Positive Linear Functional on C*-Algebra is Increasing on Hermitian Elements

Theorem

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra with identity element ${\mathbf 1}_A$.

Let $f : A \to \C$ be a positive linear functional.

Let $\le_A$ be the canonical preordering of $A$.

Let $x, y \in A$ be Hermitian such that $x \le_A y$.

Then $\map f x \le \map f y$.

Proof

From the definition of the canonical preordering, we have:

$y - x$ is positive.

From the definition of a positive linear functional we have:

$\map f {y - x} \ge 0$

Since $f$ is linear, we have:

$\map f y - \map f x \ge 0$

From Positive Linear Functional on C*-Algebra is Real on Hermitian Elements, we have $\map f x \in \R$ and $\map f y \in \R$.

Hence:

$\map f x \le \map f y$

$\blacksquare$