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An extreme dual functional has a pure-state unitary translate

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MathlibAnnex.Analysis.CStarAlgebra.exists_pureState_unitaryTranslate_of_mem_extremePoints_dual_closedBall

theorem

Expresses each extreme dual functional as a unitary translate of a pure state.

Statement

Let be a complex C*-algebra, possibly without an identity. Let be the space of bounded complex-linear functionals on , and let . Extremality is taken for real convex combinations. Write for the forced C*-unitization and identify with its canonical ideal in .

For a unitary , left multiplication preserves this ideal, and defines the dual translate

If , there are a pure state on and a unitary such that

(1)

A pure state is a positive norm-one bounded complex-linear functional that is extreme, for real convexity, in the state space. The two identities in (1) concern the same and .

Assumptions

The nonzero-algebra hypothesis ensures that an extreme element of the dual unit ball has norm one. The source also fixes the compatible C*-order used to express positivity. No identity in , separability or finite dimension is required. The unitary belongs to the forced unitization even when happens to have an identity.

Conclusion

The same pure state and unitary satisfy both functional equalities in (1). Positivity, norm one and purity concern on the original algebra. The assertion does not require or assert uniqueness of the unitary.

Main citations

  • Exact decomposition declaration: exists_pureState_unitaryTranslate_of_mem_extremePoints_dual_closedBall.
  • Ryotaro Tanaka, arXiv:2504.05551v1, §5: the paper's facial and pure-state comparison. The forced-unitization and original-algebra coefficient construction here follow the pinned source.

Lean source signature (exact)

The declaration prefix below is a contiguous source excerpt, ending before its proof assignment.

theorem exists_pureState_unitaryTranslate_of_mem_extremePoints_dual_closedBall
    {rho : StrongDual ℂ A}
    (hrho : rho ∈ (closedBall (0 : StrongDual ℂ A) 1).extremePoints ℝ) :
    ∃ phi : StrongDual ℂ A, IsPureNonUnitalState A phi ∧
      ∃ u : unitary (Unitization ℂ A),
        unitaryTranslate u rho = phi ∧ rho = unitaryTranslate (star u) phi
In the source Mathematical meaning
Exact declaration{A : Type u}, [NonUnitalCStarAlgebra A], [Nontrivial A] The nonzero complex C*-algebra. An identity is not assumed.
Exact declaration[PartialOrder A], [StarOrderedRing A] The compatible C*-order needed for positive functionals and states.
Exact declarationrho : StrongDual ℂ A The bounded complex-linear functional on the original algebra.
Exact declarationhrho : rho ∈ (closedBall ... 1).extremePoints ℝ Extremality in the entire norm-closed dual unit ball, with real convexity. It implies norm one here.
Exact declaration∃ phi : StrongDual ℂ A, ... One bounded complex-linear functional is selected in the same dual space as . Positivity is supplied by the following predicate, not by this type alone.
Exact declarationIsPureNonUnitalState A phi Extremality among positive norm-one functionals, including membership in that state space. It supplies positivity and norm one for the same .
Exact declaration∃ u : unitary (Unitization ℂ A), ... One unitary in the forced unitization is selected after the pure state; it need not be an element of .
Exact declarationunitaryTranslate u rho = phi The equality of the whole functionals , hence equality at every .
The equality with Exact declarationstar u The whole inverse-translation equality , using the same and in both equalities.
Exact surrounding source context (separate excerpts)

Source lines 35–35:

variable {A : Type u} [NonUnitalCStarAlgebra A] [Nontrivial A]

Source lines 56–56:

variable [PartialOrder A] [StarOrderedRing A]

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

Technical documentation

Exact declaration for the source guide
theorem exists_pureState_unitaryTranslate_of_mem_extremePoints_dual_closedBall
    {rho : StrongDual ℂ A}
    (hrho : rho ∈ (closedBall (0 : StrongDual ℂ A) 1).extremePoints ℝ) :
    ∃ phi : StrongDual ℂ A, IsPureNonUnitalState A phi ∧
      ∃ u : unitary (Unitization ℂ A),
        unitaryTranslate u rho = phi ∧ rho = unitaryTranslate (star u) phi := by
Back to the reading-guide item
Exact declaration for the source guide
theorem exists_pureState_unitaryTranslate_of_mem_extremePoints_dual_closedBall
    {rho : StrongDual ℂ A}
    (hrho : rho ∈ (closedBall (0 : StrongDual ℂ A) 1).extremePoints ℝ) :
    ∃ phi : StrongDual ℂ A, IsPureNonUnitalState A phi ∧
      ∃ u : unitary (Unitization ℂ A),
        unitaryTranslate u rho = phi ∧ rho = unitaryTranslate (star u) phi := by
Back to the reading-guide item
Exact declaration for the source guide
theorem exists_pureState_unitaryTranslate_of_mem_extremePoints_dual_closedBall
    {rho : StrongDual ℂ A}
    (hrho : rho ∈ (closedBall (0 : StrongDual ℂ A) 1).extremePoints ℝ) :
    ∃ phi : StrongDual ℂ A, IsPureNonUnitalState A phi ∧
      ∃ u : unitary (Unitization ℂ A),
        unitaryTranslate u rho = phi ∧ rho = unitaryTranslate (star u) phi := by
Back to the reading-guide item
Exact declaration for the source guide
theorem exists_pureState_unitaryTranslate_of_mem_extremePoints_dual_closedBall
    {rho : StrongDual ℂ A}
    (hrho : rho ∈ (closedBall (0 : StrongDual ℂ A) 1).extremePoints ℝ) :
    ∃ phi : StrongDual ℂ A, IsPureNonUnitalState A phi ∧
      ∃ u : unitary (Unitization ℂ A),
        unitaryTranslate u rho = phi ∧ rho = unitaryTranslate (star u) phi := by
Back to the reading-guide item
Exact declaration for the source guide
theorem exists_pureState_unitaryTranslate_of_mem_extremePoints_dual_closedBall
    {rho : StrongDual ℂ A}
    (hrho : rho ∈ (closedBall (0 : StrongDual ℂ A) 1).extremePoints ℝ) :
    ∃ phi : StrongDual ℂ A, IsPureNonUnitalState A phi ∧
      ∃ u : unitary (Unitization ℂ A),
        unitaryTranslate u rho = phi ∧ rho = unitaryTranslate (star u) phi := by
Back to the reading-guide item
Exact declaration for the source guide
theorem exists_pureState_unitaryTranslate_of_mem_extremePoints_dual_closedBall
    {rho : StrongDual ℂ A}
    (hrho : rho ∈ (closedBall (0 : StrongDual ℂ A) 1).extremePoints ℝ) :
    ∃ phi : StrongDual ℂ A, IsPureNonUnitalState A phi ∧
      ∃ u : unitary (Unitization ℂ A),
        unitaryTranslate u rho = phi ∧ rho = unitaryTranslate (star u) phi := by
Back to the reading-guide item
Exact declaration for the source guide
theorem exists_pureState_unitaryTranslate_of_mem_extremePoints_dual_closedBall
    {rho : StrongDual ℂ A}
    (hrho : rho ∈ (closedBall (0 : StrongDual ℂ A) 1).extremePoints ℝ) :
    ∃ phi : StrongDual ℂ A, IsPureNonUnitalState A phi ∧
      ∃ u : unitary (Unitization ℂ A),
        unitaryTranslate u rho = phi ∧ rho = unitaryTranslate (star u) phi := by
Back to the reading-guide item
Exact declaration for the source guide
theorem exists_pureState_unitaryTranslate_of_mem_extremePoints_dual_closedBall
    {rho : StrongDual ℂ A}
    (hrho : rho ∈ (closedBall (0 : StrongDual ℂ A) 1).extremePoints ℝ) :
    ∃ phi : StrongDual ℂ A, IsPureNonUnitalState A phi ∧
      ∃ u : unitary (Unitization ℂ A),
        unitaryTranslate u rho = phi ∧ rho = unitaryTranslate (star u) phi := by
Back to the reading-guide item
Exact declaration for the source guide
theorem exists_pureState_unitaryTranslate_of_mem_extremePoints_dual_closedBall
    {rho : StrongDual ℂ A}
    (hrho : rho ∈ (closedBall (0 : StrongDual ℂ A) 1).extremePoints ℝ) :
    ∃ phi : StrongDual ℂ A, IsPureNonUnitalState A phi ∧
      ∃ u : unitary (Unitization ℂ A),
        unitaryTranslate u rho = phi ∧ rho = unitaryTranslate (star u) phi := by
Back to the reading-guide item