Physlib.QuantumMechanics.OperatorAlgebra.Basic

Let $A$ be a unital complex $C^*$-algebra representing the observables of a physical system. An observable is defined as a self-adjoint element $a \in A$ such that $a^* = a$. These elements correspond to measurable physical quantities—such as position, momentum, energy, or spin—because self-adjointness ensures that their spectrum (the set of possible measurement outcomes) consists of real numbers.

Let $A$ be a unital complex $C^*$-algebra representing the observables of a physical system. A positive element is defined as an observable $a \in A$ (a self-adjoint element, $a = a^*$) such that $a \geq 0$. This condition implies that the spectrum of $a$ is contained in the non-negative real numbers $[0, \infty)$, meaning that all possible measurement outcomes of the observable are non-negative. The set of positive elements induces a partial order on the space of observables, where $a \leq b$ if and only if $b - a$ is a positive element.

Let $A$ be an operator algebra (such as a unital complex $C^*$-algebra). An effect is an observable $a \in A$ that satisfies the condition $0 \le a \le \mathbb{1}$, where $0$ is the zero element and $\mathbb{1}$ is the identity element of the algebra. This inequality is defined with respect to the standard partial ordering of self-adjoint elements (where $a \ge 0$ means $a$ is a positive element). Physically, an effect represents a measurement outcome that can occur with a probability between 0 and 1, such as a "yes/no" outcome in a quantum measurement.

Let $A$ be a unital complex $C^*$-algebra representing the observable structure of a physical system. A unitary element $U \in A$ is an element that satisfies $U^* U = U U^* = \mathbb{1}$, where $\mathbb{1}$ is the identity element of the algebra. Unitary elements represent reversible transformations of the system, such as symmetries or time evolution under a Hamiltonian, and act on observables $a \in A$ via the conjugation map $a \mapsto U a U^*$.

Let $A_1$ and $A_2$ be operator algebras, typically understood as unital complex $C^*$-algebras representing physical systems. A channel from $A_1$ to $A_2$ is a linear map $\phi : A_1 \to A_2$ that is unital, meaning it preserves the identity element ($\phi(\mathbb{1}) = \mathbb{1}$), and completely positive. In the context of mathematical physics, channels represent the most general description of the dynamics or evolution of a system, mapping observables from one algebra to another.

Given a unital complex $C^*$-algebra $A$ and a complex Hilbert space $H$, a representation of $A$ on $H$ is a unital $\ast$-homomorphism from $A$ to the algebra of bounded linear operators $\mathcal{B}(H)$.