The orbit commutant proof route
The covariant commutant formula applies to arbitrary Hilbert spaces and locally compact Hausdorff groups. Its proof combines the coefficient Hilbert algebra and its closed polar decomposition with representation amplification and a normal slice. A nonfaithful representation uses its invariant central-kernel quotient.
The sections below give the orbit lesson’s mathematical prerequisites, their exact hypotheses and the order in which they are used.
Orbit representations and orthogonal state measures · Machine-readable proof route · Component authorship and terms
GNS
n_phi is a left ideal; finite linear extension on m_phi; dense-range GNS with Lambda(xa)=pi(x)Lambda(a); pi normal, unital, faithful; finite positive contractions e_i in n_phi intersect n_phi-star increase strongly to 1.
phi normal faithful semifinite on unital M, where semifinite means ultraweak density of m_phi. Arbitrary Hilbert dimension and predual; no faithful-state/countability reduction.
WG: OA-MOD-WG-003. Domain algebra and its positive cone
WG: OA-MOD-WG-004. Linear extension without infinite subtraction
WG: OA-MOD-WG-005. Cauchy-Schwarz and the null ideal
WG: OA-MOD-WG-006. The GNS construction for an arbitrary weight
WG: OA-MOD-WG-007. Normality is a net argument
WG: OA-MOD-WG-008. Finite positive cutoffs characterize semifiniteness
WG: OA-MOD-WG-009. Density of the two-sided finite domain in the GNS space
WG: OA-MOD-WG-010. Faithfulness and the finite-weight specialization
Earlier steps: GNS.BASIC, GNS.NORMAL, BOUND.BK, BOUND.CP, HILBERT.BASIC.
ACTION
The given strongly continuous faithful covariant implementation implies norm continuity of every predual orbit and bounded joint sigma-strong-star continuity of (s,x) -> alpha_s(x).
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. Given normal unital covariant rho,V with V strongly continuous; use the invariant central-kernel quotient if rho is nonfaithful.
SUPPORT: ORBIT-BRIDGE-COVARIANT-TOPOLOGY — The given implementation supplies action continuity
F11: OA-FLOW.TOP.JOINT — Simultaneously varying the automorphism and the element
WAcurrent: 12. Normal representations
Earlier steps: BOUND.CP.
REGULAR
pi_alpha is a faithful normal unital coefficient representation on H tensor L2(G); left translations are strongly continuous unitaries; covariance holds.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. Coefficient representation faithful normal unital on arbitrary H.
F15: OA-FLOW.REG.CONSTRUCTION — Bounded coefficient operators and left translations
F15: OA-FLOW.REG.NORMALITY — Compact monotonicity handles arbitrary nets
Earlier steps: ACTION, HAAR.SECTIONS, BOUND.BK, BOUND.CP.
CONVOLUTION
K_alpha is an involutive algebra with right-coefficient convolution and Delta-dependent sharp; F_alpha has norm <= L1 norm and respects product, adjoint and coefficient module identities; kernel action is continuous compact-support and F_alpha is injective.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. K_alpha consists of compact-support strong-star continuous, norm-bounded coefficients.
F08: OA-FLOW.INT.OPERATIONS — The right-coefficient convolution
F08: OA-FLOW.INT.COVARIANT — Integrated operators and their identities
F08: OA-FLOW.INT.KERNEL — The regular action on vector sections
F08: OA-FLOW.INT.ALGEBRA — Algebra laws without formal rearrangements
Earlier steps: ACTION, REGULAR, HAAR.PRODUCT, HAAR.BASIC, BOUND.BK, BOUND.CP.
INTEGRATED.GENERATION
F_alpha(K_alpha) generates P; normalized shrinking compact-support scalar averages converge strongly to identity, recover every coefficient and every left group unitary.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. Unital covariant pair, strongly continuous implementing representation.
F08: OA-FLOW.INT.GENERATION — Recovering both parts of covariance
Earlier steps: CONVOLUTION, HAAR.BASIC, BOUND.BK.
COEFFICIENT.DOMAIN
b_phi=span(K_alpha*n_phi on the right) is a left convolution ideal; tildeLambda(b_phi) consists of continuous compact-support sections and tildeLambda is injective.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. General nsf phi and the exact GNS identities; right M-module stability is not claimed.
F09: OA-FLOW.DW.GNSDOMAIN — An algebraic domain with a continuous GNS image
Earlier steps: GNS, CONVOLUTION, HAAR.SECTIONS.
COEFFICIENT.LEFT
tildeLambda(f*g)=F_alpha(f)tildeLambda(g) and tildeLambda(x dot g)=pi_alpha(x)tildeLambda(g).
f in K_alpha, g in b_phi, x in M; Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity.
F09: OA-FLOW.DW.GNSCONVOLUTION — Convolution becomes bounded left multiplication
Earlier steps: COEFFICIENT.DOMAIN, CONVOLUTION, GNS.
COEFFICIENT.STAR
B_phi=b_phi intersect b_phi-sharp is a star algebra containing b-star dot f dot a for finite-domain a,b.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. a,b in n_phi; no right ideal assertion for n_phi.
F09: OA-FLOW.DW.COMMON — A domain on which the involution is defined twice
Earlier steps: COEFFICIENT.DOMAIN, CONVOLUTION.
COEFFICIENT.DENSITY
A_phi=tildeLambda(B_phi) is Hilbert dense; F_alpha(B_phi) has strong closure P; products A_phi*A_phi are Hilbert dense.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. General nsf phi; directed finite positive contraction net from GNS, not a countable approximate unit.
F09: OA-FLOW.DW.DENSITY — Vector density, operator generation and products
Earlier steps: GNS, REGULAR, COEFFICIENT.LEFT, COEFFICIENT.STAR, INTEGRATED.GENERATION, HAAR.SECTIONS.
STANDARD.GNS
All nsf weights have canonical GNS maps in one weight-constructed standard form; common conjugation J; SF44 supplies U_alpha-star Lambda_phi(alpha(x))=Lambda_(phi circ alpha)(x).
Weights nsf; arbitrary algebra and Hilbert dimension. Automorphism is normal; strong continuity uses ACTION plus SF12 topology.
SF: OA-MOD-SF-08 — A finite matrix weight and its four closed graphs
SF: OA-MOD-SF-09 — Canonical comparison of the positive cones
SF: OA-MOD-SF-12 — Canonical implementation of automorphisms
SF: OA-MOD-SF-13 — One Hilbert space for all n.s.f. GNS maps
Earlier steps: GNS, WH.CORRESPONDENCE, MF.COMMUTANT, CONE.GEOMETRY, ACTION.
RELATIVE.TOMITA
S_(psi,phi)=J Delta_(psi,phi)^(1/2) closed with positive injective self-adjoint Delta; mixed domain x in n_phi intersect n_psi-star maps Lambda_phi(x) to Lambda_psi(x-star), and that mixed-domain graph is a core.
phi,psi nsf, canonical common standard realization. Arbitrary Hilbert dimension and predual; no faithful-state/countability reduction.
SF: OA-MOD-SF-08 — A finite matrix weight and its four closed graphs
SF: OA-MOD-SF-09 — Canonical comparison of the positive cones
SF: OA-MOD-SF-13 — One Hilbert space for all n.s.f. GNS maps
F09: OA-FLOW.DW.IMPORT.TOMITA — Relative operators used for closability
Earlier steps: STANDARD.GNS, WH.CORRESPONDENCE, MF.COMMUTANT.
COEFFICIENT.CLOSURE
Jcal xi(s)=Delta_G(s)^(-1/2) U_s-star J xi(s-inverse) is an antiunitary involution; maximal pointwise T with fibers Delta_G(s)^(1/2) Delta_(phi circ alpha_s,phi)^(1/2) is closed; S0 subset Jcal T.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. Relative Tomita domain and standard transport identities; each individual section is strongly measurable.
F09: OA-FLOW.DW.CLOSED-FIELDS — A pointwise closedness lemma
F09: OA-FLOW.DW.RELATIVETOMITA — Closing the coefficient involution
Earlier steps: RELATIVE.TOMITA, STANDARD.GNS, COEFFICIENT.DENSITY, HAAR.BASIC.
LEFT.HILBERT
A_phi with coefficient product and sharp is a left Hilbert algebra, with left von Neumann algebra exactly P.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. The four axioms are bounded left multiplication, adjoint identity, closable involution and dense products.
F09: OA-FLOW.DW.LEFTHILBERT — What the domain construction now provides
Earlier steps: COEFFICIENT.CLOSURE, COEFFICIENT.LEFT, COEFFICIENT.STAR, COEFFICIENT.DENSITY.
WH.CORRESPONDENCE
General nsf phi has a full finite-star Hilbert algebra with exact closed involution; its left-bounded vectors are precisely Lambda(n_phi), and the associated weight recovers phi on all positives, including infinite values.
phi nsf on arbitrary M; no faithful-state restriction.
WH: OA-MOD-WH-09 — From a faithful normal semifinite weight to an algebra
WH: OA-MOD-WH-10 — Closability on the full finite-star domain
WH: OA-MOD-WH-11 — Fullness and recovery of the original weight
Earlier steps: GNS, MF.KERNELS, NORMAL.WEIGHT.SUP, DOMINATED.WEIGHT.VECTORS, BOUND.BK, BOUND.CP.
MODULAR.ORIGINAL
sigma_t^phi is the conjugation action of Delta_phi^(it); it preserves phi and n_phi and obeys Lambda_phi(sigma_t(a))=Delta_phi^(it)Lambda_phi(a).
phi nsf; full weight Hilbert algebra; Arbitrary Hilbert dimension and predual; no faithful-state/countability reduction.
MF: OA-MOD-MF-06 — The modular fundamental theorem
WH: OA-MOD-WH-11 — Fullness and recovery of the original weight
Earlier steps: MF.COMMUTANT, WH.CORRESPONDENCE, SPECTRAL.CALCULUS, BOUND.CP.
RELATIVE.IMAGINARY
Delta_(psi,phi)^(it)=[Dpsi:Dphi]_t Delta_phi^(it), with the same positive operator as RELATIVE.TOMITA.
phi,psi nsf in the common phi-standard form; balanced-matrix cocycle definition, not merely an unspecified modular cocycle.
SI: OA-MOD-SI-12. Ordinary relative GNS maps are a specialization
SI: OA-MOD-SI-14. The balanced-matrix cocycle is independent of the reference
SF: OA-MOD-SF-13 — One Hilbert space for all n.s.f. GNS maps
Earlier steps: RELATIVE.TOMITA, SPATIAL.IDENTIFICATION, MODULAR.ORIGINAL.
COCYCLE.CALCULUS
Balanced-matrix derivatives are unitary strongly-star continuous families; time cocycle, ordered chain rule, modular implementation and normal-automorphism pullback naturality hold.
All weights nsf on one arbitrary von Neumann algebra; normal automorphism or normal isomorphism for transport.
SI: OA-MOD-SI-14. The balanced-matrix cocycle is independent of the reference
CH: Reversal and paths of reference changes
F10: OA-FLOW.AWC.LAWS — Two parameters, two cocycle identities
Earlier steps: SPATIAL.IDENTIFICATION, MODULAR.ORIGINAL, SPECTRAL.CALCULUS.
TENSOR.WEIGHT
phi tensor mu is nsf, canonical on prescribed faithful spatial representation, with closed tensor Tomita operator and product modular group; automorphism pullback respects the specified tensor construction on all positive elements.
Both weights nsf; Hilbert factors arbitrary; no equality-of-weights inference solely from elementary positive tensor values.
TG: OA-MOD-TG-03 — Comparing spatial representations
TG: OA-MOD-TG-04 — Tensor Hilbert algebras and Tomita domains
TG: OA-MOD-TG-05 — All positive values and nonfaithful support corners
F10: OA-FLOW.AWC.IMPORT.WEIGHTS — Weight calculus, owned by OA-MOD
Earlier steps: WH.CORRESPONDENCE, CONE.GEOMETRY, MODULAR.ORIGINAL, SPECTRAL.CALCULUS, BOUND.CP, TENSOR.NATURALITY.
TENSOR.CONSTRUCTION
The specified tensor nsf weight has exact tensor GNS and closed Tomita polar decomposition, and its modular group is the product group.
Both weights nsf; arbitrary Hilbert dimensions; no naturality assertion included in this construction-only row.
TG: OA-MOD-TG-01 — Positive tensors and a graph core
TG: OA-MOD-TG-02 — Closed tensors, adjoints, and polar supports
TG: OA-MOD-TG-03 — Comparing spatial representations
TG: OA-MOD-TG-04 — Tensor Hilbert algebras and Tomita domains
TG: OA-MOD-TG-05 — All positive values and nonfaithful support corners
Earlier steps: WH.CORRESPONDENCE, CONE.GEOMETRY, MODULAR.ORIGINAL, SPECTRAL.CALCULUS, BOUND.CP, TENSOR.DOMAIN.
JOINT.COCYCLE
(s,t) -> [D(phi circ alpha_s):Dphi]_t is jointly sigma-strong-star continuous; pullback group/time/inverse laws M5 hold with the indicated factor order.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. General nsf phi, not assumed invariant.
F10: OA-FLOW.AWC.UNTWIST — Constructing a unitary without a field decomposition
F10: OA-FLOW.AWC.INVARIANT — An auxiliary invariant weight
F10: OA-FLOW.AWC.IDENTITY — Expressing every derivative using one curve
F10: OA-FLOW.DW.COCYCLECONT — Joint continuity at full group generality
F10: OA-FLOW.AWC.LAWS — Two parameters, two cocycle identities
Earlier steps: ACTION, STANDARD.GNS, COCYCLE.CALCULUS, TENSOR.WEIGHT, TENSOR.COCYCLE, HAAR.MULTIPLICATION, TENSOR.COMMUTANT.
COEFFICIENT.FLOW
R_t f(s)=Delta_G(s)^(it)c_t(s)sigma_t(f(s)) is a star-automorphism group of K_alpha, preserves supports/L1 norm and b_phi/B_phi.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. Exact M3-M5 and JOINT.COCYCLE; bounded coefficient products.
F12: OA-FLOW.DW.COEFFICIENTFLOW — The modular-time action on coefficients
Earlier steps: JOINT.COCYCLE, MODULAR.ORIGINAL, COCYCLE.CALCULUS, CONVOLUTION.
MODULAR.UNITARY
Q_t xi(s)=Delta_G(s)^(it)Delta_(phi circ alpha_s,phi)^(it)xi(s) defines a strongly continuous unitary group and a unique positive nonsingular tildeDelta.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. Exact relative imaginary-power identity and joint continuity; continuous compact-support vector density.
F12: OA-FLOW.DW.MODULARUNITARIES — Constructing a unitary group on the section space
Earlier steps: RELATIVE.IMAGINARY, JOINT.COCYCLE, HAAR.SECTIONS, STONE, SPECTRAL.CALCULUS.
SMOOTH.CALCULUS
For every smooth compact-support k, k(log tildeDelta) acts pointwise by k(log(Delta_G(s)Delta_relative(s))).
Self-adjoint global and pointwise generators from MODULAR.UNITARY; each target vector arbitrary, no universal exceptional set.
F12: OA-FLOW.DMO.FUNCTIONALCALCULUS — Smooth bounded functions act pointwise
Earlier steps: MODULAR.UNITARY, FOURIER.SMOOTH, HAAR.PRODUCT, SPECTRAL.CALCULUS.
REAL.POWERS
For each real r, D(tildeDelta^r) is exactly the maximal L2 pointwise domain with fiber Delta_G(s)^r Delta_relative(s)^r; action and squared-norm integral equal the fiber formula.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. Fixed r and vector, arbitrary Hilbert dimension; negative powers have actual unbounded domains.
F12: OA-FLOW.DW.REALPOWERDOMAINS — Every real exponent, with its full domain
Earlier steps: SMOOTH.CALCULUS, SPECTRAL.CALCULUS.
IMAGINARY.INVARIANCE
tildeDelta^(it) tildeLambda(f)=tildeLambda(R_t f), hence tildeDelta^(it) A_phi=A_phi.
f in b_phi, exact general finite-domain GNS identity M3 and relative imaginary powers M4.
F12: OA-FLOW.DMO.INTERTWINING — Modular time preserves the common domain
Earlier steps: COEFFICIENT.FLOW, MODULAR.UNITARY, MODULAR.ORIGINAL, GNS.
GRAPH.CORE
Any dense linear D subset D(f(P)) invariant under every e^(itP) is a graph core for f(P), for Borel f finite spectral-a.e.; in particular for positive nonsingular A and every real A^r.
Arbitrary H; no D subset D(P) or countability assumption; actual domain inclusion is required.
F07: OA-FLOW.GRAPH.MAIN — The graph-localization theorem
F07: OA-FLOW.GRAPH.POWERS — Imaginary-power invariance controls real powers
Earlier steps: SPECTRAL.CALCULUS, SPECTRAL.REDUCTION.
CLOSED.POLAR
A_phi is a graph core for tildeDelta^(1/2); closure S0=Jcal tildeDelta^(1/2), which is its polar decomposition.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. Exact half-power inclusion from D28 and REAL.POWERS; dense A_phi and IMAGINARY.INVARIANCE.
F12: OA-FLOW.DW.INVARIANTCORE — The closed involution and its polar decomposition
Earlier steps: COEFFICIENT.CLOSURE, COEFFICIENT.DENSITY, REAL.POWERS, IMAGINARY.INVARIANCE, GRAPH.CORE, SPECTRAL.CALCULUS.
MF.KERNELS
General Hilbert-algebra multiplication domains, affiliation, right-algebra density/commutant generation/product cores and WH03-04 full completion preserve the original closed involution and left algebra.
Arbitrary left Hilbert algebra (including nonunital/zero/nonseparable cases).
Building the two multiplication actions of a Hilbert algebra
Recovering the commutant from right-bounded vectors
WH: OA-MOD-WH-03 — Completing the two multiplication domains
WH: OA-MOD-WH-04 — Mixed bounded vectors and fullness
Earlier steps: BOUND.BK, HILBERT.BASIC, POLAR.CLOSED, SPECTRAL.CALCULUS.
MF.COMMUTANT
For any left Hilbert algebra with closed involution S=J Delta^(1/2), the generated left algebra satisfies JMJ=M-prime.
Arbitrary left Hilbert algebra; exact full completion and common graph domain for both half powers.
MF: OA-MOD-MF-01 — Starting data, full completion and exact inputs
MF: OA-MOD-MF-02 — The resolvent creates bounded multiplication
MF: OA-MOD-MF-03 — A common domain for the two half powers
MF: OA-MOD-MF-04 — The resolvent satisfies a weak operator equation
MF: OA-MOD-MF-05 — Fourier uniqueness recovers the commutant
MA: OA-MOD-MA-04 — Fourier uniqueness from Gaussian approximation
MA: OA-MOD-MA-06 — A spectral resolvent as a strong integral
MA: OA-MOD-MA-07 — Solving an equation expressed through unbounded pairings
Earlier steps: MF.KERNELS, POLAR.CLOSED, SPECTRAL.CALCULUS, SCALAR.ANALYSIS, BOUND.BK, HILBERT.BASIC.
BF75.I2
Jcal P Jcal=P-prime for the regular crossed product at full group/Hilbert-space generality.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. Actual coefficient left Hilbert algebra LEFT.HILBERT and its exact CLOSED.POLAR identification.
The standard crossed-product commutant
Earlier steps: LEFT.HILBERT, CLOSED.POLAR, MF.COMMUTANT, STANDARD.GNS.EQUIVALENCE.
BF75.I5
P-prime=(M-prime tensor 1 union {U_g tensor R_g}) double-prime, with R_g xi(s)=Delta_G(g)^(1/2)xi(sg).
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. Faithful standard realization, canonical strongly continuous U commuting J.
The standard crossed-product commutant
Earlier steps: BF75.I2, STANDARD.GNS.
BF77.E9
For every normal unital covariant rho,V, P_rho-prime=(rho(M)-prime tensor 1 union {V_g tensor R_g}) double-prime.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. V strongly continuous; arbitrary faithful normal representation comparison, with invariant central kernel quotient for nonfaithful rho.
The regular commutant in every covariant representation
Positive observations can be implemented by vector sums
Earlier steps: BF75.I5, POSITIVE.VECTOR.SERIES, TENSOR.SLICES, BOUND.BK, BOUND.CP.
ORBIT.1.4
For faithful multiplication D=L-infinity(Y,mu), D-prime=D and discrete Haar factor 1 yield R-prime=(pi-prime(D) union {t_g}) double-prime, t_g xi(s)=v_g xi(sg).
Original consumer: compact metrizable support Y, quasi-invariant Borel probability, countable discrete free point action. General E9 is retained upstream.
OWNED-3: 1. The support and the implemented model
Earlier steps: BF77.E9, ORBIT.MEASURE.MULTIPLICATION.
STONE
Every strongly continuous unitary real-line group on arbitrary H has a unique self-adjoint Stone generator, with exact norm-difference-quotient domain.
Strongly continuous unitary representation of the real line; arbitrary H.
SG: OA-MOD-SG-04 — Stone's theorem with the derivative domain
Earlier steps: SCALAR.ANALYSIS, SPECTRAL.CALCULUS, HILBERT.BASIC.
FOURIER.SMOOTH
FLOW12 M6-M7 smooth compact-support scalar inversion, L1 transform, and strong spectral integral identity.
k in Cc-infinity(real line); arbitrary self-adjoint P.
SUPPORT: ORBIT-BRIDGE-FOURIER — Smooth compact-support Fourier inversion
Earlier steps: SCALAR.ANALYSIS, SPECTRAL.CALCULUS.
SPECTRAL.REDUCTION
A closed subspace reducing every e^(itP) reduces every Borel spectral projection of P.
P self-adjoint on arbitrary H; no separability assumption.
SUPPORT: ORBIT-BRIDGE-SPECTRAL-REDUCTION — The spectral projections follow the unitary group
SG: OA-MOD-SG-04 — Stone's theorem with the derivative domain
SKcurrent: OA-MOD-SK-08 — Transport, reduction, and membership in an algebra
Earlier steps: STONE, SPECTRAL.CALCULUS.
HAAR.BASIC
General LCH Haar existence/uniqueness, compact cutoffs, positive normalized shrinking averages, continuous Delta and the exact inversion/right-translation formulas.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity.
HRcurrent: HR-06. Haar existence from normalized finite covers
HRcurrent: HR-07. Haar uniqueness on arbitrary groups
HRcurrent: HR-08. Open sigma compact cosets and the outer regular convention
HRcurrent: HR-09. Complete locally determined Haar measure and finite-exponent classes
GRcurrent: 3. Translations, the modular function and inversion
Earlier steps: SCALAR.ANALYSIS, HILBERT.BASIC.
HAAR.SECTIONS
H tensor L2(G) identifies with strongly measurable L2(G,H), with dense continuous compact-support sections.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. Completed locally determined Haar convention; arbitrary H.
HRcurrent: HR-03. Finite-set regularity, densities and $L^p$ approximation
HRcurrent: HR-09. Complete locally determined Haar measure and finite-exponent classes
GRcurrent: 4. Vector integrals and the Hilbert tensor identification
Earlier steps: HAAR.BASIC, SCALAR.ANALYSIS, HILBERT.BASIC.
HAAR.PRODUCT
Radon-product integration at full nonmetrizable LCH scope: compact continuous scalar/vector integrands, open/lower-semicontinuous nonnegative section formulae, and completed Borel integrable functions on sigma-finite carriers.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. Radon product rather than an asserted equality of product Borel sigma algebras; section representatives of finite-exponent vectors have sigma-compact carriers.
HRcurrent: HR-05. The Radon product, without a product-Borel assumption
GRcurrent: 4. Vector integrals and the Hilbert tensor identification
Earlier steps: HAAR.BASIC, HAAR.SECTIONS, SCALAR.ANALYSIS.
HAAR.MULTIPLICATION
Locally determined Haar L-infinity is a faithful normal multiplication MASA generated by Cc(G); integration is nsf, its standard conjugation is complex conjugation and its modular operator is 1.
Arbitrary locally compact Hausdorff G, left Haar measure, point-ultraweak action; no second countability, sigma compactness or unimodularity. Use local L-infinity, not a false identification with global outer-regular L-infinity.
SUPPORT: ORBIT-BRIDGE-HAAR-MASA — The local Haar multiplication algebra and its standard form
HRcurrent: HR-03. Finite-set regularity, densities and $L^p$ approximation
HRcurrent: HR-08. Open sigma compact cosets and the outer regular convention
HRcurrent: HR-09. Complete locally determined Haar measure and finite-exponent classes
Earlier steps: HAAR.SECTIONS, BOUND.BK, BOUND.CP, SCALAR.ANALYSIS.
TENSOR.COMMUTANT
(M-prime spatial-tensor Haar-D)-prime=M spatial-tensor Haar-D in the actual arbitrary weight-standard H tensor L2(G).
Arbitrary Hilbert dimension and predual; no faithful-state/countability reduction. M arbitrary concrete, Haar D a multiplication MASA; no standard-form or weight premise.
Tensor commutants and arbitrary representation amplification
SUPPORT: ORBIT-BRIDGE-HAAR-MASA — The local Haar multiplication algebra and its standard form
Earlier steps: SPATIAL.COMMUTATION, HAAR.MULTIPLICATION.
TENSOR.SLICES
Arbitrary-cardinal amplification commutant and normal rank-one slice membership used by BF77 E8-E9.
Arbitrary Hilbert L,K and concrete von Neumann algebra A.
SPATIALcurrent: **Theorem 5.2.**
SPATIALcurrent: **Proposition 8.1.**
SPATIALcurrent: **Theorem 9.2** (slice maps). Let \(X\in M\bar\otimes N\).
Tensor commutants and arbitrary representation amplification
Earlier steps: SPATIAL.MATRICES, SPATIAL.GENERATORS, SPATIAL.AMPLIFICATION, SPATIAL.FLIP, SPATIAL.SLICES.
POSITIVE.VECTOR.SERIES
Every normal positive functional on a concrete unital arbitrary-H von Neumann algebra is a positive vector series with summable squared norms.
Normal positive functional; normal image/inverse when transferring between faithful representations. No separability assumption.
BIcurrent: 10. Positive normal functionals as sums of vector functionals
WAcurrent: 12. Normal representations
Positive observations can be implemented by vector sums
Earlier steps: BOUND.BK, BOUND.CP, HILBERT.BASIC.
POLAR.CLOSED
General closed antilinear involution S has exact-domain polar S=J Delta-half, F=J Delta-minus-half, Delta=FS, SF=Delta-inverse, and J Delta^z J=Delta^(-conjugate(z)). Ordinary linear closed graphs/adjoints and T-star-T half-power form domain are included.
Arbitrary closed densely defined operators and left Hilbert algebras; no vector-model, state or separability premise.
Exact domain instantiation for the preserved commutant proof
Closing an involution and recovering its modular data
Recovering operators from energy forms
Earlier steps: BOUND.BK, HILBERT.BASIC, SPECTRAL.CALCULUS.
SPECTRAL.CALCULUS
Arbitrary-H Borel spectral calculus with actual power/log domains, transport, affiliation, and a common graph approximation for both positive and negative half powers.
General self-adjoint operators; common graph domain V=D(A-half) intersect D(A-minus-half), no countability reduction.
Exact domain instantiation for the preserved commutant proof
Spectral calculus with its domains retained
Earlier steps: BOUND.BK, HILBERT.BASIC, SCALAR.ANALYSIS.
TENSOR.DOMAIN
Existing TG1-2 instantiated: the closed positive tensor has its full spectral integral domain and rectangular graph cutoffs, including zero factors and cancellation domains.
Arbitrary Hilbert dimension and predual; no faithful-state/countability reduction. Both factors positive self-adjoint; ordinary algebraic tensor is a graph core for the closed tensor.
The tensor-domain formulas and their graph core
Earlier steps: SPECTRAL.CALCULUS, HILBERT.BASIC.
TENSOR.NATURALITY
OT1: tensor nsf weights transport under the normal spatial tensor isomorphism on every positive element, including infinite values, via full Hilbert-algebra/GNS and multiplier-vector transport.
Both factor maps normal unital star isomorphisms; all factor weights nsf; arbitrary dimensions.
ROOTtensor: ORBIT-TENSOR-TRANSPORT — Naturality on the entire positive cone
The tensor-domain formulas and their graph core
Earlier steps: TENSOR.CONSTRUCTION, TENSOR.DOMAIN, WH.CORRESPONDENCE.
TENSOR.COCYCLE
OT2-3: the matrix/tensor diagonal weight agrees on all positives, and the same-second-factor balanced-matrix cocycle equals the first-factor cocycle tensor identity.
phi1,phi2 nsf on M and the same nsf mu on N; intrinsic SI64-66 cocycle convention; no countability assumption.
ROOTtensor: ORBIT-TENSOR-DIAGONAL — A diagonal tensor weight has the prescribed two corners
ROOTtensor: ORBIT-TENSOR-COCYCLE — Cancellation of the common second factor
The tensor-domain formulas and their graph core
Earlier steps: TENSOR.CONSTRUCTION, TENSOR.DOMAIN, COCYCLE.CALCULUS, WH.CORRESPONDENCE.
GNS.BASIC
The general finite left ideal and linear finite extension give a dense GNS space and bounded left representation, before its normality conclusion.
phi normal weight; semifiniteness/faithfulness only where WG explicitly requires them. Arbitrary Hilbert dimension and predual; no faithful-state/countability reduction.
WG: OA-MOD-WG-003. Domain algebra and its positive cone
WG: OA-MOD-WG-004. Linear extension without infinite subtraction
WG: OA-MOD-WG-005. Cauchy-Schwarz and the null ideal
WG: OA-MOD-WG-006. The GNS construction for an arbitrary weight
Earlier steps: BOUND.BK, BOUND.CP, HILBERT.BASIC.
NW.CONVEX
Exact arbitrary locally convex separation, Banach-Alaoglu/Hilbert weak-ball compactness and Krein-Smulian bounded-slice criterion used by NW06-10.
General Banach dual M=(M_*)-star and real selfadjoint convex sets; no separability assumption.
Predual, normal-weight and cone arguments for the orbit proof
Earlier steps: BOUND.CP, HILBERT.BASIC, BOUND.BK.
NORMAL.WEIGHT.SUP
Every normal weight is the supremum on all positives of its dominated bounded normal positive functionals, including zero and infinite values.
Arbitrary normal weights on arbitrary M; dominated family need not be directed.
Detecting normal weights by finite observations
Predual, normal-weight and cone arguments for the orbit proof
Earlier steps: GNS.BASIC, NW.CONVEX, BOUND.CP.
GNS.NORMAL
WG007 increasing-net strong convergence and NW11 give normality of every GNS vector pullback by positive-functional norm closure, and hence normality of the GNS representation.
phi arbitrary normal weight, before any semifiniteness restriction.
WG: OA-MOD-WG-007. Normality is a net argument
Predual, normal-weight and cone arguments for the orbit proof
Earlier steps: GNS.BASIC, NORMAL.WEIGHT.SUP, BOUND.CP.
DOMINATED.WEIGHT.VECTORS
OW02-03 give a bounded positive commutant contraction and exact implementing vector for omega<=phi; the general finite contraction net supplies all-positive order and the polar target-range equality.
phi normal semifinite, including nonfaithfulness; omega bounded normal positive; finite domination constant. Arbitrary Hilbert dimension and predual; no faithful-state/countability reduction.
OW: OA-MOD-OW-02. What a comparison operator remembers
OW: OA-MOD-OW-03. The vector implementing a finite observation
Predual, normal-weight and cone arguments for the orbit proof
Earlier steps: GNS, NORMAL.WEIGHT.SUP, BOUND.CP, POLAR.CLOSED.
STANDARD.GNS.EQUIVALENCE
Every prescribed axiomatic standard form is unitarily equivalent to the standard form of any nsf weight; the unitary preserves algebra, J and cone, and transports canonical automorphism implementations.
Arbitrary standard forms and normal unital star isomorphism; arbitrary cardinal directed support corners, no faithful state on the whole M.
SE: OA-MOD-SE-10 — Patching the unique comparisons
SE: OA-MOD-SE-11 — Transporting positive functionals and automorphisms
WH: OA-MOD-WH-13 — Every von Neumann algebra has such a weight
Predual, normal-weight and cone arguments for the orbit proof
Earlier steps: STANDARD.GNS, STANDARD.CORNER.PATCHING, CONE.GEOMETRY, GNS.
SPATIAL.MATRICES
Arbitrary basis columns, operator matrices, entrywise commutation and bounded finite-subset strong truncations.
Arbitrary Hilbert dimension and predual; no faithful-state/countability reduction. Concrete von Neumann algebras; no modular or weight premise.
SPATIALcurrent: **Proposition 1.3.**
SPATIALcurrent: **Proposition 2.1.**
Tensor commutants and arbitrary representation amplification
Earlier steps: HILBERT.BASIC, BOUND.BK.
SPATIAL.GENERATORS
Normal amplification and extension, generated spatial tensor algebra, von Neumann amplification and matrix algebra.
Arbitrary Hilbert dimension and predual; no faithful-state/countability reduction. Concrete von Neumann algebras; no modular or weight premise.
SPATIALcurrent: **Proposition 3.1.**
SPATIALcurrent: **Theorem 5.2.**
Tensor commutants and arbitrary representation amplification
Earlier steps: SPATIAL.MATRICES, BOUND.BK, BOUND.CP.
SPATIAL.AMPLIFICATION
(A tensor identity)-prime=A-prime spatial-tensor B(L), with finite-subset matrix proof.
Arbitrary Hilbert dimension and predual; no faithful-state/countability reduction. Concrete von Neumann algebras; no modular or weight premise.
Tensor commutants and arbitrary representation amplification
Earlier steps: SPATIAL.MATRICES, SPATIAL.GENERATORS.
SPATIAL.FLIP
Associativity, flip, normal leg maps and spatial transport.
Arbitrary Hilbert dimension and predual; no faithful-state/countability reduction. Concrete von Neumann algebras; no modular or weight premise.
SPATIALcurrent: **Proposition 8.1.**
Tensor commutants and arbitrary representation amplification
Earlier steps: SPATIAL.MATRICES, SPATIAL.GENERATORS.
SPATIAL.SLICES
Normal vector slices belong to the opposite tensor leg; norm and module rules; arbitrary Hilbert dimensions.
Arbitrary Hilbert dimension and predual; no faithful-state/countability reduction. Concrete von Neumann algebras; no modular or weight premise.
SPATIALcurrent: **Theorem 9.2** (slice maps). Let \(X\in M\bar\otimes N\).
Tensor commutants and arbitrary representation amplification
Earlier steps: SPATIAL.MATRICES, BOUND.CP, SPATIAL.AMPLIFICATION.
SPATIAL.COMMUTATION
Full arbitrary-H commutation (M spatial-tensor N)-prime=M-prime spatial-tensor N-prime.
Arbitrary Hilbert dimension and predual; no faithful-state/countability reduction. Concrete von Neumann algebras; no modular or weight premise.
SPATIALcurrent: **Proposition 6.1.** Let \(M\) be a von Neumann algebra on \(H\).
Tensor commutants and arbitrary representation amplification
Earlier steps: SPATIAL.MATRICES, SPATIAL.GENERATORS, SPATIAL.AMPLIFICATION, HILBERT.BASIC, BOUND.BK.
STANDARD.CORNER.PATCHING
SE01-09: full standard-cone support corners, relative closed graph-core recognition, directed finite joins and density of their corner union.
SE: OA-MOD-SE-01 — Axioms and the geometric facts available before comparison
SE: OA-MOD-SE-02 — Compression and the commutant
SE: OA-MOD-SE-03 — Every projection has a faithful cone corner
SE: OA-MOD-SE-04 — The two cyclic domains are graph cores
SE: OA-MOD-SE-05 — Recognizing a modular conjugation by its exact graph
SE: OA-MOD-SE-06 — The corner seen by a positive vector
SE: OA-MOD-SE-07 — Closing one order ideal gives its entire supported face
SE: OA-MOD-SE-08 — Projections and all closed cone faces
SE: OA-MOD-SE-09 — Finding a full-support vector on each small corner
CONE.GEOMETRY
Weight-constructed natural cone is self-dual, spans H, has pointwise J-fixedness, unique cone-preserving commutant unitary, and every bounded normal positive functional has a unique cone vector with square-root norm estimate.
The positive cone of a standard representation
Predual, normal-weight and cone arguments for the orbit proof
SPATIAL.IDENTIFICATION
For a general nsf numerator and nsf commutant denominator, the coefficient form is closed/dense and equals the complete rectangular linking-weight Tomita energy form; modular conjugation actions identify the relative ordinary two-weight graph with the spatial derivative.
SI: OA-MOD-SI-02. The opposite weight and its modular data
SI: OA-MOD-SI-03. Unitary and antiunitary changes of representation
SI: OA-MOD-SI-04. Every finite rectangular intertwiner has a vector
SI: OA-MOD-SI-05. A diagonal weight separates four graph corners
SI: OA-MOD-SI-06. The first GNS column is the original representation
SI: OA-MOD-SI-07. The relative map has exactly the coefficient form
SI: OA-MOD-SI-12. Ordinary relative GNS maps are a specialization
SI: OA-MOD-SI-14. The balanced-matrix cocycle is independent of the reference
Earlier steps: SPATIAL.DIRECTFORM.
BOUND.BK
Bounded bicommutant/commutant framework, positive square roots and support/inverse order, four-unitary span, bounded-net limits and von Neumann polar/Douglas facts.
BK01–08 and the established bounded operator foundations
BOUND.CP
Concrete arbitrary-H Banach preduals, summable vector-pair functionals, positive normal-map criterion, and normal functional decomposition.
CP06–12: arbitrary-H concrete preduals and bounded supports
HILBERT.BASIC
Arbitrary-H completion, orthogonal projection, adjoints and bounded sesquilinear-form representation.
Hilbert-space completion, projection and Riesz representation
SCALAR.ANALYSIS
Scalar Lebesgue finite-measure integration, monotone/dominated convergence, absolute Fubini, change of variables/integration by parts, Gaussian transform MA03, and scalar Cauchy/residue tools for MA06-07.
Modular analytic scalar integrals and their measure prerequisites
Scalar integration, monotone and dominated convergence, and Fubini
Scalar measure and spectral foundations
ORBIT.MEASURE.MULTIPLICATION
Original metric probability-space multiplication MASA, Radon-Nikodym transport, support/density and existing state-integral contracts.
The orbit lesson and its named earlier state-integral and measure proofs
SPATIAL.BOUNDEDVECTORS
Bounded vectors, coefficient ideal and finite-energy polarization
Exact hypotheses and complete proof domains in the linked programme section.
Which vectors define bounded intertwiners?
Earlier steps: GNS.BASIC, BOUND.BK, HILBERT.BASIC.
WEIGHT.SUPPORTS
Finite-domain, null and supported faithful semifinite weight corners
Exact hypotheses and complete proof domains in the linked programme section.
The projection of the finite domain
Earlier steps: GNS.BASIC, BOUND.BK.
FORMS.CORES
Canonical closure, exact form cores and resolvent order
Exact hypotheses and complete proof domains in the linked programme section.
The completion test and canonical closure
Earlier steps: HILBERT.BASIC, SPECTRAL.CALCULUS, FORMS.FACTORIZATION.
FORMS.FACTORIZATION
Bounded contraction factorization inside the algebra
Exact hypotheses and complete proof domains in the linked programme section.
A factorization inside the algebra
Earlier steps: BOUND.BK, HILBERT.BASIC.
SPATIAL.DIRECTFORM
Full direct spatial energy construction, closure, null space, exact domains and form order
Exact hypotheses and complete proof domains in the linked programme section.
Conventions and the actual prerequisites
Earlier steps: SPATIAL.BOUNDEDVECTORS, WEIGHT.SUPPORTS, FORMS.CORES, NORMAL.WEIGHT.SUP, BOUND.CP, SPECTRAL.CALCULUS.
Human source context
Masamichi Takesaki, Theory of Operator Algebras II, Definition X.1.3 and Theorem X.1.21 / Corollary X.1.22(i), printed pages 240 and 253–255; publisher edition. The quotient qualification under a literal nonfaithful reading is stated in the covariant proof; the concrete commutant identity and the abstract quotient crossed product are distinct assertions.
Kawamura, Takemoto and Tomiyama, State extensions in transformation group C*-algebras, Section 1, provides a freely accessible comparison. This edition proves its own represented-algebra norm bound, purity, orbit equivalence and orthogonal-measure statements.
Downstream spatial convergence
The convergence consequence SI15 requires the spatial convergence theorem SS09. The SC01–10 and SI08/SI14 arguments use the direct-form and modular results linked above.
Further prerequisite results
The following results retain their exact hypotheses and domains. The general BF15 argument retains its action-topology and weak compact convex hull inputs. CI06 states the scalar multiplier’s closed-strip continuity explicitly.
TT1-IV.8.Ex1
8. The direct integral Hilbert space
TT1-V.6.1
6. Closed subspaces of a Hilbert space
TT1-V.7.1
TT1-V.7.2
TT1-V.7.3
The ergodic dichotomy and effective quotient
TT1-V.7.4
Bounded coefficient operators and left translations
Compact monotonicity handles arbitrary nets
TT1-V.7.5
Ultraweak L^1 averages and matrix coefficients
Locating the regular algebra in one tensor product
The normal comparison and its generators
Relabeling the coefficient system
TT2-A.7
TT2-A.8
Representation with the exact square-root domain
Nondense forms, lower semicontinuity, and symmetry
The completion test and canonical closure
Closedness and lower semicontinuity on all vectors
Closability and lower semicontinuity on the given domain
Cores determine the domain, not just a dense set of vectors
TT2-A.9
Problems with solutions — Problem 5
When a form sum equals an operator sum
TT2-A.10
Resolvents as energy minimizers
One resolvent inequality recovers the whole form order
A factorization inside the algebra
TT2-A.11
The domain of an increasing limit
Strong convergence of resolvents, including nets
Logarithms and imaginary powers
TT2-A.12
Compatible pairs, intersections, and sums
Evaluation spaces and quotient completeness
Interpolating a bounded linear map