Compatible modular weights and the full positive cone

A modular flow can signal the existence of an operator-valued weight, but equality on a finite GNS ideal does not determine what happens at infinite scalar values. The construction here descends a trace-case map through two compatible crossed products. A single positive spectral density and an onto extended-positive average make the descent work on every positive input. Inner products are linear in the first variable.

The reconstruction statement and its exact inputs

Let \(N\subset M\) be a unital inclusion of von Neumann algebras, with no countability assumption. Let \(\varphi\) on \(N\) and \(\psi\) on \(M\) be faithful normal semifinite weights. Suppose

Here \(n\in N\) and \(t\in\mathbb R\). The target is a unique faithful normal semifinite operator-valued weight \(E:M_+\to\widehat N_+\) with

The equality is in \([0,\infty]\), so (OE.2) includes positive operators of infinite \(\psi\)-weight. The converse says that an existing faithful normal semifinite \(E\), composed with any faithful normal semifinite \(\varphi\), produces a compatible pair.

Four exact external contracts remain visible. Scalar harmonic facts supply vector-valued real-line Plancherel, compact approximate identities, and the Weyl commutant. Forward modular restriction is used only for an already existing faithful normal semifinite operator-valued weight. The all-normal trace representation includes weights with an infinite part. Positive Tonelli governs the normal extensions of the dual orbit integrals. The catalog records these as open prerequisites. None of these providers is the theorem (OE.2) or the false all-positive reverse implication of IX.4.20.

A concrete common crossed-product pair

Represent \(M\) faithfully and normally on the GNS space of \(\psi\). Put \(U_t=\Delta_\psi^{it}\), so \(U_t mU_t^*=\sigma_t^\psi(m)\). On \(\mathcal H=L^2(\mathbb R,H_\psi)\), define

Equation (OE.1) makes this the same regular action for \(N\). Set

The unitary field \((V\xi)(r)=U_r\xi(r)\) conjugates \(\pi(m)\) to the constant operator \(m\otimes1\); in particular both representations in (OE.4) are faithful and normal. Conjugation by \(D_s\) defines the shared dual action \(\theta_s\), with \(\theta_s(\pi(m))=\pi(m)\) and \(\theta_s(\lambda(t))=e^{-ist}\lambda(t)\).

Here is the fixed-point check that the descent needs. The operators \((R_t\xi)(r)=\xi(r+t)\) and \((V_t\xi)(r)=U_t\xi(r+t)\) are related by \(VV_tV^*=1\otimes R_t\). Each \(V_t\) commutes with \(P\). If \(X\in P\) is fixed by every \(\theta_s\), then \(VXV^*\) commutes with all multiplication phases \(1\otimes D_s\) and all translations \(1\otimes R_t\). Their Weyl commutant is scalar on the second tensor factor, so \(VXV^*=a\otimes1\). It also commutes with the conjugates of constant \(M'\)-operators. At the continuous fiber \(r=0\) this gives \(a\in M''=M\). Thus \(P^\theta=\pi(M)\), and the same argument gives \(Q^\theta=\pi(N)\). Applying it to the spectral projections and infinite-part projection of an extended positive element gives the corresponding fixed-point identities in \(\widehat P_+\) and \(\widehat Q_+\).

Dual averages and one normalized trace density

For \(X\in P_+\), integrate its positive dual orbit:

The integral lies in \(\widehat M_+\). The integral is the increasing supremum of compact-interval integrals, evaluated on positive normal functionals. The fixed-point result of OE-02 places it in the displayed cone. Interchanging two positive suprema proves normality; invariance of the orbit integral proves \(M\)-bimodularity. A continuous nonnegative orbit with zero integral vanishes at zero, so \(T_P\) is faithful. The same construction gives \(T_Q:Q_+\to\widehat N_+\).

For \(f\in C_c(\mathbb R)\), vector-valued Plancherel with the Lebesgue convention in (OE.5) gives

The coefficient vectors \(\lambda(f)m\) span an ultraweakly dense subspace of \(P\). Equation (OE.6), bimodularity, and finite \(\psi\)-cutdowns show that \(T_P\) and \(W_P=\widehat\psi\circ T_P\) are semifinite. The same holds for \(T_Q\) and \(W_Q=\widehat\varphi\circ T_Q\). Their normality and faithfulness follow directly from the orbit construction and the scalar weights. This reasoning works with arbitrary Hilbert-space multiplicity: a single compactly supported vector orbit has a separable closed span, where scalar Plancherel applies coordinatewise.

The forward modular restriction contract, applied only to the existing \(T_P\) and \(T_Q\), identifies the modular actions of \(W_P,W_Q\) on \(M,N\). The dual action commutes with \(\operatorname{Ad}\lambda(t)\): its scalar phase on \(\lambda(t)\) cancels under conjugation. Moving this conjugation through the positive orbit integral, and restricting it to the fixed algebra, gives

Here the hat denotes the action on the extended positive cone of \(M\). It is covariance, and need not be invariance. Modular invariance of \(\psi\), also on its extended cone, now gives

The same two steps hold for \(Q\), with \(\varphi\) in place of \(\psi\). Thus it is the scalar weights \(W_P,W_Q\) that are invariant under each inner conjugation. Their unitary centralizer criterion puts every \(\lambda(t)\) in both scalar centralizers. Generation of \(P,Q\) therefore yields \(\sigma_t^{W_P}=\operatorname{Ad}\lambda(t)\) and the analogous formula for \(W_Q\). Inverse centralizer-density perturbation then constructs uniquely normalized faithful normal semifinite traces \(\tau_P,\tau_Q\) such that

Both traces obey \(\tau\circ\theta_s=e^{-s}\tau\). To see the sign, let \(h\) be the positive injective affiliated operator with \(h^{it}=\lambda(t)\). The dual action has \(\theta_s(h)=e^{-s}h\). Perturbing the dual-invariant \(W\) by \(h^{-1}\) gives the trace and the displayed scaling. A direct Fourier check for \(M=\mathbb C\) gives \(h(p)=e^{-p}\) and \(\tau(F)=\int e^pF(p)\,dp\), whose translated weight scales by \(e^{-s}\). The normalization in (OE.7) is stronger than mere equality of modular automorphism groups.

Tracial induction without a semifinite restriction

For arbitrary semifinite \(Q\subset P\) with faithful normal semifinite traces \(\tau_Q,\tau_P\), define for \(X\in P_+\) a normal weight on \(Q\) by

Here \(y\in Q_+\). This weight can be nonsemifinite even when both traces are semifinite. For example, the usual trace of \(1\) on \(B(\ell^2)\) is infinite when restricted to \(\mathbb C1\). The all-normal trace representation therefore supplies a unique \(S(X)\in\widehat Q_+\) with

This holds for every \(y\in Q_+\). The dot denotes extended-positive trace pairing; it is not an ordinary product with a possibly unbounded operator.

Uniqueness in (OE.9) transfers additivity and homogeneity in \(X\) to \(S\). Normality follows by swapping the supremum of an increasing positive net with the normal trace pairing. For \(a\in Q\), cyclicity in the pairing shows \(\chi_{a^*Xa}(y)=\chi_X(aya^*)\); hence \(S(a^*Xa)=a^*S(X)a\). Setting \(y=1\) in (OE.9) gives

This identity holds on all \(P_+\). Thus \(S\) is faithful. To verify semifiniteness, start with \(\tau_P(X)<\infty\). The infinite spectral projection of \(S(X)\) vanishes. Its bounded spectral cuts \(q_k\in Q\) increase strongly to \(1\), and \(S(q_kXq_k)=q_kS(X)q_k\in Q_+\). These compressions converge ultraweakly to \(X\). The finite \(\tau_P\)-cone is ultraweakly dense, proving that the bounded-output domain of \(S\) is dense. No increasing-compression assertion is needed.

The same pairing proves uniqueness among trace-preserving normal operator-valued weights. Applied to the \(P,Q\) of OE-02 and their scaling traces, it also gives equivariance:

Indeed \(\theta_{-s}^Q S\theta_s^P\) has the same scalar trace identity (OE.10), because the two trace-scaling factors cancel. Consequently \(S(\pi(M)_+)\subset\widehat N_+\): equivariance fixes every spectral projection of the output. Define \(E=S|_{M_+}\), identifying \(M,N\) with their images under \(\pi\). It is normal and \(N\)-bimodular.

The equality at infinity

The shared \(\lambda(t)\in Q\subset P\) has one spectral generator \(h\), not two unrelated affiliated densities. In both algebras, (OE.7) and the fixed-reference cocycle theorem identify the entire scalar weights as \(W_P=(\tau_P)_h\) and \(W_Q=(\tau_Q)_h\). Let \(h_k=h\wedge k\in Q_+\), and write \(C_k(Z)=h_k^{1/2}Zh_k^{1/2}\) on either algebra's positive cone. Normality of the density perturbations, \(Q\)-bimodularity of \(S\), and (OE.10) give, for every \(X\in P_+\),

Every equality is in \([0,\infty]\). The spectral cuts turn an unbounded density into bounded bimodule sandwiches; they do not require trace measurability or finite scalar values.

Write bars for the normal extensions to extended positive cones. Positive Tonelli applied to (OE.5) and equivariance (OE.11) gives the commutative descent square

For precision, this means equality after evaluation by every positive normal functional on \(Q\): both sides integrate the same nonnegative dual-orbit pairing. The canonical embeddings \(\widehat M_+\hookrightarrow\widehat P_+\) and \(\widehat N_+\hookrightarrow\widehat Q_+\) are the fixed-point spectral-pair embeddings of OE-02.

The public full-range proof for a faithful normal semifinite operator-valued weight says \(\overline T_P(\widehat P_+)=\widehat M_+\). It starts with the ultraweakly dense finite-output ideal, gives each positive output in that ideal a positive bounded-input lift by bimodule factorization, resolves \(1\) by a possibly uncountable family of such outputs, and then lifts spectral increments of an arbitrary extended positive target. That proof cannot be replaced by a countable analytic test on the finite cone.

Take any \(y\in M_+\), including \(\psi(y)=\infty\), and choose \(X\in\widehat P_+\) with \(\overline T_P(X)=y\). Normal extension of (OE.12) and the square (OE.13) then give

This is (OE.2) on the whole positive cone. It uses an onto lift at the infinite values; the finite-GNS reverse calculation in IX.4.20 does not provide one.

Faithfulness, semifiniteness, and direct uniqueness

Equation (OE.14) and faithfulness of \(\psi\) make \(E\) faithful. For semifiniteness, take \(y\in M_+\) with \(\psi(y)<\infty\). Then \(E(y)\) has no infinite spectral part. If \(q_k=1_{[0,k]}(E(y))\), the bimodule rule gives \(E(q_kyq_k)=q_kE(y)q_k\in N_+\), and \(q_kyq_k\to y\) ultraweakly. The finite \(\psi\)-cone itself is ultraweakly dense: finite-weight positive contractions \(e_i\uparrow1\) strongly give \(e_iye_i\to y\) and \(\psi(e_iye_i)\leq\|y\|\psi(e_i^2)<\infty\). Hence the bounded-output domain of \(E\) is ultraweakly dense. The compressions are not asserted to increase.

Suppose \(E_1,E_2\) both satisfy (OE.2) for the same \(\varphi,\psi\). For \(c\in\mathfrak n_\varphi\), their bimodule identities imply

These are the extended quadratic forms of \(E_1(y)\) and \(E_2(y)\) on the dense left \(N\)-module \(\Lambda_\varphi(\mathfrak n_\varphi)\). To see why this domain suffices, take the finite-energy spectral projections of either form. Their bounded cuts send this module into its finite form domain, where form norm is bounded by a multiple of Hilbert norm. The cuts approximate every finite-energy vector in form norm, so the finite part of this module is a form core. Equality in (OE.15), including infinite values, gives the same core and the same closed form. Extended-positive spectral-pair uniqueness therefore gives \(E_1(y)=E_2(y)\) for every \(y\).

For the converse direction, start with an existing faithful normal semifinite \(E\), choose faithful normal semifinite \(\varphi\), and put \(\psi=\widehat\varphi\circ E\). Scalar cutdowns show \(\psi\) is normal faithful semifinite. The forward modular restriction theorem for this existing \(E\) gives (OE.1). It does not use compatible-pair reconstruction.

Dependency and source boundary

The theorem proved by the chain OE-02–06 is relative to its named providers. The concrete averages, trace case, common-density argument, full-range lift, and form-core uniqueness are displayed because each controls a different possible loss of an infinite value. The all-normal trace representation is stronger than the semifinite-density statement in PT-08. The public OVW-03 full-range property is stronger than density of the bounded-output ideal. The positive Tonelli identity is an identity of extended maps, not only of ordinary bounded operators. OE-01 is proved relative to those contracts; not every foundation of those providers is proved in these lessons.

The target statement is Takesaki, Theory of Operator Algebras II, IX.4 Theorem 4.18, printed p. 223/PDF p. 243; its printed proof continues on pp. 225–227/PDF pp. 245–247. This course route uses the trace construction of Haagerup, Operator Valued Weights in von Neumann Algebras I, Theorems 1.12, 2.7 and 4.7(1), and II, Theorem 5.1/Lemma 5.2. IX.4.20 has a false all-positive reverse implication, corrected in What the analytic strong sum actually proves through Analytic rank-one operators with the wrong infinite value. Nothing in OE-05 invokes that reverse. The argument does not import the converse modular theorem that would presuppose OE-01.