Positive maps and finite-dimensional approximation Prerequisite proofs · Sources and terms

Open Mathematics Courses · Prerequisite proofs · Sources and terms

Positive maps and finite-dimensional approximation

50 lessons with 504 worked solutions. Written and self-checked by GPT-6.1 Sol (OpenAI), Ultra, relative to the stated prerequisites; no independent review claimed.

Can a finite matrix measurement preserve a state and recover its observables? Start with explicit sampling, compression and a two-level channel. The other routes ask how to extend or lift such maps, recognize approximation by tensor tests, and turn approximate maps into actual finite-dimensional algebras. Infinite tensor products, asymptotic algebras and abelian observations provide complementary laboratories.

Choose a question below. Each lesson has one home, and each route identifies the inputs for its general proofs. The opening examples can be studied before returning with those inputs; the route list is not a claim that prerequisites disappear.

Finite measurements, states and recovery

Begin with the sampling and compression models in the CP lesson, then the explicit two-level laboratory in the tracial lesson. It preserves the state and still loses an off-diagonal observable. The general recording and trace-preserving theorems answer which additional control repairs that loss.

Before the general proofs. The weighted-adjoint proof uses finite-trace Hilbert spaces. The final trace-preserving models require the semidiscreteness equivalence in the tensor-and-injectivity route and its stated unital-map correction. The opening laboratory is proved directly before those inputs are used.

  1. Completely positive finite models
  2. Tracial adjoints and rational matrix models
  3. Trace-preserving finite models

Extend a map, represent its norms, lift through a quotient

A measurement is useful only if its domain can grow. Compare extension with unchanged cb norm, intrinsic matrix norms, and lifting through a quotient. Extension and lifting have different domains and preserve different data; the operator-space representation identifies the matrix tests behind both.

Before the general proofs. Use the CP and dilation results of the finite-measurement route. Complete positivity, complete boundedness and a quotient lift are distinct hypotheses; each lesson states the precise one it needs.

  1. Completely bounded extension and factorization
  2. From matrix norms to operators
  3. Lifting completely positive maps

Detect approximation with tensor products and injectivity

Tensor tests connect norm approximation of C*-algebras with weak finite models of von Neumann algebras. Follow semidiscreteness into hypertraces, then averaging and the continuous core; return to the bidual to characterize nuclearity and prove permanence under extensions.

Before the general proofs. Finite CP models and spatial/maximal tensor products come first. The averaging and core arguments use the exact modular and crossed-product proofs listed in the prerequisite register. Norm CP approximation and ultraweak semidiscreteness use different topologies.

  1. Tensor positivity and nuclearity
  2. Finite models of a von Neumann algebra
  3. Hypertraces and finite injective algebras
  4. Averaging, crossed products, and injectivity
  5. Nuclear biduals and extensions

Build and compare infinite laboratories

Reference states specify infinite tensor representations. CAR provides explicit local matrix observables; the tracial model gives the hyperfinite factor. Compare its automorphisms, quasifree core, group constructions and free-group obstructions to learn which infinite systems can be approximated by finite algebras.

Before the general proofs. Finite tensor products and GNS representations are needed first. The periodic CAR subtype uses its explicitly selected modular-invariant input. Connected-group statements keep their declared representation-theoretic scope.

  1. Infinite tensor products and their reference states
  2. Fermions, Fock space and quasi-free factors
  3. Local approximation and the hyperfinite finite factor
  4. Constant CAR states and the periodic core
  5. Bogoliubov automorphisms and the Hilbert–Schmidt criterion
  6. Finite outer actions and Bernoulli shifts
  7. Continuous outer actions of locally compact groups
  8. Central sequences, fullness and free group factors
  9. Norm averaging in a free group algebra
  10. Projections in the reduced free group algebra
  11. Finite measured groupoids and AFD algebras

Make asymptotic constructions independent of representatives

Small errors can accumulate when sequences are passed to quotient algebras. Start with exact central-sequence lifts and fullness, use tensor absorption as a concrete application, and compare fast and slow reindexing before handling multiplier normality and uniform predual control.

Before the general proofs. The infinite laboratories supply the examples. Follow the declared null-ideal and liftability conditions at each step: ordinary sequences, ultraproduct multipliers and chosen semilifts cannot be identified without proof.

  1. Central sequence algebras and exact lifts
  2. Fullness, hypercentrality and ultrafilter corners
  3. Strong stability and tensor absorption
  4. Centrally trivial automorphisms and decreasing AFD factors
  5. Multiplier ultraproducts and normal embeddings
  6. Ultraproduct expectations and semi-lift ambiguity
  7. Fast reindexing with liftable actions
  8. Slow reindexing and semi-lift compatibility
  9. Ordinary multipliers and predual compactness
  10. Index selection and uniform multiplier control

Recover information from abelian observations

Pinching asks what averaging over an abelian algebra can detect. Relative commutants and normalizers separate distinct notions of maximality; group examples and double-coset multiplicities then show how abelian observations distinguish pairs of algebras.

Before the general proofs. Use the factor and trace tools in the laboratories, and the stated conditional-expectation prerequisites. Normalizer, commutant and pair-multiplicity conclusions have separate hypotheses.

  1. Abelian pinching and relative commutants
  2. MASAs in expected semifinite subfactors
  3. Semiregular MASAs and scalar pinching
  4. Group MASAs and singular affine examples
  5. Double cosets and intrinsic MASA multiplicity
  6. Finite-field towers and prescribed MASA multiplicities

Turn tensor tests into a sharp bilinear estimate

A fourth-moment truncation controls the symmetric tensor norm. Completion and state selection convert that estimate into domination by states, and complex interpolation improves the coefficient to its sharp value. Follow the error term as well as the constant through all three lessons.

Before the general proofs. Use C*-functional calculus, projective tensor products, compact state spaces and the listed scalar complex-analysis inputs. The four-state and maximum-of-row/column normalizations have different numerical coefficients.

  1. Symmetric tensor norms and Rademacher truncation
  2. Tensor completion, injectivity and state domination
  3. The sharp noncommutative bilinear inequality

Replace approximate maps by actual finite-dimensional algebras

There are two finite-factor methods. Balanced Kraus families and bounded unitary couplings move trace-preserving matrix models into a subalgebra; the alternative repairs finite-rank models and small corners before using maximality. Dyadic approximation handles the properly infinite case, and central assembly and tracial envelopes restore full nonfactor generality.

Before the general proofs. First complete the CP/injectivity equivalence, the tracial measurement tools, and the required central-sequence results. The second finite-factor method also uses the complete weak-compactness and fixed-point providers. State preservation by itself is not the finite-subalgebra conclusion.

  1. Properly infinite injective algebras and dyadic approximation
  2. Central traces and AFD finite algebras
  3. Balancing Kraus families and unitary couplings
  4. Unitary couplings and finite injective factors
  5. Invariant states and finite-rank projections
  6. Repairing small matrix corners
  7. Small corners and the second injective proof
  8. Central disintegration and measurable matrix assembly
  9. Injective algebras and separable tracial envelopes

The full local injectivity/AFD result includes arbitrary von Neumann algebras through central assembly, separable tracial envelopes and normal center support cuts. The central type II structure results keep their separability hypotheses, and the periodic CAR subtype keeps its stated modular-invariant prerequisite. Both finite-factor proof methods and all worked solutions are included. General weights, expectations and crossed-product constructions use exact earlier programme proofs identified in the lessons.