PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.JB.GeneratedByOne.PosInvertibility

Let $C(a)$ be the closed commutative associative real Banach algebra generated by the identity $1$ and an element $a$ in a normed Jordan algebra. For any $z \in C(a)$, this function defines the element obtained by evaluating the binomial series for $\sqrt{1-z}$: $$\sum_{n=0}^\infty \binom{1/2}{n} (-z)^n.$$ This series converges in the norm of $C(a)$ because its coefficients are related to Catalan numbers, which ensures convergence even at the boundary of the unit disk.

Let $B_{1/2}(x) = \sum_{n=0}^\infty \binom{1/2}{n} x^n$ be the formal power series over the real numbers $\mathbb{R}$ representing the binomial expansion with exponent $1/2$. Then the square of this power series is equal to $1 + x$: $$\left( \sum_{n=0}^\infty \binom{1/2}{n} x^n \right)^2 = 1 + x.$$ This formal identity serves as the algebraic basis for defining square roots in a Jordan-Banach algebra via the binomial series.

For any natural number $n$, the $n$-th Catalan number $C_n$ satisfies the following telescoping identity: $$\frac{C_n}{4^n} = 2 \left( \frac{1}{4^n} \binom{2n}{n} - \frac{1}{4^{n+1}} \binom{2n+2}{n+1} \right),$$ where $\binom{2n}{n}$ denotes the $n$-th central binomial coefficient.

For any natural number $N$, the sum of the first $N$ normalized Catalan numbers is given by: $$\sum_{n=0}^{N-1} \frac{C_n}{4^n} = 2 \left( 1 - \frac{1}{4^N} \binom{2N}{N} \right)$$ where $C_n$ is the $n$-th Catalan number and $\binom{2N}{N}$ is the $N$-th central binomial coefficient. This result follows from the telescoping nature of the terms, providing a uniform bound used for the convergence of the square root binomial series.

The sequence defined by $n \mapsto \frac{C_n}{4^n}$ is summable, where $C_n$ denotes the $n$-th Catalan number. That is, the infinite series $$\sum_{n=0}^{\infty} \frac{C_n}{4^n}$$ converges.

For any real number $a$ and natural number $n$, the generalized binomial coefficient $\binom{a}{n}$ satisfies the recurrence relation $$\binom{a}{n + 1} = \binom{a}{n} \frac{a - n}{n + 1}$$ where $\binom{a}{n}$ is defined for real $a$.

For any natural number $n$, the $(n+1)$-th Catalan number $C_{n+1}$ satisfies the recurrence relation $$C_{n+1} = \frac{2(2n + 1)}{n + 2} C_n,$$ where $C_n$ denotes the $n$-th Catalan number.

For any natural number $n$, the generalized binomial coefficient $\binom{1/2}{n+1}$ satisfies the identity $$\binom{1/2}{n+1} = \frac{(-1)^n C_n}{2 \cdot 4^n},$$ where $C_n$ denotes the $n$-th Catalan number. This signed identity relates the half-binomial coefficients to Catalan numbers and is used to establish the scalar majorant for the binomial series of the square root function $\sqrt{1-x}$ at the endpoint of convergence.

For any natural number $n$, the absolute value of the generalized binomial coefficient $\binom{1/2}{n+1}$ is equal to $\frac{C_n}{2 \cdot 4^n}$, where $C_n$ denotes the $n$-th Catalan number.

The sequence of the absolute values of the generalized binomial coefficients $\binom{1/2}{n}$ is summable. That is, the series $\sum_{n=0}^{\infty} \left| \binom{1/2}{n} \right|$ converges.

Let $E$ be a nontrivial normed Jordan algebra and let $C(a)$ be the norm-closed Jordan subalgebra generated by $1$ and $a \in E$. For any $z \in C(a)$ with $\|z\| \leq 1$, the series of the norms of the half-binomial terms $$\sum_{n=0}^{\infty} \left\| \binom{1/2}{n} (-z)^n \right\|$$ is summable.

Let $E$ be a nontrivial normed Jordan algebra and let $C(a)$ be the norm-closed Jordan subalgebra generated by the identity $1$ and an element $a \in E$. For any $z \in C(a)$ belonging to the closed unit ball (i.e., $\|z\| \le 1$), the binomial series $$\sum_{n=0}^{\infty} \binom{1/2}{n} (-z)^n$$ is summable in $C(a)$.

Let $E$ be a normed Jordan algebra and $C(a)$ be the norm-closed Jordan subalgebra generated by an element $a \in E$ and the identity $1$. For any $z \in C(a)$ and $n \in \mathbb{N}$, the $n$-th term of the Cauchy product of the binomial series $\sum_{k=0}^\infty \binom{1/2}{k} (-z)^k$ with itself is equal to the $n$-th term of the expression $1 - z$. Specifically, $$\sum_{k+l=n} \left( \binom{1/2}{k} (-z)^k \right) \cdot \left( \binom{1/2}{l} (-z)^l \right) = \begin{cases} 1 & \text{if } n = 0 \\ -z & \text{if } n = 1 \\ 0 & \text{if } n \ge 2 \end{cases}$$ where $\binom{1/2}{k}$ denotes the generalized binomial coefficient.

Let $E$ be a nontrivial normed Jordan algebra and $C(a)$ be the norm-closed Jordan subalgebra generated by the identity $1$ and an element $a \in E$. For any element $z \in C(a)$ such that $\|z\| \leq 1$, the binomial series for the square root, defined as $$\sqrt{1-z} := \sum_{n=0}^\infty \binom{1/2}{n} (-z)^n,$$ satisfies the identity $$\left( \sqrt{1-z} \right)^2 = 1 - z.$$

Let $E$ be a nontrivial Jordan-Banach algebra (JB-algebra) and $a \in E$. Let $C(a)$ denote the norm-closed Jordan subalgebra generated by the identity element $1$ and $a$. If $y \in C(a)$ is a non-negative element ($y \geq 0$), then there exists an element $s \in C(a)$ such that $s^2 = y$.

Let $E$ be a nontrivial normed Jordan algebra (specifically a JB algebra) and let $a$ be an element of $E$. Let $C(a)$ denote the norm-closed Jordan subalgebra generated by the identity $1$ and $a$. For any element $y \in C(a)$, $y$ is non-negative ($0 \le y$) if and only if there exists an element $s \in C(a)$ such that $s^2 = y$. This result is known as the local exact-cone theorem for a closed one-generator ordered JB algebra.

Let $E$ be a nontrivial Jordan-Banach algebra (JB-algebra) and $a$ be an element of $E$. Let $C(a)$ denote the norm-closed Jordan subalgebra generated by the identity element $1$ and $a$. If an element $y \in C(a)$ is both invertible and non-negative ($y \ge 0$), then there exists a positive real number $\varepsilon > 0$ such that $\varepsilon \cdot 1 \le y$.

Let $E$ be a nontrivial JB algebra (a normed Jordan algebra) and $a \in E$. Let $C(a)$ denote the norm-closed Jordan subalgebra generated by the identity $1$ and the element $a$. For any element $x \in C(a)$, it is not the case that both $\|x\| \cdot 1 - x$ and $\|x\| \cdot 1 + x$ are invertible in $C(a)$. In other words, at least one of these two elements must be non-invertible.

Let $C(a)$ be the norm-closed Jordan subalgebra generated by the identity $1$ and an element $a$ in a normed Jordan algebra. For any element $y \in C(a)$, if there exists a sequence $(s_n)_{n \in \mathbb{N}}$ in $C(a)$ and an element $r \in C(a)$ such that $s_n$ converges to $r$ and the sequence of squares $(s_n^2)_{n \in \mathbb{N}}$ converges to $y$, then $y$ has a square root in $C(a)$ (i.e., there exists some $x \in C(a)$ such that $x^2 = y$).

Let $C(a)$ be the norm-closed Jordan subalgebra generated by the identity element $1$ and an element $a$ in a normed Jordan algebra. For any element $y \in C(a)$, if the norm of the difference between the identity and $y$ satisfies $\|1 - y\| < 1$, then $y$ is invertible in $C(a)$.

Let $C(a)$ be the norm-closed Jordan subalgebra generated by the identity $1$ and an element $a$ in a normed Jordan algebra. For any element $y \in C(a)$, if there exists a real number $\epsilon > 0$ such that $\epsilon \cdot 1 \leq y$, then $y$ is a unit (invertible) in $C(a)$.

Let $E$ be a nontrivial normed Jordan algebra and let $a \in E$. Let $C(a)$ denote the norm-closed Jordan subalgebra generated by the identity $1$ and $a$. For any non-negative element $y \in C(a)$ (i.e., $y \geq 0$), $y$ is invertible in $C(a)$ if and only if it lies in the interior of the positive cone, which is equivalent to the existence of a real number $\epsilon > 0$ such that $\epsilon \cdot 1 \leq y$.

Let $C(a)$ be the norm-closed Jordan subalgebra generated by the identity $1$ and an element $a$ in a normed Jordan algebra. For any $x \in C(a)$ and any real numbers $c, d \in \mathbb{R}$ with $d \neq 0$, the element $(x - c \cdot 1)^2 + d^2 \cdot 1$ is a unit (invertible) in $C(a)$.

Let $C(a)$ be the norm-closed Jordan subalgebra generated by the identity $1$ and an element $a$ in a normed Jordan algebra. For any element $x \in C(a)$ and real numbers $c, d \in \mathbb{R}$ with $d \neq 0$, the evaluation of the polynomial $P(X) = (X - c)^2 + d^2 \in \mathbb{R}[X]$ at $x$ is an invertible element (a unit) in $C(a)$.

Let $C(a)$ be the norm-closed Jordan subalgebra generated by the identity $1$ and an element $a$ in a JB-algebra. For any element $s \in C(a)$, if the square $s^2$ is invertible (i.e., it is a unit in the commutative ring $C(a)$), then there exists a non-negative element $t \in C(a)$ ($t \geq 0$) such that $t$ is the inverse of $s^2$, satisfying $t \cdot s^2 = 1$.