Modular theory and weights
Weights, semicyclic representations and the Tomita–Takesaki theory: finite domains, standard forms, modular groups and KMS states, Connes cocycles, operator-valued weights, and noncommutative integration.
280 readings, including the course guide. Choose a topic below or browse the full list.
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Closing involutions and the modular theorem
Follow the closed involution through the two multiplication actions, bounded approximation and analytic domains to the modular commutant theorem.
- Closing an involution and recovering its modular data
- Building the two multiplication actions of a Hilbert algebra
- Right Hilbert algebra density and commutants
- The full left Hilbert algebra obtained by dualizing twice
- Weights and the Hilbert spaces of multiplication
- Analytic kernels for unbounded modular operators
- Approximating multiplication without losing its bound
- The modular group and its analytic algebra
Relative modular and spatial derivatives
Use standard forms and closed positive forms to construct spatial energies, relative Tomita operators and their support corners.
- Recovering operators from energy forms
- The positive cone of a standard representation
- Recovering a representation from its positive cone
- Spatial comparison on an arbitrary representation
- Constructing spatial energy from finite observations
- Adding spatial energies and changing the reference weight
- Spatial energy and modular time
- Supported GNS representations and the balanced cocycle
- Balanced weights and relative Tomita blocks
- Recovering a weight from spatial energy
Modular flows, KMS and centralizers
Characterize modular time by its boundary condition, identify the fixed algebra, and study invariant weights and equilibrium states.
- The modular group and its analytic algebra
- The KMS boundary condition determines the modular group
- Fixed observations, density weights and modular time
- Recognizing invariant weights by their densities
- Upper imaginary time and finite weight multipliers
- Finite pieces and invariant observations for modular flows
- From a C*-modular condition to the GNS von Neumann algebra
- The geometry of KMS states and their central densities
- Central decompositions and disjoint equilibrium representations
Changing weights and Connes cocycles
Compare weights through supported relative data, prove composition and reconstruction, then study domains and continuity.
- Supported GNS representations and the balanced cocycle
- Spatial energy and modular time
- Balanced weights and relative Tomita blocks
- Changing the reference of a weight cocycle
- Relative modular flows and corner composition
- Reconstructing a weight from a modular cocycle
- Partial cocycles and their exact mixed KMS domains
- Cocycle matrices and entire strip domains
- Compact-time continuity along automorphism orbits
- Norm continuity of relative modular time
Measurable operators and trace integration
Construct measurable operators with their domains, extend the trace, prove the integration and duality results, and pass to general nontracial Lp spaces.
- Finite trace domains, central parts and amplification
- Closed operators through their graph projections
- Operators recovered from small trace defects
- Spectral layers in a semifinite trace algebra
- Trace densities and noncommutative integration
- The trace norm and bounded densities
- Square-integrable pairings and strong involution
- Spectral tails, singular values and trace order
- Homogeneous operators in the continuous core
- Nontracial Lp spaces for arbitrary von Neumann algebras
Expectations and operator-valued weights
Begin with contractive retractions and the modular invariance criterion. Continue through extended positives, operator-valued integration, commutant duality and expectation index.
- Contractive retractions and conditional expectations
- Tracial multiplication actions and the preserving expectation
- Conditional expectations from modular invariance
- Averaging and limits of normal expectations
- Corner weights tested on all positive energies
- When countable sums detect extended positive energy
- Finite calculus and composition of operator-valued weights
- Detecting and determining operator-valued weights
- Modular invariants of operator-valued weights
- Commutant duality for operator-valued weights
- Compatible modular weights and operator-valued reconstruction
- Expectation index and the basic construction
Plancherel weights, tensor products and disintegration
Construct group weights and tensor weights, then assemble measurable Hilbert spaces and weight fields with their stated countability hypotheses.
- Haar convolution as left and right Hilbert algebras
- Regular representations, Fourier algebras, and Borel group measures
- Tensor operators and tensor weights
- Measurable Hilbert fields and their diagonal commutant
- Central decomposition from countable operator equations
- Countable cores, multiplier domains and measurable choices
- Compatible measurable GNS fields and both representation transports
- Measurable weight fields and their integrals
- The realization boundary in measurable weight fields
- Recovering algebras from measurable fibres
- Separable GNS spaces and disintegration of C*-weights
All readings
- Modular theory through weights, time and reconstruction
- Finite domains and representations of general weights
- Weights and semicyclic representations: the exact opening package
- Comparing weights through finite-energy vectors
- A bounded-operator kernel for weight arguments
- Finite domains, null directions, and support corners
- Concrete preduals from Hilbert tensors
- Detecting normal weights by finite observations
- The predual-valued map of a normal weight
- Countable amplification and ultraweak bicommutant closure
- Normal positive maps and their preadjoints
- The vector retained by a comparison map
- A faithful weight on the GNS commutant
- Spectral calculus with its domains retained
- Banach holomorphy, generators, and resolvent limits
- Recovering operators from energy forms
- Closing energy domains and comparing resolvents
- Compatible pairs and complex interpolation
- Closing an involution and recovering its modular data
- Building the two multiplication actions of a Hilbert algebra
- Haar convolution as left and right Hilbert algebras
- Polynomial cutoffs recover an operator's range
- Right Hilbert algebra density and commutants
- The full left Hilbert algebra obtained by dualizing twice
- Weights and the Hilbert spaces of multiplication
- Changing algebra coordinates and scaling a weight
- Analytic kernels for unbounded modular operators
- Approximating multiplication without losing its bound
- The modular group and its analytic algebra
- The positive cone of a standard representation
- Recovering a representation from its positive cone
- Order and concavity of positive implementing vectors
- The modulus and phase of a vector
- Positive vectors at half and quarter modular time
- A bounded algebra inside its positive cone
- Recovering Jordan multiplication from the positive cone
- Implementing Jordan symmetries on standard forms
- Positive changes of scale on a standard cone
- Inner implementers from spectral tails
- Bounded generators of cone symmetries
- Why an everywhere-defined derivation is bounded
- Recovering an algebra from an oriented cone
- Cone symmetries without a global faithful state
- Positivity seen through two components
- Unbounded products in the standard positive cone
- Canonical L2 and standard implementations
- Adding square roots of normal functionals
- Supported GNS representations and the balanced cocycle
- Contractive retractions and conditional expectations
- Spatial comparison on an arbitrary representation
- Constructing spatial energy from finite observations
- Adding spatial energies and changing the reference weight
- Spatial energy and modular time
- The KMS boundary condition determines the modular group
- From a C*-modular condition to the GNS von Neumann algebra
- Closed graphs and bounded operator cutoffs
- Operators recovered from small trace defects
- When a small trace defect must vanish
- What a functional can detect in measure
- Functions, cutoffs and actual domains
- Convergence on large domains
- Spectral tails, singular values and trace order
- Measurable operators between representations
- Cyclic families and countable detection
- Closed operators through their graph projections
- Finite trace domains, central parts and amplification
- Averaging, central traces and normality
- Relative tensor products and fusion
- Fusion over a common spectral point
- Reconstructing a weight from a modular cocycle
- Recovering a weight from spatial energy
- Fixed observations, density weights and modular time
- Tracial multiplication actions and the preserving expectation
- Conditional expectations from modular invariance
- Corner weights tested on all positive energies
- Trace densities and noncommutative integration
- Spectral layers in a semifinite trace algebra
- Recognizing invariant weights by their densities
- Inner modular flow and rigidity of finite weight data
- Weight domination and exact half-strip endpoints
- Partial cocycles and their exact mixed KMS domains
- Weak compactness and second adjoints
- Forced C*-unitization
- Finite weight domains and GNS
- Bounded functionals and cyclic vectors
- States and the positive spanning family
- The universal C*-bidual
- Surjective isometries and normality
- Analytic generators and exact finite weight domains
- Upper imaginary time and finite weight multipliers
- Commuting modular flows and relative half-strip multipliers
- Central cocycles and common finite domains
- Normal-functional polar decomposition and invariant pairs
- Finite corners, faithful traces, and affine group fixed points
- Ergodic factors and trace normalization
- Semifinite corners and bounded modular operators
- Modular partitions and the full finite domain
- Tensor operators and tensor weights
- Compatible measurable GNS fields and both representation transports
- Measurable weight fields and their integrals
- The realization boundary in measurable weight fields
- Separable GNS spaces and disintegration of C*-weights
- Changing the reference of a weight cocycle
- Compact-time continuity along automorphism orbits
- Norm continuity of relative modular time
- Detecting a von Neumann algebra through a separable subalgebra
- Lower semicontinuous C*-weights
- A supported inverse weight cocycle
- Comparing supported weights and transporting their cuts
- Uniform geometry of faithful weights
- Homogeneous operators in the continuous core
- When countable sums detect extended positive energy
- Finite calculus and composition of operator-valued weights
- Detecting and determining operator-valued weights
- Modular invariants of operator-valued weights
- Commutant duality for operator-valued weights
- Compatible modular weights and operator-valued reconstruction
- Countable modular sums and infinite-weight boundaries
- Joint spectral measures and commutator estimates in standard form
- Measurable Hilbert fields and their diagonal commutant
- Central decomposition from countable operator equations
- Countable cores, multiplier domains and measurable choices
- Recovering algebras from measurable fibres
- Regular representations, Fourier algebras, and Borel group measures
- Six laboratories on expectations and operator-valued averages
- Compatible traces and the coupling operator
- A module, its standard corner and its double
- Traces on modules and full correspondences
- Nontracial Lp spaces for arbitrary von Neumann algebras
- Full Hilbert-algebra norms and weighted rank-one models
- Closed GNS graphs, cyclic vectors and bounded modular time
- The geometry of KMS states and their central densities
- Measures on a compact convex space
- Central decompositions and disjoint equilibrium representations
- Finite pieces and invariant observations for modular flows
- Modular time from a faithful normal state
- Positive vectors from a faithful state
- Scalar integrals and Hilbert representation
- Hereditary closure and positive duality
- When pointwise norm continuity becomes uniform
- Completeness of the automorphism group
- Inner and outer automorphisms
- Balanced weights and relative Tomita blocks
- The Hilbert–Schmidt standard form and its modular domains
- Trace defects and graph cores
- Averaging and limits of normal expectations
- The trace norm and bounded densities
- Square-integrable pairings and strong involution
- Conditional expectations and type III
- Comparing semifinite traces
- Cocycle matrices and entire strip domains
- Extended center-valued traces
- Positive equivalence and atomic expectations
- Semifinite conjugations and vector supports
- Amplified traces and concrete representations
- Vector implementation and spatiality
- Relative modular flows and corner composition
- Null ideals and faithful quotients of modular weights
- Spectral localization and products for real flows
- GNS filters, modular spectra, and cocycle invariance
- Ambient maximal abelian algebras from modular spectral gaps
- Entire orbits and compact Fourier support
- Expectation index and the basic construction
- Right action and the centralizer of a positive functional
- Abelian densities and the relative KMS strip
- Finite type I algebras and expected subalgebras
- Unitary loops and trace winding
- Asymptotic intertwining in standard form
- Square-zero modular intertwiners
- Orthogonal assembly and modular halving
- Matrix blocks and approximate product states
- Weighted tensor splitting from summable errors
- Modular halving and tensor absorption
- Spectral cuts and localized intertwiners
- Global assembly of approximate intertwiners
- From local intertwiners to approximately inner automorphisms
- Small centralizer defects and scalar averaging
- Modular averaging with controlled tensor moments
- Modular integral algebras and covariance
- Modular automorphisms are approximately inner
- Discrete modular crossings and their cores
- Opposite dual circles and irrational tensor products
- Approximate innerness and absorption descent
- Equivariant absorption from central matrix units
- Central sequences in modular crossed products
- Tracial absorption in constant matrix products
- Central freeness and relative commutants
- Aperiodic comparison and crossed products
- Modular periods of integer crossed products
- Type III crossings and circle actions
- Powers models and uniqueness in type III₁
- Relation weights and change of measure
- Expected diagonals and invariant traces
- Measurable corners and modular spectra
- Relation symmetries and weight covariance
- Relation cores and lifted symmetries
- Relation factors, traces and corners
- Orbit quotients and modular types
- Groupoid reductions and modular matrix blocks
- Measured comparison, returns and modular amplification
- Dyadic approximation and modular weights
- Cohomology, lifting and modular reweighting
- Ancillary actions and modular transport
- Groupoid exercises and modular consequences
- Haar amenability and modular averaging
- Amenable extensions and radical obstructions
- Measured amenability and finite approximation
- Infinite products and modular time
- Inductive limits and quasi-free factors
- Spectral cuts and irreducible centralizers
- Narrow modular spectra and projection displacement
- Projection partitions and dominant centralizers
- Bicentralizer structure and injective factors
- Directed quotients and approximate innerness
- Homogeneous groupoid matchings and weighted arrays
- Ratio products and cyclic groupoid reductions
- Generating weighted arrays and groupoid isomorphisms
- Type III1 normalization by weighted arrays
- Hypercentral sequences in the type III1 factor
- Matching weighted matrix units by approximately inner maps
- Approximately inner automorphisms of the type III1 factor
- Centrally trivial automorphisms and modular time
- Outer conjugacy and modular invariants
- Finite-index bounded vectors and module bases
- Separable modular subalgebras and preserving expectations
- The range of the index of an expectation
- Centralizer partitions and abelian approximation
- Bilinear state bounds and sharp finite matrix examples
- State-weighted adjoints and modular compatibility
- Rational state records and exact coherence loss
- Trace-preserving models and unitary rounding
- Balanced Kraus completion and coupling energy
- Unitary couplings and finite-factor approximation
- Injective corners and AFD descent
- Small-corner construction and finite injectivity
- Strong approximation and matrix compressions
- Markov expectations and coefficient recovery
- Finite chains of Jones projections
- Small expectations of projections
- Free-group norm averages with arbitrary substitute words
- Residual subgroups and uniform group-factor estimates
- Residual tails and tracial ultraproducts
- A continuum of separable type II₁ factors
- Free-tree representations and projections
- Dyadic subfactors and ultrafilter limits
- Null sequences and hypercentral commutants
- Local projection tests for outer automorphisms
- Fixed central sequences determine an automorphism
- Centre-valued characteristic data
- Quotient cocycles and reconstruction
- Finite matrix averaging and cohomology
- Core continuity and cocycle comparison
- Approximate first cohomology
- Two-sided intertwining and cocycle limits
- Characteristic classes and subgroup alignment
- Cyclic normalization and Bernoulli relations
- Irrational modular times and tail witnesses
- Modular returns and comparison through the discrete core
- Odometer carries and realization of a measured flow
- Classification by the flow of weights
- Full-group approximation of relation normalizers
- Return bands and semifinite components
- Orbit equivalence from the associated flow
- Equivariant discrete decomposition
- Module triviality and inner approximation
- Central triviality and the core-inner kernel
- Classification by factors, relations and flows
- A continuous section of the module map
- Joint obstructions and outer conjugacy
- Modular time and the outer center
- Crossed products from modular return data
- Pullbacks of the intrinsic invariant
- Modularly invariant Powers subfactors
- Amenable actions on AFD factors
- Ultraproduct corners and hypercentral tests
- Bicentralizers and the general hypercentral identity
- Bratteli reflection and the exceptional graph
- Factorization, hereditary corners and quotient lifts
- Representation fields and quasi-state duality
- Pure states and common maximal abelian subalgebras
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