Positive linear maps #
This file defines positive linear maps as a linear map that is also an order homomorphism.
Implementation notes #
We do not define PositiveLinearMapClass
to avoid adding a class that mixes order and algebra.
One can achieve the same effect by using a combination of LinearMapClass
and OrderHomClass
.
We nevertheless use the namespace for lemmas using that combination of typeclasses.
Notes #
More substantial results on positive maps such as their continuity can be found in
the Analysis/CStarAlgebra
folder.
A positive linear map is a linear map that is also an order homomorphism.
- toFun : E₁ → E₂
Instances For
Notation for a PositiveLinearMap
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Reinterpret an element of a type of positive linear maps as a positive linear map.
Equations
- PositiveLinearMapClass.toPositiveLinearMap f = { toLinearMap := ↑f, monotone' := ⋯ }
Instances For
Reinterpret an element of a type of positive linear maps as a positive linear map.
Equations
- PositiveLinearMapClass.instCoeToLinearMap = { coe := fun (f : F) => PositiveLinearMapClass.toPositiveLinearMap f }
Equations
- PositiveLinearMap.instFunLike = { coe := fun (f : E₁ →ₚ[R] E₂) => f.toFun, coe_injective' := ⋯ }
Define a positive map from a linear map that maps nonnegative elements to nonnegative elements
Equations
- PositiveLinearMap.mk₀ f hf = { toLinearMap := f, monotone' := ⋯ }