PhyslibAlpha.ProbabilisticTheory.CStarAlgebra.Commutative

For a character $\chi$ in the character space of a commutative $C^*$-algebra $A$, this definition represents $\chi$ as a $\mathbb{C}$-star-algebra homomorphism from $A$ to $\mathbb{C}$. For any element $a \in A$, the value of this homomorphism is given by the evaluation of the character $\chi$ on $a$.

Let $A$ be a commutative $C^*$-algebra. For any character $\chi$ in the character space of $A$ over $\mathbb{C}$ and any element $a \in A$, the value of $\chi$ viewed as a $\mathbb{C}$-star-algebra homomorphism (denoted as $\text{character}(\chi)$) applied to $a$ is equal to the evaluation of the character $\chi$ on $a$, written as $\chi(a)$.

Let $A$ be a commutative $C^*$-algebra. For any two elements $a, b \in A$, if $\chi(a) = \chi(b)$ for every character $\chi$ in the character space of $A$ (the set of nonzero $\mathbb{C}$-algebra homomorphisms from $A$ to $\mathbb{C}$), then $a = b$.

Let $A$ be a commutative $C^*$-algebra and $a \in A$ be a self-adjoint element. For any character $\chi$ in the character space of $A$ (the set of non-zero multiplicative linear functionals), the value $\chi(a)$ is a real number, expressed formally as $\chi(a) = \text{Re}(\chi(a))$.

For any two elements $a$ and $b$ in a commutative $C^*$-algebra $A$, this definition constructs an element $\text{inf}(a, b) \in A$ such that for every character $\chi$ in the character space of $A$, the value of the Gelfand transform of $\text{inf}(a, b)$ at $\chi$ is the minimum of the real parts of the values of the Gelfand transforms of $a$ and $b$ at $\chi$. That is, $\chi(\text{inf}(a, b)) = \min(\text{Re}(\chi(a)), \text{Re}(\chi(b)))$. In the case where $a$ and $b$ are self-adjoint, this corresponds to the pointwise minimum of their Gelfand transforms.

Let $A$ be a commutative $C^*$-algebra. For any elements $a, b \in A$ and any character $\chi$ in the character space of $A$, the value of the character $\chi$ at the infimum $a \sqcap b$ is given by the minimum of the real parts of $\chi(a)$ and $\chi(b)$. That is, $$\chi(a \sqcap b) = \min(\text{Re}(\chi(a)), \text{Re}(\chi(b)))$$ where the resulting real value is viewed as a complex number.

Let $A$ be a commutative $C^*$-algebra. For any two elements $a, b \in A$, the element $\text{inf}(a, b)$ is self-adjoint. This element $\text{inf}(a, b)$ is defined such that for every character $\chi$ in the character space of $A$, its value is given by the minimum of the real parts of the values of the Gelfand transforms of $a$ and $b$, i.e., $\chi(\text{inf}(a, b)) = \min(\operatorname{Re}(\chi(a)), \operatorname{Re}(\chi(b)))$.

Let $A$ be a commutative $C^*$-algebra. An element $a \in A$ is non-negative ($0 \le a$) if and only if for every character $\chi$ in the character space of $A$, the value of the character at $a$ is non-negative ($0 \le \chi(a)$).

Let $A$ be a commutative $C^*$-algebra. For any two self-adjoint elements $a, b \in A$, the partial order $a \le b$ holds if and only if for every character $\chi$ in the character space of $A$ (the set of non-zero algebra homomorphisms from $A$ to $\mathbb{C}$), the real part of $\chi(a)$ is less than or equal to the real part of $\chi(b)$, i.e., $\operatorname{Re}(\chi(a)) \le \operatorname{Re}(\chi(b))$.

Let $A$ be a commutative $C^*$-algebra and let $A_{sa}$ denote the set of its self-adjoint elements (also known as observables). Then $A_{sa}$ satisfies the Riesz decomposition property: for any nonnegative observables $a, b, c \in A_{sa}$ such that $c \le a + b$, there exist $c_1, c_2 \in A_{sa}$ such that $c = c_1 + c_2$ with $0 \le c_1 \le a$ and $0 \le c_2 \le b$. This decomposition is achieved by considering the values of the observables character by character over the character space of the algebra.

Let $A$ be a commutative $C^*$-algebra. The space of self-adjoint elements of $A$ (its observables), denoted by $\text{selfAdjoint } A$, is a classical system. This means that the dual cone of positive linear functionals on $\text{selfAdjoint } A$ forms a lattice, or equivalently, the state space of the algebra is a Choquet simplex where every state is uniquely decomposable into pure states.