PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.JB.GeneratedByOne.Closed

For an element $a$ in a normed Jordan algebra $E$ over $\mathbb{R}$, this definition defines $C_J(a)$ as the topological closure of the $\mathbb{R}$-submodule $J[a]$ generated by the identity element $1$ and $a$. The submodule $J[a]$ is the span of the set of Jordan powers $\{a^n\}_{n \in \mathbb{N}}$, where $a^0 = 1$.

For any element $a$ in a normed Jordan algebra $E$ over $\mathbb{R}$, the $\mathbb{R}$-submodule $J[a]$ generated by the Jordan powers $\{a^n\}_{n \in \mathbb{N}}$ (where $a^0 = 1$) is contained within its topological closure $C_J(a)$.

For any element $a$ in a normed Jordan algebra $E$, the identity element $1$ belongs to the closed Jordan subalgebra $C_J(a)$ generated by $1$ and $a$.

For any element $a$ in a normed Jordan algebra $E$, $a$ is an element of the closed Jordan subalgebra $C_J(a)$ generated by the identity element $1$ and $a$.

Let $E$ be a normed Jordan algebra. For any element $a \in E$, let $C_J(a)$ denote the topological closure of the subalgebra generated by the identity element $1$ and $a$. For any elements $x, y \in E$, if $x \in C_J(a)$ and $y \in C_J(a)$, then their Jordan product $x \cdot y$ is also an element of $C_J(a)$.

Let $E$ be a normed Jordan algebra and $a$ be an element in $E$. Let $C_J(a)$ denote the closed $\mathbb{R}$-submodule generated by the identity element $1$ and $a$, which is the topological closure of the span of the Jordan powers $\{a^n\}_{n \in \mathbb{N}}$. For any elements $x, y, z \in C_J(a)$, the Jordan product is associative, satisfying $(x * y) * z = x * (y * z)$.

Let $E$ be a normed Jordan algebra and let $a \in E$. The set $C_J(a)$, defined as the topological closure of the $\mathbb{R}$-submodule generated by the identity element $1$ and the element $a$ (the span of the Jordan powers $\{a^n\}_{n \in \mathbb{N}}$), is a closed set in $E$.

Let $E$ be a normed Jordan algebra that is complete (i.e., a JB-algebra). For any element $a \in E$, the subalgebra $C_J(a)$—defined as the topological closure of the $\mathbb{R}$-submodule generated by the identity element $1$ and $a$—is complete.

For an element $a$ in a normed Jordan algebra $E$, this definition denotes the type representing the norm-closed Jordan subalgebra $C(a)$ generated by the identity $1$ and $a$. Although the multiplication in the ambient algebra $E$ is generally non-associative, the multiplication restricted to this subtype is associative, making $C(a)$ a commutative real Banach algebra.

The closed Jordan subalgebra $C(a)$ generated by the identity $1$ and an element $a$ is equipped with a multiplication operation. For any $x, y \in C(a)$, the product $x \cdot y$ is defined as the Jordan product inherited from the ambient normed Jordan algebra $E$.

Let $E$ be a normed Jordan algebra and $a \in E$. Let $C(a)$ be the norm-closed Jordan subalgebra generated by the identity $1$ and $a$. For any elements $x, y \in C(a)$, the product $x \cdot y$ computed within $C(a)$ is equal to the Jordan product $x \cdot y$ of their corresponding elements in the ambient algebra $E$.

For an element $a$ in a normed Jordan algebra $E$, the closed Jordan subalgebra $C(a)$ generated by the identity $1$ and $a$ is equipped with a multiplicative identity element, denoted as $1$. This element corresponds to the identity element of the ambient algebra $E$.

Let $E$ be a normed Jordan algebra and $a \in E$. Let $C(a)$ be the closed Jordan subalgebra generated by the identity element $1$ and $a$. The identity element $1$ of the subalgebra $C(a)$ is equal to the identity element $1$ of the ambient algebra $E$.

The closed Jordan subalgebra $C(a)$ generated by an element $a$ and the identity $1$ in a normed Jordan algebra $E$ forms a commutative ring. While the Jordan product in the ambient algebra $E$ is generally non-associative, its restriction to $C(a)$ is associative, satisfying $(x \cdot y) \cdot z = x \cdot (y \cdot z)$ for all $x, y, z \in C(a)$, alongside the standard axioms of a commutative ring.

The closed Jordan subalgebra $C(a)$ generated by an element $a$ and the identity $1$ in a normed Jordan algebra $E$ is an algebra over the real numbers $\mathbb{R}$. This algebra structure is inherited from the ambient algebra $E$, where the scalar multiplication $c \cdot x$ for $c \in \mathbb{R}$ and $x \in C(a)$ is defined by the scalar multiplication in $E$, and the unit map is given by $c \mapsto c \cdot 1$.

For an element $a$ in a normed Jordan algebra $E$, the closed Jordan subalgebra $C(a)$ generated by the identity $1$ and $a$ forms a normed ring. The norm on $C(a)$ is inherited from $E$ and is submultiplicative, satisfying $\|x \cdot y\| \le \|x\| \cdot \|y\|$ for all $x, y \in C(a)$, where the product is the restricted Jordan product.

For an element $a$ in a normed Jordan algebra $E$, the closed Jordan subalgebra $C(a)$ generated by the identity $1$ and $a$ is a normed algebra over the real numbers $\mathbb{R}$. This means the $\mathbb{R}$-algebra structure on $C(a)$ is compatible with the norm inherited from $E$, specifically satisfying the property $\|\lambda \cdot x\| = |\lambda| \|x\|$ for all scalars $\lambda \in \mathbb{R}$ and elements $x \in C(a)$.

Let $E$ be a normed Jordan algebra. If $E$ is a complete space, then for any element $a \in E$, the closed Jordan subalgebra $C(a)$ generated by the identity $1$ and $a$ is also a complete space.

Let $E$ be a normed Jordan algebra and $a \in E$. Let $C(a)$ be the norm-closed Jordan subalgebra generated by $1$ and $a$. For any element $x \in C(a)$ and any natural number $n \in \mathbb{N}$, the $n$-th power of $x$ computed within the commutative ring structure of $C(a)$ is equal to the $n$-th Jordan power of $x$ computed in the ambient algebra $E$. That is, $x^n = x^{[n]}$, where $x^{[n]}$ is defined by $x^{[0]} = 1$ and $x^{[n+1]} = x \circ x^{[n]}$.