PhyslibAlpha.QuantumMechanics.OperatorAlgebra.Observables.Effects

Let $A$ be a unital $C^*$-algebra and $a \in A$ be an observable (a self-adjoint element). Then $a$ is an effect, defined as satisfying the operator inequality $0 \le a \le 1$, if and only if the spectrum of $a$, denoted $\sigma(a)$, is a subset of the closed interval $[0, 1]$.

Let $A$ be an operator algebra and $a \in A$ be an observable (a self-adjoint element). Given a continuous function $f: \mathbb{R} \to \mathbb{R}$ and a proof that $f$ maps the spectrum of $a$, denoted $\sigma(a)$, into the closed interval $[0, 1]$, this definition constructs an effect from the element $f(a)$ obtained via the continuous functional calculus. Specifically, the resulting element satisfies the operator inequality $0 \le f(a) \le \mathbb{1}$.

Given an effect $E$ in an operator algebra $A$, its complement is the effect defined by $\mathbb{1} - E$, where $\mathbb{1}$ is the identity element of the algebra. Since $E$ satisfies the operator inequality $0 \le E \le \mathbb{1}$, its complement also satisfies $0 \le \mathbb{1} - E \le \mathbb{1}$.

For any effect $E$ in an operator algebra $A$, the complement of the complement of $E$ is equal to $E$. Given that the complement of an effect $E$ is defined as $\mathbb{1} - E$ (where $\mathbb{1}$ is the identity element of the algebra), this identity states that $\mathbb{1} - (\mathbb{1} - E) = E$.