PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.JB.GeneratedByOne.Uniform

Let $a$ be an element of a normed Jordan algebra, and let $C(a)$ be the norm-closed Jordan subalgebra generated by the identity $1$ and $a$. For any element $x \in C(a)$, the norm of its square satisfies $\|x^2\| = \|x\|^2$.

Let $E$ be a normed Jordan algebra and $a \in E$. Let $C(a)$ denote the norm-closed Jordan subalgebra generated by $a$ and the identity element $1$. For any element $x \in C(a)$, the spectral radius of $x$ (with respect to the scalar field $\mathbb{R}$) is less than or equal to its norm, denoted as $\rho(x) \leq \|x\|$.

Let $a$ be an element of a normed Jordan algebra, and let $C(a)$ be the norm-closed Jordan subalgebra generated by the identity $1$ and $a$. For any element $x \in C(a)$ and any natural number $n \in \mathbb{N}$, the norm of the power $x^{2^n}$ satisfies the equality $$\|x^{2^n}\| = \|x\|^{2^n}$$ where $\|\cdot\|$ denotes the norm and the power is taken using the associative multiplication within $C(a)$.