Modular theory through weights, time and reconstruction

A von Neumann algebra can carry integration without a chosen trace. This course asks how a weight produces a Hilbert space, how multiplication and involution produce modular time, and how comparison recovers information about weights, representations and subalgebras.

The intended reader knows C-star algebras, von Neumann algebra preduals, Hilbert-space operators, spectral measures and weak topologies. Finite calculations provide entry points. General results retain their stated net, support and closed-domain hypotheses; measurable-field results state their own separability and measure-space assumptions.

Choose a question and follow its proof homes

The modules below are questions with linked proof homes. They are branches rather than a demand to read every file in the displayed order. Follow a proof's named inputs before using it. In particular, use the closed-form branch before an unbounded comparison; use the Hilbert-algebra and field foundations before the measurable-core arguments; and use standard form or the spatial construction before the relative results that require them.

A first pass can follow weight representations, closed involution, modular time and the matrix density problem. A field-focused pass can follow Hilbert fields, closed graph assembly, countable algebra cores, the fullness boundary, Tomita integration and inverse disintegration. Both passes return to the same complete proof homes below.

Choose an observation and its representation

Start with the question this guide develops: a weight supplies integration, its multiplication and involution supply a modular clock, and comparison asks which features survive a change of weight or representation.

This guide gives the routes and shared notation.

The full guide is the home of this opening question.

Construct the Hilbert space of a weight

Begin with the semicyclic GNS construction and its null quotient. At a dominated-form comparison, use the form-domain tools in the next branch first; the comparison operator lives in the GNS commutant, while the weight lives on the algebra.

A finite positive value, a finite-star GNS vector and a finite linear weight domain must remain distinct.

Full treatments:

Control an unbounded operator by its closed graph

Use closed positive forms, form domains and cores when an unbounded operator first appears. The Banach holomorphy and interpolation tools are used where analytic continuation or strip estimates need them.

Specify the ambient Hilbert space, both graph coordinates, and the domain of every closed product.

Full treatments:

Recover modular time from multiplication and involution

Follow left multiplication and the closable involution through the right algebra and bounded approximation to the full modular theorem. The closed Tomita graph supplies the polar factors; analytic regularization then gives the domains for weight identities and KMS.

An elementary modular eigenvalue calculation is a preview. The commutant identity and full domain theorems require their own proofs.

Full treatments:

Remove dependence on a cyclic vector

Construct the natural cone and standard form, then compare corners and representations. Canonical implementations remove the choice of cyclic vector.

Order of cone vectors and order of the associated normal functionals are separate assertions.

Full treatments:

Predict what changes when the density changes

Start with the worked matrix calculation in Fixed observations, density weights and modular time. Test fixedness on the correct finite domain, construct density weights from the positive cone, then compute their clock through a closed graph and separately justified spectral corners. Trace and invariant-density branches follow their respective hypotheses.

The matrix frequencies distinguish fixed observations from the clock. A density with a kernel has modular time on its support corner.

Full treatments:

Compare two weights without choosing a common trace

When two weights have no common trace, use spatial form construction and modular identification before the weight-change and cocycle results. Keep the support, converse, half-strip and continuity arguments at the actual comparisons where they are needed.

A spatial derivative, a bounded GNS comparison map and a cocycle have different types.

Full treatments:

Recover a subalgebra by integration and expectation

Ask when integration recovers a subalgebra. Begin with contractive positive maps, then the modular criterion for a weight-preserving expectation. Extended positives and operator-valued weights supply the larger integration problem and its commutant and compatibility arguments.

Semifiniteness of a restricted weight is a hypothesis to check, not a consequence of merely naming the subalgebra.

Full treatments:

Compose systems and compare their Hilbert spaces

Use bounded vectors to compose modules by a relative tensor product. Read the unit and associativity proofs with the reference-weight comparison. The coupling operator is a representation-sensitive application.

Weight comparison must transport the coefficients and their domains, as well as the completed Hilbert space.

Full treatments:

Assemble and disassemble measurable systems

Start with measurable Hilbert fields and central decomposition. In Recovering algebras from measurable fibres, follow the scalar tests, graph assembly, and linked rational-core and multiplier arguments. Identify the global algebra before asking about fullness. Tomita integration is the analytic bridge to inverse central disintegration and its two uniqueness questions. Compatible GNS and weight fields then use their own typed transports.

A measurable graph, an integrable graph norm and essentially bounded multiplication are three tests. Recovering original fibre fullness needs the stated defect hypothesis.

Full treatments:

Pass between C-star observations and normal representations

Pass from C-star observations to normal representations through states, finite-domain constructions and the universal bidual. Use the lower-semicontinuity and extension theorems before disintegrating a C-star weight.

A C-star weight and its normal extension need separate domain and topology statements.

Full treatments:

Read harmonic analysis through the weight construction

Read the convolution Hilbert algebra before its Plancherel and Fourier-algebra consequences. The nonunimodular modular operator is part of the weight construction.

The left regular representation, coefficient functions and scalar integration have different domains and continuity conditions.

Full treatments:

Use one domain and sign convention

Unless a result states otherwise, a von Neumann algebra may be nonseparable and need not be sigma-finite. Normality means preservation of bounded increasing nets in the positive cone, or ultraweak continuity for a bounded linear map when the equivalence has been established. A sequential statement is explicitly sequential.

Write \(\mathfrak n_\varphi=\{x:\varphi(x^*x)<\infty\}\), \(\mathfrak m_\varphi=\operatorname{span}\mathfrak n_\varphi^*\mathfrak n_\varphi\). A semifinite weight has an ultraweakly dense finite domain; a state is a separate, bounded normalization. Positivity and additivity take values in \([0,\infty]\), with \(0\cdot\infty=0\). No expression subtracts two infinite weight values.

We use first-variable-linear inner products, \(\sigma_t^\varphi(x)=\Delta_\varphi^{it}x\Delta_\varphi^{-it}\) in a faithful weight representation, and \([D\psi:D\varphi]_t\) for the change from \(\varphi\) to \(\psi\). All eventual strip formulas must agree with that convention. The weight-preserving automorphism group, the commutant comparison operator, and the Connes cocycle have different types and are not interchangeable notation.

Every unbounded operator is specified by its graph/domain and ambient Hilbert space. Products mean a densely defined composition followed by closure only when density and closability have been proved. Imaginary powers of a nonsingular positive operator are unitary; a nonfaithful weight requires support projections and cannot silently use the faithful formula at zero.

Identify the results each construction must prove

The following are source-neutral contracts for the principal constructions. Only the contracts with linked authored proofs may be used as locally proved facts.

B1 / involution. For a concrete von Neumann algebra \(M\subseteq B(H)\) and a cyclic separating vector \(\Omega\), define the conjugate-linear map \(S_0:x\Omega\mapsto x^*\Omega\) on \(M\Omega\). Prove that its graph is closable by constructing the companion conjugate-linear Tomita map \(F_0\) on \(M'\Omega\) and proving \(F_0\subseteq S_0^*\); specify the convention for conjugate-linear adjoints. Prove density using the cyclic/separating equivalence and describe the closed involution, its inverse graph, and its polar decomposition. The cyclic-vector model does not close general-weight B2–B4.

B2 / Hilbert algebras. Start with an involutive algebra in a Hilbert space, bounded left multipliers, the adjoint identity, closable involution, and dense products. Construct right bounded vectors and the commutant Hilbert algebra, the full completion and its uniqueness. Prove every assertion about multiplication domains, involutions, bounded-vector closures, and graph cores. For Tomita algebras include entire analytic modular orbits with the source's Hilbert and multiplier norm controls. Direct integrals retain the exact measurable-field and countability assumptions; these are not assumed globally.

B3 / fundamental theorem. For a full left Hilbert algebra with its left von Neumann algebra \(M\), construct \(S=J\Delta^{1/2}\), prove \(J^2=1\), \(J\Delta J=\Delta^{-1}\), \(JMJ=M'\), and \(\Delta^{it}M\Delta^{-it}=M\). Prove strong continuity of the modular unitaries and identify invariant cores. No finite-dimensional spectral calculation is a substitute for these commutant and domain assertions.

B4 / weights. Construct the finite positive cone, hereditary definition algebra, null ideal, semicyclic GNS map, and normal representation. Prove the normality equivalences (net monotonicity, complete additivity for arbitrary sigma-strongly summable positive families, lower semicontinuity, supremum of dominated normal functionals) on arbitrary algebras. Construct the opposite weight and prove its faithfulness, semifiniteness, and normality. Establish the weight/Hilbert-algebra correspondence and inverse identifications, including the closedness properties of the GNS map in the exact source topology. Include locally compact, possibly nonunimodular Plancherel theory, Fourier-algebra duality and its technical estimates; separately retain the C*-algebra weight variants. The WG, DW, NW, HA, RD, WH, OW, PF and LW units provide local proof candidates for their specified parts. These units do not prove their foundational prerequisites or cover the full source range and its exercises.

B5 / modular automorphisms and KMS. For an n.s.f. weight on arbitrary \(M\), prove a unique sigma-weakly continuous modular automorphism group preserving the weight, with the specified implementation. Construct analytic elements using Gaussian smoothing and prove which finite-domain intersections are graph cores. State the KMS boundary identities first on typed analytic finite-domain pairs, then prove the exact source characterization; all strip orientation, conjugation and half-power formulas require an explicit check of their conventions. No bounded-state KMS proof closes the weight theorem.

B6 / centralizers and perturbations. Identify the fixed-point algebra of \(\sigma^\varphi\) with the appropriate finite-domain trace-commutation condition. Prove the source-range affiliated perturbation theorem with a positive self-adjoint operator affiliated with the centralizer, support/kernel conditions explicit. Do not claim that the restriction of an arbitrary n.s.f. weight to its centralizer is automatically semifinite. Specify extended integrals and bounded spectral truncations; do not infer monotonicity of noncommuting sandwiches.

B6 / relative and cocycle data. First for positive normal functionals in a standard form, construct the relative Tomita map, its support projections, graph closure and relative modular operator. Extend to weights by the source's full construction. For two n.s.f. weights on \(M\), prove strongly continuous unitaries \(u_t=[D\psi:D\varphi]_t\in M\),

u_{s+t}=u_s\sigma_s^\varphi(u_t),\qquad \sigma_t^\psi=\operatorname{Ad}(u_t)\circ\sigma_t^\varphi.

Prove the chain rule \([D\chi:D\varphi]_t=[D\chi:D\psi]_t[D\psi:D\varphi]_t\), reference-weight independence of the spatial construction, and the source's converse and perturbation characterizations. Nonfaithful variants must state partial isometries/support conventions separately. OA-FLOW owns action cocycle/cohomology applications; it imports this weight-change theorem.

B7 / natural cone. For an n.s.f. GNS representation, construct the closed natural cone from the correctly typed finite-domain vectors (including the \(\Delta^{1/4}\) domain). Prove self-duality, \(J\xi=\xi\) on the cone, and invariance under \(xJxJ\). Prove standard-form existence and the unique unitary carrying one standard form to another along a normal *-isomorphism. Prove the bijection between normal positive functionals and cone vectors, its exact continuity estimates, the canonical implementation of automorphisms, and all support and order subtleties. Hilbert-cone order must not be equated without proof to order of the corresponding functionals.

B8 / measurable operators. Fix a von Neumann algebra with an n.s.f. trace \(\tau\); define affiliation by commutant-unitary invariance of the graph. In this construction every operator is closed and densely defined. Such an affiliated operator is \(\tau\)-measurable when its domain is \(\tau\)-dense: for every \(\varepsilon>0\) a projection \(e\) satisfies \(eH\subseteq D(T)\), \(\tau(1-e)<\varepsilon\). Prove the equivalent spectral-tail criteria, the closure of sums/products, and the complete topological *-algebra structure for neighborhoods controlled by \(\|Te\|\) and \(\tau(1-e)\). Define positive integration by spectral truncation and prove the trace extension, Lp domains/norms, Holder and duality statements only in their exact source range. Nonfinite trace, noncommutative products, and unbounded domains stay visible.

B9 / spatial derivative. Given \(M\subseteq B(H)\), an n.s.f. weight \(\psi'\) on \(M'\), and a normal semifinite weight \(\varphi\) on \(M\), define the \(\psi'\)-bounded vectors by bounded extension of

R_{\psi'}(\xi)\Lambda_{\psi'}(x')=x'\xi, \quad x'\in\mathfrak n_{\psi'}.

Prove their density and that \(R_{\psi'}(\xi)R_{\psi'}(\xi)^*\in M_+\). Construct and close the quadratic form \(q(\xi)=\varphi(R_{\psi'}(\xi)R_{\psi'}(\xi)^*)\) on its finite part; prove the precise core theorem and identify its positive self-adjoint representative \(d\varphi/d\psi'\). For faithful weights prove modular implementation, reciprocity, and the products of imaginary powers yielding Connes cocycles; for nonfaithful numerator retain the correct support. These are not closed by the bounded GNS comparison operator in OA-MOD-DW.

B9 / fusion. For normal right/left \(N\)-modules \(H_N\), \({}_NK\), construct bounded intertwiner coefficients in a fixed standard form. Prove positivity of the matrix-valued coefficients, form the null quotient of the algebraic tensor domain, and complete. Prove the density of elementary vectors, natural commuting outer actions, independence of the n.s.f. reference weight, associator, pentagon and unit unitaries. The resulting representation of the algebraic tensor product of outer commutants need not be injective; the standard module for \(N=\mathbb C\oplus\mathbb C\) is a required regression case. Source faithfulness assumptions on modules must be preserved or explicitly generalized with proof.

B9 / expectations. For an inclusion \(N\subseteq M\), prove the positivity, bimodularity and complete positivity consequences of a contractive retraction in the exact unital/nonunital source setting. For n.s.f. \(\varphi\) and a unital von Neumann subalgebra on which \(\varphi\) restricts semifinitely, prove the equivalence between modular invariance and existence of the unique normal \(\varphi\)-preserving conditional expectation. Specify its GNS projection, faithfulness and restrictions. The trace-preserving case is an exact specialization, not coverage of the modular criterion.

B9 / operator-valued weights. Define the extended positive cone \(\widehat N_+\) as extended positive elements evaluated on normal positive functionals, including infinite parts. For \(T:M_+\to\widehat N_+\), specify additivity, positive homogeneity, \(N\)-bimodularity, normality, faithfulness and semifiniteness, together with \(\mathfrak n_T\), \(\mathfrak m_T\), and the finite-range extension. Prove composition with scalar weights, domains and normality/semifiniteness transfer under the exact hypotheses, and the source's existence, duality and Radon–Nikodym results. General expectation/dual-weight/index machinery appearing in subfactor chapters is covered here; special classification applications belong to the D-family courses. Crossed-product dual weights stay with OA-FLOW.

Read statements at their proved scope

A linked proof supplies its stated conclusion at its named mathematical inputs. A preview is not a proof of a general theorem. Exact source comparisons and prerequisite checks remain separate from the reader's route.

The fullness discussion is a concrete example: the full-fibres implication is unconditional, the reverse implication has an analytic-defect hypothesis, and the CH counterexample is a separate conditional investigation. The scalar field calculation proves its particular model directly. These are different statements, each retained at its own scope.

These lessons are working mathematical texts. Not every prerequisite they use is proved within the course.