PhyslibAlpha.QuantumMechanics.OperatorAlgebra.Observables.Positive

Given an operator algebra $A$, a positive observable is an observable $a \in A$ (a self-adjoint element) that is non-negative, denoted by $a \ge 0$. In the context of $C^*$-algebras, this condition implies that the spectrum of $a$ is contained in $[0, \infty)$.

Let $A$ be a $C^*$-algebra. An observable $a \in A$ (a self-adjoint element) is positive, denoted as $0 \leq a$, if and only if there exists an observable $b \in A$ such that $a = b^2$.

Given an observable $a$ in a $C^*$-algebra $A$, its positive part $a^+$ is the positive observable (a self-adjoint element with non-negative spectrum) defined by applying the function $f(t) = \max(0, t)$ to $a$ via the continuous functional calculus.

Let $A$ be a $C^*$-algebra and $a \in A$ be an observable (a self-adjoint element). The negative part of $a$, denoted as $a^-$, is a positive observable (meaning $a^- = (a^-)^*$ and $a^- \ge 0$). This element is defined via the continuous functional calculus such that the observable $a$ can be decomposed as $a = a^+ - a^-$, where $a^+$ is the positive part of $a$ and the two parts are orthogonal ($a^+ a^- = 0$).

Let $A$ be a $C^*$-algebra. For any observable $a \in A$ (a self-adjoint element), let $a^+$ denote its positive part and $a^-$ denote its negative part. Then the observable $a$ is equal to the difference of these two parts: $$a = a^+ - a^-$$ where $a^+$ and $a^-$ are the positive observables defined via the continuous functional calculus.

Let $A$ be a $C^*$-algebra and $a \in A$ be an observable (a self-adjoint element). Let $a^+$ denote the positive part of $a$ and $a^-$ denote the negative part of $a$. Then the positive and negative parts are orthogonal, meaning $a^+ a^- = 0$.

Let $A$ be a complex $C^*$-algebra and let $a \in A$ be an observable (a self-adjoint element). Let $a^+$ and $a^-$ denote the positive and negative parts of $a$, respectively, as defined via the continuous functional calculus. Then the product of the negative part and the positive part is zero: $a^- a^+ = 0$.

Let $A$ be a $C^*$-algebra and $a \in A$ be an observable (a self-adjoint element). Suppose there exist positive observables $b, c \in A$ (self-adjoint elements with non-negative spectra) such that $a$ can be decomposed as $a = b - c$. If $b$ and $c$ are orthogonal, such that $bc = 0$, then $b$ is the positive part of $a$ (denoted $a^+$) and $c$ is the negative part of $a$ (denoted $a^-$).