PhyslibAlpha.ProbabilisticTheory.Channel.Normal

Let $E_1$ and $E_2$ be ordered vector spaces over $\mathbb{R}$. A positive linear map $\phi: E_1 \to E_2$ is said to be **normal** if it preserves the suprema of directed sets. That is, for any directed set $D \subseteq E_1$ that has a supremum, the image $\phi(D)$ also has a supremum and satisfies $\phi(\sup D) = \sup \phi(D)$.

Let $E$ be an ordered vector space over the real numbers $\mathbb{R}$. The identity positive linear map $\text{id}_E: E \to E$ is normal. That is, for any directed set $D \subseteq E$ that has a supremum $\sup D$, the image of the set under the identity map also has a supremum and satisfies $\text{id}_E(\sup D) = \sup (\text{id}_E(D))$.

Let $E_1, E_2$, and $E_3$ be ordered vector spaces over the real numbers $\mathbb{R}$. Let $\phi: E_1 \to E_2$ and $\psi: E_2 \to E_3$ be positive linear maps. If $\phi$ and $\psi$ are normal maps (that is, they preserve the suprema of directed sets), then their composition $\psi \circ \phi$ is also normal.

Let $E$ and $F$ be ordered vector spaces over $\mathbb{R}$. Let $\phi: E \to F$ be a normal positive linear map (meaning $\phi$ preserves the suprema of directed sets). Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of elements in $E$ such that $f_n \ge 0$ for all $n$. If the sequence of partial sums $\sum_{n=0}^{N-1} f_n$ has a least upper bound $x \in E$, then the sequence of partial sums of the images $\sum_{n=0}^{N-1} \phi(f_n)$ has $\phi(x)$ as its least upper bound.

Let $E$ and $F$ be ordered vector spaces. A channel $\phi: E \to F$ (defined as a unital positive linear map) is said to be **normal** if its underlying positive linear map preserves the least upper bounds (suprema) of directed sets. Specifically, if a directed set $D \subseteq E$ has a supremum $\sup D$, then the image set $\phi(D)$ also has a supremum and satisfies $\phi(\sup D) = \sup \phi(D)$.

The identity map $\text{id}_E$ on an ordered vector space $E$ is normal. That is, for any directed set $D \subseteq E$ that has a least upper bound $\sup D$, the identity map preserves this bound such that $\text{id}_E(\sup D) = \sup \text{id}_E(D)$.

Let $E, F$, and $G$ be ordered vector spaces. Let $\phi: E \to F$ and $\psi: F \to G$ be normal channels (unital positive linear maps that preserve the least upper bounds of directed sets). Then their composition $\psi \circ \phi: E \to G$ is also a normal channel.