varepsilon>0\),

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:

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

HL-08--09 give

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

Its spectral tails give

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

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

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

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

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

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

Hence

This proves

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

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.