Packages faithfulness together with equality of the represented image and the compact operators.
Statement
Let be a complex -algebra, with no unit assumed, let be a complex Hilbert space, and let be a -representation. The predicate below asserts that identifies exactly with the compact operators on .
Definition
The three conditions are
Equivalently, is faithful and ; hence is -isomorphic to through the specified representation.
Assumptions
No separability, irreducibility or singleton-spectrum hypothesis is part of this definition. A unit is not assumed.
Conclusion
The output is a property of the given representation , not merely an assertion that some abstract -isomorphism exists.
The complete definition is copied from the pinned source.
/-- A nonunital star representation that is injective and has precisely the
compact operators as its range. This is the unbundled form of a star
isomorphism with `K(H)`. -/
def IsCompactOperatorModel [NonUnitalCStarAlgebra A]
(e : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) : Prop :=
Function.Injective e ∧
(∀ a : A, IsCompactOperator (e a)) ∧
∀ T : H →L[ℂ] H, IsCompactOperator T → ∃ a : A, e a = T
/-- A nonunital star representation that is injective and has precisely the
compact operators as its range. This is the unbundled form of a star
isomorphism with `K(H)`. -/
def IsCompactOperatorModel [NonUnitalCStarAlgebra A]
(e : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) : Prop :=
Function.Injective e ∧
(∀ a : A, IsCompactOperator (e a)) ∧
∀ T : H →L[ℂ] H, IsCompactOperator T → ∃ a : A, e a = T
/-- A nonunital star representation that is injective and has precisely the
compact operators as its range. This is the unbundled form of a star
isomorphism with `K(H)`. -/
def IsCompactOperatorModel [NonUnitalCStarAlgebra A]
(e : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) : Prop :=
Function.Injective e ∧
(∀ a : A, IsCompactOperator (e a)) ∧
∀ T : H →L[ℂ] H, IsCompactOperator T → ∃ a : A, e a = T
/-- A nonunital star representation that is injective and has precisely the
compact operators as its range. This is the unbundled form of a star
isomorphism with `K(H)`. -/
def IsCompactOperatorModel [NonUnitalCStarAlgebra A]
(e : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) : Prop :=
Function.Injective e ∧
(∀ a : A, IsCompactOperator (e a)) ∧
∀ T : H →L[ℂ] H, IsCompactOperator T → ∃ a : A, e a = T
/-- A nonunital star representation that is injective and has precisely the
compact operators as its range. This is the unbundled form of a star
isomorphism with `K(H)`. -/
def IsCompactOperatorModel [NonUnitalCStarAlgebra A]
(e : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) : Prop :=
Function.Injective e ∧
(∀ a : A, IsCompactOperator (e a)) ∧
∀ T : H →L[ℂ] H, IsCompactOperator T → ∃ a : A, e a = T
/-- A nonunital star representation that is injective and has precisely the
compact operators as its range. This is the unbundled form of a star
isomorphism with `K(H)`. -/
def IsCompactOperatorModel [NonUnitalCStarAlgebra A]
(e : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) : Prop :=
Function.Injective e ∧
(∀ a : A, IsCompactOperator (e a)) ∧
∀ T : H →L[ℂ] H, IsCompactOperator T → ∃ a : A, e a = T
/-- A nonunital star representation that is injective and has precisely the
compact operators as its range. This is the unbundled form of a star
isomorphism with `K(H)`. -/
def IsCompactOperatorModel [NonUnitalCStarAlgebra A]
(e : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) : Prop :=
Function.Injective e ∧
(∀ a : A, IsCompactOperator (e a)) ∧
∀ T : H →L[ℂ] H, IsCompactOperator T → ∃ a : A, e a = T