PhyslibAlpha.ProbabilisticTheory.StarAlgebra.Jordan

Let $A$ be a ring equipped with a star operation (involution). For any two self-adjoint elements $a, b \in A$ (where $a^* = a$ and $b^* = b$), their anticommutator is defined as $ab + ba$. This operation maps a pair of self-adjoint elements to another self-adjoint element.

Let $A$ be a star-ring. For any two self-adjoint elements $a, b \in \text{selfAdjoint } A$, the anticommutator of $a$ and $b$ (as an element of $A$) is equal to $ab + ba$.

Let $A$ be a ring with a star-involution (a star-ring) that is also a real star-module. Let $\text{selfAdjoint}(A)$ denote the set of self-adjoint elements $x \in A$ such that $x^* = x$. For any $a \in \text{selfAdjoint}(A)$, the Jordan product of the identity element $1$ and $a$ is equal to $a$: $$1 \circ a = a$$ where the Jordan product (or anticommutator) is defined as $x \circ y = xy + yx$, and $1$ is the identity element of the Jordan algebra structure on $\text{selfAdjoint}(A)$ (which, for this unscaled product, corresponds to $\frac{1}{2}$ of the identity in $A$).

Let $A$ be a star-ring that is also a module over $\mathbb{R}$. For any self-adjoint element $a \in \text{selfAdjoint } A$, the Jordan product of $a$ with the identity element $1$ is equal to $a$, written as $a \circ 1 = a$. Here, the Jordan product $\circ$ is the anticommutator $a \circ b = ab + ba$, and $1$ is the multiplicative identity of the Jordan algebra.

Let $A$ be a ring equipped with a star operation, where scalar multiplication by real numbers satisfies the standard compatibility conditions with the ring multiplication (i.e., it commutes and satisfies the scalar tower property). For any two self-adjoint elements $a, b \in A$ and any real number $c$, the anticommutator of $c \cdot a$ and $b$ is equal to $c$ times the anticommutator of $a$ and $b$. That is, $$\{c \cdot a, b\} = c \cdot \{a, b\}$$ where $\{x, y\} = xy + yx$ denotes the anticommutator.

Let $A$ be a star-ring. For any self-adjoint elements $a, b \in A$ and any real scalar $c \in \mathbb{R}$, the anticommutator of $a$ and the scalar product $c \cdot b$ satisfies the identity $\{a, c \cdot b\} = c \cdot \{a, b\}$, where $\{x, y\} = xy + yx$ denotes the anticommutator of $x$ and $y$.

Let $A$ be a ring with a star operation (involution). For any self-adjoint elements $a, b \in A$, let $\{a, b\} = ab + ba$ denote their anticommutator. Then the following identity holds: $$\{\{a, b\}, \{a, a\}\} = \{a, \{b, \{a, a\}\}\}$$ This identity ensures that the space of self-adjoint elements forms a Jordan algebra under the symmetrized product.

Let $A$ be a ring with a star-involution that is also a module over the real numbers $\mathbb{R}$. The Jordan product of two observables $a, b \in \text{Observable } A$ is defined as the symmetrized product $a \circ b = \frac{1}{2}(ab + ba)$. This operation ensures the result is also an observable (a self-adjoint element).