PhyslibAlpha.AlgebraicFramework.CStarAlgebra.Jordan

Let $A$ be a unital $C^*$-algebra and $\text{selfAdjoint}(A)$ be the set of its self-adjoint elements. For any $a \in \text{selfAdjoint}(A)$, the Jordan product of the identity element $1$ and $a$ satisfies $1 \circ a = a$. Here, the Jordan product is defined as $x \circ y = \frac{1}{2}(xy + yx)$.

For any self-adjoint element $a$ in a unital $C^*$-algebra $A$, the Jordan square $a \circ a$ (defined by the normalized Jordan product $a \circ b = \frac{1}{2}(ab + ba)$) is equal to the ordinary associative square $a^2$ in $A$.

Let $A$ be a unital $C^*$-algebra. For any self-adjoint element $a \in \text{selfAdjoint}(A)$, its Jordan square $a * a$ (defined as $a \circ a = \frac{1}{2}(aa + aa) = a^2$) is non-negative, satisfying $0 \le a * a$.

Let $A$ be a unital $C^*$-algebra. The space of self-adjoint elements $\text{selfAdjoint}(A) = \{a \in A \mid a^* = a\}$, equipped with the normalized Jordan product $a \circ b = \frac{1}{2}(ab + ba)$, is a Jordan order-unit space. This means that $\text{selfAdjoint}(A)$ carries a compatible structure of both a Jordan algebra and an order-unit space, where the identity element $1 \in A$ acts as the order unit and the square of any element $a \circ a$ is non-negative ($a \circ a \ge 0$).

Let $A$ be a unital $C^*$-algebra. For any self-adjoint elements $a, b \in \text{selfAdjoint}(A)$, the norm of their Jordan product $a \circ b = \frac{1}{2}(ab + ba)$ satisfies the inequality $$\|a \circ b\| \leq \|a\| \|b\|.$$

Let $A$ be a unital $C^*$-algebra. The space of self-adjoint elements, $\text{selfAdjoint } A = \{ a \in A \mid a^* = a \}$, is a normed Jordan algebra over $\mathbb{R}$. The algebraic structure is defined by the physically normalized Jordan product $a \circ b = \frac{1}{2}(ab + ba)$, and the norm is the standard operator norm inherited from $A$, satisfying $\|a \circ b\| \leq \|a\| \|b\|$.

Let $A$ be a $C^*$-algebra. The space of self-adjoint elements of $A$, denoted as $\text{selfAdjoint } A = \{ a \in A \mid a^* = a \}$, is a complete metric space. This completeness arises because the self-adjoint part is the closed fixed-point set of the continuous star operation $(\cdot)^*$ within the complete space $A$.

Let $A$ be a unital $C^*$-algebra and let $\text{selfAdjoint } A = \{ a \in A \mid a^* = a \}$ be the set of its self-adjoint elements. For any $a \in \text{selfAdjoint } A$, the norm of its square is equal to the square of its norm, i.e., $$\|a^2\| = \|a\|^2$$ where $\|\cdot\|$ denotes the operator norm inherited from $A$. (Note: in the context of the Jordan algebra $\text{selfAdjoint } A$, the square $a^2$ refers to the Jordan product $a \circ a$, which coincides with the ordinary associative square).

Let $A$ be a unital $C^*$-algebra and let $\text{selfAdjoint } A = \{ a \in A \mid a^* = a \}$ be the set of its self-adjoint elements. For any $a, b \in \text{selfAdjoint } A$, the norm of the square of $a$ satisfies the inequality $$\|a^2\| \leq \|a^2 + b^2\|$$ where $\|\cdot\|$ denotes the operator norm inherited from $A$.

Let $A$ be a unital $C^*$-algebra. The space of self-adjoint elements, $\text{selfAdjoint } A = \{ a \in A \mid a^* = a \}$, forms a JB-algebra. The algebraic structure is given by the physically normalized Jordan product $a \circ b = \frac{1}{2}(ab + ba)$, and the norm is the operator norm inherited from $A$. Under this structure, $\text{selfAdjoint } A$ is a complete normed Jordan algebra satisfying the JB-condition: $\|a^2\| \leq \|a^2 + b^2\|$ for all $a, b \in \text{selfAdjoint } A$.

Let $A$ be a unital $C^*$-algebra and let $\text{selfAdjoint } A = \{ a \in A \mid a^* = a \}$ be the Jordan algebra of its self-adjoint elements equipped with the product $a \circ b = \frac{1}{2}(ab + ba)$. For any $a, b \in \text{selfAdjoint } A$, the Jordan quadratic representation $U_a(b)$, defined abstractly as $U_a(b) = 2 a \circ (a \circ b) - (a \circ a) \circ b$, is equal to the associative triple product $aba$ in $A$.

Let $A$ be a unital $C^*$-algebra and $p \in A$ be a self-adjoint element. Then $p$ is a Jordan projection (meaning $p \circ p = p$ with respect to the Jordan product $a \circ b = \frac{1}{2}(ab + ba)$) if and only if $p$ is an idempotent element in the associative algebra $A$ (meaning $p^2 = p$).

In a unital $C^*$-algebra $A$, two self-adjoint elements $p$ and $q$ are Jordan-orthogonal (meaning their Jordan product $p \circ q$ is zero) if and only if their anticommutator vanishes, i.e., $pq + qp = 0$. For the physically normalized product $a \circ b = \frac{1}{2}(ab + ba)$, this condition realizes the ordinary operator orthogonality $pq = 0$.