MATHLIBANNEX / CANONICAL DECLARATION CARD

Irreducible dimension equal to a fixed number

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MathlibAnnex.Analysis.CStarAlgebra.forall_irreducible_finrank_eq_iff_component_compact_homeomorph

theorem

Characterizes one fixed irreducible dimension by finite-dimensional compact component models.

Statement

Let be a nonzero complex C*-algebra, and let .

For a complex C*-algebra , write for its geometric structure space. Its points are the unions of kernels of norm-one bounded complex-linear functionals supporting a maximal proper face of the closed unit ball of ; faces use real convexity. Write for the hull-kernel closure on :

These are the geometric structure and its closure, with the norm-one description justified by the support-functional norm theorem.

For , let be its finite-chain component: the points reachable from by a finite chain in which adjacent points are equal or form a nonclosed two-point set. This uses the linked-pair relation and its generated components, rather than topological connectedness. Give the induced closure

For a complex Hilbert space , let be the algebra of compact operators on . In particular, denotes closure on , where is itself the algebra to which the geometric structure construction is applied.

The following conditions are equivalent:

  • For every irreducible -representation on a complex Hilbert space ,
(1)
  • For every , there are a nonzero finite-dimensional complex Hilbert space with and a bijection satisfying
(2)

The same model has dimension and supports the homeomorphism . The number is fixed for all components, while and may depend on . Equality (2) expresses continuity of the same and for arbitrary subsets; denotes the image of a set.

Assumptions

The algebra is nonzero; no assumption is made. Representations are -homomorphisms into bounded operators, without a requirement of unitality. Irreducibility includes a nonzero action, but does not require faithfulness. All Hilbert spaces are complex and complete. Finite-dimensionality is a conclusion on the representation side and a requirement on the model side. It is separate from the numerical dimension equation. No separability hypothesis is added.

Conclusion

Every irreducible representation space is finite-dimensional of the same dimension exactly when each geometric component has a compact-operator model of that dimension. For nonzero , both conditions fail at , as specified by the zero-dimension exclusion and the equivalence above. This does not add positivity of as a hypothesis.

Main citations

The fixed-dimension characterization specializes the general dimension-condition theorem to equality with . The finite-dimensional compact-model condition retains both finiteness and the numerical condition on the same target. The same-map upgrade theorem supplies full continuity without replacing the finite comparison or its inverse. The independent-universe dimension transport and the exclusion of zero dimension specify those two scope points.

The finite-homeomorphism fields, closure continuity, and homeomorphism fields specify the map conditions used here. The closure-transport identity expresses a homeomorphism by equality of closures of arbitrary sets.

Lean source signature (exact)

The preceding variable declaration introduces and its nontriviality. The local spectral order is used for positivity and pure states. The include hA in retains nonzero among this theorem's parameters, although the main excerpt does not repeat these ambient binders. Both separate excerpts are shown in the source context below.

HomogeneousCharacterization.lean. The contiguous declaration prefix ends before the proof assignment.

theorem forall_irreducible_finrank_eq_iff_component_compact_homeomorph (n : ℕ) :
    (∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
      ∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →
        FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔
    ∀ I : NormedSpace.GeometricStructureSpace ℂ A,
      ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
        ((NormedSpace.geometricStructureClosure ℂ A).subspace
          ((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n)

The component condition uses the following separate exact definition, including its complete right-hand side: HomogeneousCharacterization.lean.

def HasFiniteDimensionalCompactHomeomorph.{uH, uX} {X : Type uX}
    (c : ClosureSpace X) (P : ℕ → Prop) : Prop :=
  ∃ (H : Type uH) (instN : NormedAddCommGroup H),
    letI := instN
    ∃ instI : InnerProductSpace ℂ H,
      letI := instI
      ∃ instC : CompleteSpace H,
        letI := instC
        ∃ instT : Nontrivial H,
          letI := instT
          FiniteDimensional ℂ H ∧ P (Module.finrank ℂ H) ∧
            Nonempty (Homeomorph c
              (NormedSpace.geometricStructureClosure ℂ
                (ContinuousLinearMap.compactOperatorStarSubalgebra ℂ H)))
In the source Mathematical meaning
Exact declarationAmbient inputs
variable {A : Type u} [NonUnitalCStarAlgebra A] [hA : Nontrivial A]
attribute [local instance] _root_.CStarAlgebra.spectralOrder
_root_.CStarAlgebra.spectralOrderedRing		 The nonzero complex C*-algebra . The two local instances give its compatible spectral order, used for positivity and pure states. They do not introduce another choice of order. The separate include hA in shown below retains nontriviality in this theorem; no unit is required.
Exact declarationRepresentation quantifiers
(∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →		 Every complex Hilbert space and every irreducible -representation . Here source L is the representation space called in the Statement. Its norm, complex inner product and completeness are required by the three bracketed inputs. Irreducibility includes a nonzero action, not faithfulness.
Exact declarationRepresentation condition
FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔		 The representation space is finite-dimensional and satisfies , as in (1). Both clauses are required. The parameter (n : ℕ) in the main declaration fixes one natural number throughout the equivalence.
Exact declarationComponent quantifier
∀ I : NormedSpace.GeometricStructureSpace ℂ A,		 Every point . The model space and comparison map required after this quantifier may depend on . The number remains fixed.
Exact declarationComparison applied to the component
ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
((NormedSpace.geometricStructureClosure ℂ A).subspace
((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n)		 Apply the displayed comparison definition to the closure induced on . geometricStructureClosure ℂ A is the closure on ; its component I is the point set . subspace makes that set a closure space, with
for . This closure is the argument c of the definition above, and its underlying set is X. The final argument (fun m => m = n) assigns the condition on the model dimension.
Exact declarationModel and inner product
∃ (H : Type uH) (instN : NormedAddCommGroup H),
letI := instN
∃ instI : InnerProductSpace ℂ H,
letI := instI		 The condition requires a model space , called H in this definition, with the following structures. instN gives its normed additive group, and instI its complex inner product. Each following letI uses that choice on the same space, first to form InnerProductSpace ℂ H, then to interpret the remaining Hilbert-space conditions.
Exact declarationCompleteness and nonzero model
∃ instC : CompleteSpace H,
letI := instC
∃ instT : Nontrivial H,
letI := instT		 instC requires this same to be complete; instT requires . Their letI declarations retain these structures when forming bounded operators, the compact-operator algebra and its geometric closure. They do not choose a second space.
Exact declarationDimension of the model
FiniteDimensional ℂ H ∧ P (Module.finrank ℂ H) ∧		 This same model must be finite-dimensional. Substituting gives . The conjunction separately retains finite-dimensionality and the numerical equation. It does not replace them by the finrank equation alone.
Exact declarationHomeomorphism existence
Nonempty (Homeomorph c
(NormedSpace.geometricStructureClosure ℂ
(ContinuousLinearMap.compactOperatorStarSubalgebra ℂ H)))		 Nonempty requires an element of this Homeomorph type: one bijection with both continuity fields below. Its target closure is on the geometric structure of the compact-operator algebra on the same model , not on itself.
Exact declarationHomeomorphism and forward field
structure Homeomorph (c : ClosureSpace α) (d : ClosureSpace β) extends α ≃ β where
continuous_toFun : c.Continuous d toFun		 extends α ≃ β supplies the bijection, extracted as e.toEquiv. Its forward map toFun is . In this application, and , so the first field says
for every .
Exact declarationInverse field of the same homeomorphism
continuous_invFun : d.Continuous c invFun		 invFun, also written e.toEquiv.symm, is the inverse of that same . The second field says
for every . Together the two fields give the arbitrary-set closure equality (2).
Exact declarationContinuity definition
def Continuous (c : ClosureSpace α) (d : ClosureSpace β) (f : α → β) : Prop :=
∀ s : Set α, f '' c.closure s ⊆ d.closure (f '' s)		 For a map , continuity means
for every . Source α, β, c, d, f denote ; the double prime denotes the set image . Unlike FinitelyContinuous, this definition has no finiteness restriction.

Size scope: the principal equivalence fixes the universe of and quantifies both representation spaces and compact-model spaces in that same universe. The separately cited transport theorem allows an independently specified Hilbert-space universe; it does not change the quantifiers of this declaration.

Exact surrounding source context (separate excerpts)

HomogeneousCharacterization.lean.

variable {A : Type u} [NonUnitalCStarAlgebra A] [hA : Nontrivial A]
attribute [local instance] _root_.CStarAlgebra.spectralOrder
  _root_.CStarAlgebra.spectralOrderedRing

HomogeneousCharacterization.lean.

include hA in

Read exact source with highlighted declaration · Raw UTF-8 source · LFH Companion ·

Technical documentation

Exact declaration for the source guide
theorem forall_irreducible_finrank_eq_iff_component_compact_homeomorph (n : ℕ) :
    (∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
      ∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →
        FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔
    ∀ I : NormedSpace.GeometricStructureSpace ℂ A,
      ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
        ((NormedSpace.geometricStructureClosure ℂ A).subspace
          ((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n) :=
Back to the reading-guide item
Exact declaration for the source guide
theorem forall_irreducible_finrank_eq_iff_component_compact_homeomorph (n : ℕ) :
    (∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
      ∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →
        FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔
    ∀ I : NormedSpace.GeometricStructureSpace ℂ A,
      ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
        ((NormedSpace.geometricStructureClosure ℂ A).subspace
          ((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n) :=
Back to the reading-guide item
Exact declaration for the source guide
theorem forall_irreducible_finrank_eq_iff_component_compact_homeomorph (n : ℕ) :
    (∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
      ∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →
        FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔
    ∀ I : NormedSpace.GeometricStructureSpace ℂ A,
      ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
        ((NormedSpace.geometricStructureClosure ℂ A).subspace
          ((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n) :=
Back to the reading-guide item
Exact declaration for the source guide
theorem forall_irreducible_finrank_eq_iff_component_compact_homeomorph (n : ℕ) :
    (∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
      ∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →
        FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔
    ∀ I : NormedSpace.GeometricStructureSpace ℂ A,
      ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
        ((NormedSpace.geometricStructureClosure ℂ A).subspace
          ((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n) :=
Back to the reading-guide item
Exact declaration for the source guide
theorem forall_irreducible_finrank_eq_iff_component_compact_homeomorph (n : ℕ) :
    (∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
      ∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →
        FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔
    ∀ I : NormedSpace.GeometricStructureSpace ℂ A,
      ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
        ((NormedSpace.geometricStructureClosure ℂ A).subspace
          ((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n) :=
Back to the reading-guide item
Exact declaration for the source guide
theorem forall_irreducible_finrank_eq_iff_component_compact_homeomorph (n : ℕ) :
    (∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
      ∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →
        FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔
    ∀ I : NormedSpace.GeometricStructureSpace ℂ A,
      ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
        ((NormedSpace.geometricStructureClosure ℂ A).subspace
          ((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n) :=
Back to the reading-guide item
Exact declaration for the source guide
theorem forall_irreducible_finrank_eq_iff_component_compact_homeomorph (n : ℕ) :
    (∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
      ∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →
        FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔
    ∀ I : NormedSpace.GeometricStructureSpace ℂ A,
      ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
        ((NormedSpace.geometricStructureClosure ℂ A).subspace
          ((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n) :=
Back to the reading-guide item
Exact declaration for the source guide
theorem forall_irreducible_finrank_eq_iff_component_compact_homeomorph (n : ℕ) :
    (∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
      ∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →
        FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔
    ∀ I : NormedSpace.GeometricStructureSpace ℂ A,
      ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
        ((NormedSpace.geometricStructureClosure ℂ A).subspace
          ((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n) :=
Back to the reading-guide item
Exact declaration for the source guide
theorem forall_irreducible_finrank_eq_iff_component_compact_homeomorph (n : ℕ) :
    (∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
      ∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →
        FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔
    ∀ I : NormedSpace.GeometricStructureSpace ℂ A,
      ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
        ((NormedSpace.geometricStructureClosure ℂ A).subspace
          ((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n) :=
Back to the reading-guide item
Exact declaration for the source guide
theorem forall_irreducible_finrank_eq_iff_component_compact_homeomorph (n : ℕ) :
    (∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
      ∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →
        FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔
    ∀ I : NormedSpace.GeometricStructureSpace ℂ A,
      ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
        ((NormedSpace.geometricStructureClosure ℂ A).subspace
          ((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n) :=
Back to the reading-guide item
Exact declaration for the source guide
theorem forall_irreducible_finrank_eq_iff_component_compact_homeomorph (n : ℕ) :
    (∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
      ∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →
        FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔
    ∀ I : NormedSpace.GeometricStructureSpace ℂ A,
      ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
        ((NormedSpace.geometricStructureClosure ℂ A).subspace
          ((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n) :=
Back to the reading-guide item
Exact declaration for the source guide
theorem forall_irreducible_finrank_eq_iff_component_compact_homeomorph (n : ℕ) :
    (∀ (L : Type u) [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L],
      ∀ pi : NonUnitalCStarRepresentation A L, pi.IsIrreducible →
        FiniteDimensional ℂ L ∧ Module.finrank ℂ L = n) ↔
    ∀ I : NormedSpace.GeometricStructureSpace ℂ A,
      ClosureSpace.HasFiniteDimensionalCompactHomeomorph.{u, u}
        ((NormedSpace.geometricStructureClosure ℂ A).subspace
          ((NormedSpace.geometricStructureClosure ℂ A).component I)) (fun m => m = n) :=
Back to the reading-guide item