Homogeneous operators in the continuous core

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.

Read the general core and density construction, LP-01--04 first. It proves the fixed algebra, normalized scaling trace, supported density correspondence, localizer pairing and coherent change of reference for arbitrary von Neumann algebras. We identify its supported symbols and Lebesgue spectral representations explicitly below. The orbit integral uses \(ds\), with the same trace normalization as LP-01. The analytic, Banach, product and duality arguments then use the named measurable-operator, trace, complex-analysis and Hahn--Banach proofs. HL-12 proves the supplementary state-dependent interpolation identification using the reflexivity already established in HL-10.

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 MT-09.

Equivariant copies of the translation algebra are weights

The core and its supported symbols. Choose a faithful normal semifinite reference \(\chi\). By LP-01, the regular core has a faithful normal semifinite orbit weight \(E\), fixed algebra \(M\), and trace \(\tau\) satisfying (HL.1). LP-04 constructs the normal trace-preserving equivariant change from a \(\psi\)-chart to a \(\chi\)-chart. It fixes \(M\) and sends the translation unitary to \([D\psi:D\chi]_t\lambda_t^\chi\). Its proof uses a multiplication unitary in the regular representation and verifies the ordered cocycle law; it proves equality of the actual traces, not just proportionality.

One may therefore regard the core as pairs \((\chi,A)\) modulo these chart maps. The identity and composition laws in LP-04 make this an equivalence relation. Move finitely many representatives to one chart to add, multiply or take adjoints. Those operations, the norm and the ultraweak topology are independent of chart because every transition is a normal isomorphism with normal inverse. A single chart identifies the resulting algebra with its complete von Neumann algebra. The fixed copy of \(M\), \(\theta\) and \(\tau\) agree in all charts.

This construction is functorial for normal isomorphisms. If \(\rho:M\to N\) is one, put \(\chi'=\chi\circ\rho^{-1}\). The map \(\Lambda_\chi(x)\mapsto\Lambda_{\chi'}(\rho(x))\) is an isometry on the GNS domain, onto a dense domain; it extends to a unitary. It intertwines the closed Tomita operators since it intertwines their defining involutions and their graph closures. Hence it intertwines the modular groups. Applying it pointwise in the regular Hilbert space gives the normal core isomorphism \[ \pi_\chi(x)\longmapsto\pi_{\chi'}(\rho(x)),\qquad \lambda_t^\chi\longmapsto\lambda_t^{\chi'}. \] It intertwines \(\theta\) and its positive integral \(E\), so it transports the dual weight \(\chi\circ E\) to \(\chi'\circ E\). It also transports the generator of \(\lambda\); the trace construction in LP-01 therefore proves trace preservation. Naturality of the balanced matrix cocycle, proved in SI-14, makes these maps commute with every chart transition. Composition and identities hold on the displayed generators and hence on the whole core. Here functoriality concerns isomorphisms; no core map for an arbitrary inclusion is being asserted.

For a normal semifinite weight \(\varphi\), including a nonfaithful one, LP-02 constructs the unique positive affiliated operator \(h_\varphi\) with \[ \varphi\circ E=\tau_{h_\varphi},\qquad \theta_s(h_\varphi)=e^{-s}h_\varphi, \qquad s(h_\varphi)=s(\varphi). \] The theorem proves the converse as well. Its proof descends the fixed relative cocycle by CX-10, lifts it by OR-03, and handles nonfaithful supports by the complementary-corner theorem; a citation is not being used in place of that earlier written argument.

The identification with the weight symbol requires a support check. Let \(e=s(\varphi)\). Complete \(\varphi|_{eMe}\) by a faithful normal semifinite weight on \((1-e)M(1-e)\), using CW-06, and call the diagonal faithful weight \(\bar\varphi\). In its chart, \(h_{\bar\varphi}^{it}=\lambda_t^{\bar\varphi}\). Since \(e\) is in the centralizer of \(\bar\varphi\), it is a constant reducing projection for both the regular coefficient action and these translations. Bimodularity of \(E\) gives, on every positive \(A\), \[ \varphi(E(A)) =\bar\varphi(E(eAe)) =\tau_{h_{\bar\varphi}}(eAe) =\tau_{e h_{\bar\varphi}}(A). \] The last equality follows with bounded spectral regularizations, which commute with \(e\), and then normal increasing evaluation. Density uniqueness in PT-08 proves \(h_\varphi=e h_{\bar\varphi}\), extended by zero on the complement. Thus its supported imaginary powers are \(\varphi^{it}:=e\lambda_t^{\bar\varphi}\); they have initial and final projection \(e\), including \(\varphi^{i0}=e\). Density uniqueness and LP-04 show that the symbol is independent of the completion and chart. Regular covariance and modular restriction to the centralizer corner give \[ h_\varphi^{it}x h_\varphi^{-it}=\sigma_t^\varphi(x) \quad(x\in eMe). \] The zero weight gives zero density and zero supported symbols.

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 construction above 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

Here is a proof that (HL.5) is defined on Lebesgue classes and is faithful. On the nonzero support put \(Q=-\log h_\varphi\), with spectral measure \(P\). Grade one gives \(\theta_s(Q)=Q+s\), so \(\theta_s(P(B))=P(B-s)\) for every Borel set \(B\). For a positive normal functional \(\nu\) on the corner, scalar Tonelli gives \[ \int_{\mathbb R}\nu(P(B-s))\,ds=|B|\nu(e). \] Indeed the inner indicator is \(1_B(r+s)\), and its integral in \(s\) is the Lebesgue measure \(|B|\), independently of \(r\). The finite scalar spectral measure has total mass \(\nu(e)\). If \(|B|=0\), the left integrand is a continuous nonnegative function of \(s\), since the dual action is spatially strongly continuous. It vanishes at zero. Every positive normal \(\nu\) therefore annihilates \(P(B)\), so \(P(B)=0\). Conversely, if \(P(B)=0\), all its translates are zero. Choosing \(\nu(e)>0\) in the same equality forces \(|B|=0\). Thus the spectral null sets are exactly the Lebesgue null sets. Spectral calculus consequently gives the faithful normal map (HL.5), onto its range. The unitary-generator theorem PT-03 also shows that this spectral algebra is generated by its imaginary powers. 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 \(E\) of LP-01. A normalized localizer satisfies \(E(a)=1\). Existence has an explicit bounded witness: if \(d\) is the injective reference density in LP-01, put \(a_0=d^{-1}1_{(1,\infty)}(d)\). For each scalar \(\ell>0\), \(\int_{s<\log\ell}e^s\ell^{-1}\,ds=1\); spectral calculus and positive normal evaluation give \(E(a_0)=1\).

If \(h=h_\varphi\) and \(x\in M_+\), write \(b_x=x^{1/2}a x^{1/2}\) and \(c_x=x^{1/2}h x^{1/2}\). Since \(E\) is bimodular, \(E(b_x)=x\). The equality \(\varphi\circ E=\tau_h\) proved in LP-02 therefore gives the first line below on the full positive cone:

Both lines mean the normal trace pairing of the corresponding extended positive form and may be infinite; its finite domain need not be dense. 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 PT-08. 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 \(f_0\in A_+\). The all-normal trace correspondence CW-11, applied to Lebesgue integration on \(A\), represents \(\mu\) by an extended measurable density \(g\), including its possible infinite part:

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. The \(L^1\) translation estimate of BI-03 makes these convolutions converge weak-star to each bounded truncation. Constants are weak-star closed, so each truncation is constant; their increasing limit 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 MT-11 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 support-preserving bijection LP-02 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 positive decomposition argument in LP-03 now extends \(D\) to a well-defined complex linear map on \(M_*\): it expresses every normal functional as four positive ones using the summable vector coefficients of CP04--06, and proves independence of that expression by positive additivity. 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 \(h_\omega\). Density uniqueness in LP-02 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.

Continuity on the closed strip. The same family is continuous in the predual norm even where a real exponent becomes zero. Here is a measure-topology argument that also covers a nonfaithful density. Near \(z=it_0\), restrict to \(0\leq\Re z\leq1/2\). Write \(e_R=1_{(0,R]}(h)\) and cut the second factor into its spectral regions \([0,\eta)\), \([\eta,R]\) and \((R,\infty)\), where \(0<\eta<1<R\). Omitting the high part of the first or second density changes the product by operators with left or right trace support at most \(\tau(1_{(R,\infty)}(h))\) or \(\tau(1_{(R,\infty)}(k))\). These tend to zero as \(R\to\infty\). After the first cut, the low part of the second factor has operator norm at most \[ \max(1,R^{1/2})\eta^{1/2}. \] On the remaining region, \(e_Rh^z\) is uniformly bounded and strongly continuous up to \(z=it_0\), by spectral dominated convergence on the support of \(h\). The operator \(1_{[\eta,R]}(k)k^{1-z}\) is norm continuous and has a fixed finite-trace support \(q=1_{[\eta,R]}(k)\).

If bounded operators \(B_n=B_nq\) converge strongly to zero and are uniformly bounded, normality of the finite functional \(a\mapsto\tau(qaq)\) gives \(\tau(B_n^*B_n)\to0\). The spectral estimate \(\tau(1_{(\delta,\infty)}(|B_n|))\leq\delta^{-2}\tau(B_n^*B_n)\) then gives convergence in measure. Apply this to the middle-region differences, first keeping \(R,\eta\) fixed, then letting \(\eta\downarrow0\) and \(R\to\infty\). This proves measure continuity at the left boundary. Apply the same argument to \(F(z)^*=k^{1-\bar z}h^{\bar z}\) at the right boundary. The differences of the original products have grade one; HL-06 therefore upgrades their measure convergence to predual norm convergence. The auxiliary cuts do not claim to preserve grades, and no interpolation-space convergence has been inferred from these cuts.

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.

The state-dependent compatible couple. 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 map \(T\mapsto F_{T d^{1-p}}\) and its onto isometry are proved in HL-12, after the independent duality proof of HL-10. The Banach argument above does not use that later interpolation theorem. 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

The norming, finite-coordinate density and closed-range separation steps below use NP1 and NP2 in their full normed-space and locally convex forms. They do not use Banach–Alaoglu or Krein–Smulian.

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 supported-symbol calculation in HL-01, here with full support, proves the modular implementation

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.

The state-dependent interpolation isometry

Fix a faithful normal state \(\chi\) and let \(d=h_\chi\). We identify \(\mathscr H(1)\) with \(M_*\) by HL-06. Thus \[ X_0=Md,\qquad \|xd\|_{X_0}=\|x\|_\infty, \qquad X_1=\mathscr H(1)=M_*. \tag{HL.66} \] Indeed, \(F_{xd}(y)=\chi(yx)=J_\chi(x)(y)\). Since \(\int d=1\), the inclusion \(X_0\subset X_1\) is contractive. Its injectivity also follows from faithfulness, by testing at \(y=x^*\). The sum space of CI-01 is exactly \(X_1\), with its predual norm.

Theorem. For \(0<p<1\), the map \[ I_p:\mathscr H(p)\longrightarrow[X_0,X_1]_p, \qquad I_p(T)=T d^{1-p} \tag{HL.67} \] is an onto linear isometry. In predual notation the image is \(F_{T d^{1-p}}\). The interpolation space uses the norm-continuous, boundary-decaying class of CI-03, with no change of definition.

The proof uses the Banach and duality results of HL-08--10 already established without this identification. In particular, every real-grade space \(\mathscr H(p)\), \(0<p<1\), is reflexive. We give the boundary approximation and the surjectivity argument explicitly.

The compatible sum, boundary-decaying strip space and complete evaluation quotient are supplied by CI01, CI03 and CI04. Norm extension, norming and convex separation use NP1 and NP2.

The embedded space and its endpoint ball

Hölder's inequality makes \(I_p\) a contraction into \(X_1\). It is injective even though \(d^{p-1}\) need not be measurable. If \(T d^{1-p}=0\), put \(q_n=1_{[1/n,n]}(d)\). The bounded operator \(d^{p-1}q_n\) is measurable, so associativity of closed measurable products gives \(Tq_n=0\). This equality implies \(q_nH\subset\ker T\): approximate a vector in \(q_nH\) using the domain of the unclosed composition, and use closedness of \(T\). Since \(q_n\uparrow1\), the closed kernel is all of \(H\). Hence \(T=0\).

Give \(Y=I_p(\mathscr H(p))\subset X_1\) its transported norm. Then \(Y\) is a reflexive Banach space, continuously embedded in \(X_1\), and \(xd=I_p(xd^p)\), so \(X_0\subset Y\).

We also need the unit ball of \(X_0\) to be closed in \(X_1\). Suppose \(x_n d\to a\) in predual norm and \(\|x_n\|_\infty\leq1\). The normal functionals \[ \eta_y(x)=\chi(yx),\qquad y\in M, \] have norm-dense linear span in \(M_*\). Otherwise Hahn--Banach and \((M_*)^*=M\) would give a nonzero \(x\in M\) with \(\chi(yx)=0\) for every \(y\); taking \(y=x^*\) contradicts faithfulness. For each \(y\), \(\eta_y(x_n)=F_{x_nd}(y)\) converges. Uniform boundedness and this norm density imply that \(\omega(x_n)\) converges for every \(\omega\in M_*\). Its limit is a bounded linear functional on \(M_*\) of norm at most one, hence is evaluation at an \(x\in M\) with \(\|x\|_\infty\leq1\). Testing at every \(y\) shows \(a=xd\). This proves the claimed closedness. Scaled balls are closed as well.

Smoothing an admissible strip inside the smaller endpoint

Write \(k_n(t)=\sqrt{n/\pi}\,e^{-nt^2}\). For \(f\in\mathcal F(X_0,X_1)\), define in \(X_1\) \[ g_n(z)=\int_{\mathbb R}k_n(t)f(z+it)\,dt. \tag{HL.68} \] These integrals exist by approximation with step functions and an integrable norm bound. On the two boundaries, convolution is also an integral in \(X_j\), and

It preserves continuity and vanishing at infinity. Interior holomorphy follows by integrating the Cauchy formula on a disk and interchanging integrals under the Gaussian bound. Thus \(g_n\in\mathcal F\).

The two boundary functions of \(f\) are uniformly continuous, since they belong to \(C_0(\mathbb R;X_j)\). Splitting the convolution into \(|t|<\delta\) and its complement proves \(\|g_n-f\|_{\mathcal F}\to0\).

More is true: \(g_n(z)=x_n(z)d\), where \(x_n\) is an entire operator-norm-holomorphic \(M\)-valued function bounded on the closed strip. To see this, contour shifting in \(X_1\) gives \[ g_n(z)=\sqrt{n/\pi} \int_{\mathbb R}e^{n(z-it)^2}f(it)\,dt. \tag{HL.69} \] For completeness, apply the scalar Cauchy theorem to \(e^{n(z-\zeta)^2}\ell(f(\zeta))\) between the vertical lines \(0\) and \(\Re z\), first using an inner line and then continuity to reach the boundary. On the horizontal edges the Gaussian tends uniformly to zero, while \(f\) is bounded in \(X_1\). Let their heights tend to infinity. All \(\ell\in X_1^*\) give the identity, hence so does Hahn--Banach. The right side of (HL.69) is an \(X_0\)-valued integral. Its kernels and their complex derivatives have integrable bounds on compact sets. It is therefore entire in \(X_0\), and \[ \|x_n(z)\|_\infty \leq e^{n(\Re z)^2}\sup_t\|f(it)\|_{X_0}. \] This proves the assertion and supplies the specific analytic approximation needed below.

The lower norm estimate

Consider any \(g\in\mathcal F\) of the form \(g(z)=x(z)d\), with \(x\) operator-norm holomorphic and bounded on the closed strip and continuous there. To estimate \(\|x(p)d^p\|_p\), take a unit vector of the complementary grade, written \(R=u h^{1-p}\), where \(\int h=1\) and \(u\in M\) is its polar partial isometry. Define \[ H(z)=\int u h^z d^{1-z}x(1-\bar z)^*. \tag{HL.70} \] HL-07, including its closed-strip continuity proof, and bounded module multiplication make \(H\) bounded and continuous on the closed strip and holomorphic inside it. The reflected adjoint of \(x\) is holomorphic. At the left boundary \(b_t=u h^{it}d^{-it}\) is a contraction in \(M\), so \[ |H(it)|=\left|\int b_t d x(1+it)^*\right| \leq\|x(1+it)d\|_1. \] At the right boundary, (HL.44) gives \[ |H(1+it)|\leq\|x(it)\|_\infty. \] Three lines, evaluated at \(1-p\), therefore bounds \(\left|\int R(x(p)d^p)^*\right|\) by \(\|g\|_{\mathcal F}\). The norming pairing of HL-08--10, and invariance of the norm under adjoints, give \[ \|g(p)\|_Y=\|x(p)d^p\|_p\leq\|g\|_{\mathcal F}. \tag{HL.71} \]

Apply this estimate to the smoothings \(g_n\) above and to \(g_n-g_m\). Their values at \(p\) are Cauchy in \(Y\). Completeness of \(Y\), its continuous injection into \(X_1\), and \(g_n(p)\to f(p)\) in \(X_1\) identify the limit with \(f(p)\). Consequently \[ [X_0,X_1]_p\subset Y, \qquad \|a\|_Y\leq\|a\|_{[X_0,X_1]_p}. \tag{HL.72} \] The actual representatives \(x_n(p)d^p\) converge in their grade-\(p\) norm. Hölder gives convergence of their products with \(d^{1-p}\) in grade-one norm, consistent with closed measurable multiplication.

Admissible representatives and the reverse inclusion

Let \(T=u h^p\ne0\), put \(m=\int h>0\), and set \[ A_T(z)=m^{p-z}u h^z d^{1-z}. \tag{HL.73} \] Then \(A_T(p)=I_p(T)\). It is bounded and predual-norm continuous on the closed strip and holomorphic inside it. Its lower boundary belongs to \(X_0\), and its endpoint bounds are exactly \[ \sup_t\|A_T(it)\|_{X_0}\leq m^p, \qquad \sup_t\|A_T(1+it)\|_1\leq m^p=\|T\|_p. \] The lower boundary need not be operator-norm continuous. We therefore do not assert \(A_T\in\mathcal F\).

Convolve \(A_T\) with \(k_n\) as in (HL.68), calling the result \(B_n\), and put \[ a_n(z)=e^{(z^2-p^2-1)/n}B_n(z). \] The lower-boundary integral belongs to \(X_0\) with norm at most \(m^p\): its approximating finite weighted averages lie in that ball, which we proved closed in \(X_1\). The same observation for integrals against a complex kernel gives the bound by \(m^p\) times its \(L^1\)-norm. In particular, translation of the Gaussian kernel proves continuity of \(B_n(it)\) in \(X_0\). The other boundary is norm continuous by the same argument in \(X_1\). The exponential factor supplies boundary decay, has modulus at most one on the strip, and gives \(a_n(p)=e^{-1/n}B_n(p)\). Interior holomorphy and sum-norm continuity follow as for (HL.68). Thus \[ a_n\in\mathcal F,\qquad \|a_n\|_{\mathcal F}\leq\|T\|_p,\qquad a_n(p)\longrightarrow I_p(T)\quad\hbox{in }X_1. \]

To turn this into the required interpolation conclusion we use reflexivity, not a claim that the interpolation ball is closed in the predual. Let \(j:Y\hookrightarrow X_1\) be the inclusion. The subspace \(j^*(X_1^*)\) is norm dense in \(Y^*\). Indeed, if its norm closure were proper, Hahn--Banach would give a nonzero element of \(Y^{**}\) annihilating it. Reflexivity writes that element as evaluation at \(y\in Y\); then every \(\ell\in X_1^*\) annihilates \(j(y)\). This contradicts injectivity of \(j\).

By (HL.72), the sequence \(a_n(p)\) is bounded in \(Y\) by \(\|T\|_p\). Its convergence in \(X_1\) gives convergence against \(j^*(X_1^*)\). The density just proved and the uniform \(Y\)-bound give weak convergence in \(Y\) to \(I_p(T)\). Hahn--Banach separation of a closed convex set now shows that finite convex combinations of any tail approximate \(I_p(T)\) in \(Y\)-norm. Each such combination still has interpolation norm at most \(\|T\|_p\). Scaling and the zero vector show that, for every \(y\in Y\) and \(\varepsilon>0\), there is a \(c\in[X_0,X_1]_p\) with \[ \|c\|_{[X_0,X_1]_p}\leq\|y\|_Y, \qquad \|y-c\|_Y<\varepsilon. \]

Fix \(0<q<1\). Apply this approximation successively to the remaining error, making the error after step \(k\) at most \(q^k\|y\|_Y\). The resulting series of interpolation elements has total interpolation norm at most \(\|y\|_Y/(1-q)\). Completeness from CI-04 gives its sum, and the continuous injection into \(Y\) identifies the sum with \(y\). Letting \(q\downarrow0\) proves the reverse of (HL.72). This proves the onto isometry (HL.67).

One may also obtain actual strip representatives from the series: choose a representative of each term with a summable, arbitrarily small excess over its interpolation norm and sum in the complete strip space of CI-03. Their values have grade one, their inverse images have grade \(p\), and Hölder controls the closed products throughout. This establishes admissible representatives with boundary norm arbitrarily close to \(\|T\|_p\); it does not assert that the infimum defining the norm is attained.

Products with several varying grades

The analytic product requirement can be stated with its exact grades. Take positive grade-one densities \(h_1,\ldots,h_N\) in \(\mathscr H(1)_+\), contractions \(u_1,\ldots,u_N\in M\), nonnegative numbers \(a_j,b_j\) with \(\sum_j a_j=\sum_j b_j=1\), and real \(q_j\) with \(\sum_jq_j=0\). Put \[ \alpha_j(z)=a_j+(b_j-a_j)z+iq_j, \qquad P(z)=\prod_{j=1}^N u_jh_j^{\alpha_j(z)}. \] Supported imaginary powers are used at zero real part. For every point of the closed strip the product has grade one, and HL-09 gives the exact bound \[ \|P(s+it)\|_1 \leq\prod_{j=1}^N \left(\int h_j\right)^{(1-s)a_j+sb_j}. \] Zero inputs are treated separately; a zero density makes its supported factor zero. In the open strip, each nonconstant exponent has positive real part, so the measure derivative argument of HL-07 applies factor by factor. An exponent with \(a_j=b_j=0\) is constant and its factor is bounded. The product rule in measure and the constant total grade make \(P\) predual-norm holomorphic.

For boundary continuity, cut off the high spectral tail of each density. The differences have trace rank bounded by the sum of those tail traces, also after multiplication by the other measurable factors. At each boundary some exponent has positive real part. Keep one such factor in a finite spectral band \([\eta,R]\); its omitted low part is small in operator norm, uniformly near that boundary after all the high cuts. The remaining bounded factors are strongly and strongly-adjoint continuous. The middle product has the form \(C(z)qD(z)\) with \(\tau(q)<\infty\). For uniformly bounded strongly-adjoint-continuous \(C,D\), its differences tend to zero in tracial \(L^2\): apply normality of \(\tau(q\,\cdot\,q)\) to \((C(z)-C(z_0))q\) and \(q(D(z)-D(z_0))\), and use the bounded multiplier estimate. Hence they tend to zero in measure. Remove the auxiliary cuts and use grade one to obtain predual-norm continuity. This proves the stated multilinear analytic families and their endpoint bounds without changing grades in the approximating strip construction above.

Source comparison. Kosaki, Applications of the complex interpolation method to a von Neumann algebra: non-commutative \(L^p\)-spaces (1984), Theorem 9.1 and Remark 9.2, printed pp. 57--59, give this identification as the left-embedding case, with the source's \(L^r\) exponent \(r=1/p\). The smoothing/reflexivity mechanism is compared with Theorem 1.5, Lemma 1.6 and Theorem 1.8, printed pp. 33--36. The proof above supplies the contour smoothing, lower estimate, closed endpoint ball and convergent series argument here; these source references replace none of those steps.

HL-01--11 cover the eleven XII.6 callback groups. Equations (HL.17) and (HL.51) retain the source corrections. The 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. The core and density inputs are the written LP01--04 proofs, with the supported-symbol and spectral-class arguments in HL-01 and the localizer evaluation in HL-02. HL-12 gives the state-dependent interpolation isometry, its admissible strip construction and finite-product analytic families, with exact source comparison to Kosaki's Theorem 9.1.