PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.CStarAlgebra.Basic

The self-adjoint part of a C⋆-algebra is a JB-algebra

i. Overview

The self-adjoint elements of a unital C⋆-algebra, with the Jordan product `a ∘ b = ½ (a b + b a)`, form a JB-algebra. With this normalization the Jordan square is the ordinary square and the quadratic representation is `U_a b = a b a`, and Jordan projections are the ordinary projections. The instances are scoped to `JB`.

ii. Key results

`JB.instIsJordanOrderUnit`, `JB.instJBAlgebra` : the Jordan order-unit and JB-algebra structures.
`JB.mul_self_eq` : the Jordan square is the ordinary square.
`JB.quadRep_eq_conj` : `U_a b = a b a`.
`JB.isJordanProjection_iff_isIdempotentElem` : Jordan projections are the ordinary projections.

iii. Table of contents

A. C⋆ order compatibility
B. The JB-algebra instance
C. Squares, `U_a = aba` and projections

A. C⋆ order compatibility

B. The JB-algebra instance

C. Squares, `U_a = aba` and projections

13 declarations

theorem

$1 \circ a = a$ for self-adjoint $a$

#one_mul

Let $A$ be a unital $C^*$ -algebra. For any self-adjoint element $a \in \text{selfAdjoint}(A)$ , the normalized Jordan product of the unit $1$ and $a$ is equal to $a$ . That is, $1 \circ a = a$ , where the Jordan product is defined as $x \circ y = \frac{1}{2}(xy + yx)$ .

theorem

The Jordan square $a \circ a$ equals the associative square $a^2$

#mul_self_eq

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

theorem

The Jordan square of a self-adjoint element is non-negative ( $0 \le a \circ a$ )

#mul_self_nonneg

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

instance

The self-adjoint part of a unital $C^*$ -algebra is a Jordan order-unit space

#instIsJordanOrderUnit

Let $A$ be a unital $C^*$ -algebra. The space of self-adjoint elements of $A$ , denoted by $\text{selfAdjoint } A$ , equipped with the normalized Jordan product $a \circ b = \frac{1}{2}(ab + ba)$ , is a Jordan order-unit space. This means that $\text{selfAdjoint } A$ is simultaneously a commutative Jordan algebra and an order-unit space (where the identity $1$ of the algebra acts as the order unit), satisfying the compatibility condition that the Jordan square of any element is non-negative ( $a^2 \geq 0$ ).

theorem

$\|a \circ b\| \leq \|a\| \|b\|$ for self-adjoint elements $a, b$

#norm_mul_le'

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

instance

$\text{selfAdjoint}(A)$ is a normed Jordan algebra

#instNormedJordanAlgebra

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 the real numbers. The multiplication is given by the Jordan product $a \circ b = \frac{1}{2}(ab + ba)$ , and the norm $\|\cdot\|$ is inherited from the algebra $A$ . This structure satisfies the Jordan identity $(a \circ b) \circ a^2 = a \circ (b \circ a^2)$ and the submultiplicative property $\|a \circ b\| \leq \|a\| \|b\|$ .

instance

The self-adjoint part of a $C^*$ -algebra is complete

#instCompleteSpace

Let $A$ be a $C^*$ -algebra. The space of self-adjoint elements of $A$ , defined as $\{a \in A \mid a^* = a\}$ , is a complete space with respect to the norm topology.

theorem

$\|a^2\| = \|a\|^2$ for self-adjoint elements

#norm_mul_self'

Let $A$ be a $C^*$ -algebra and $\text{selfAdjoint}(A)$ 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., $\|a^2\| = \|a\|^2$ . Here, the product $a^2$ corresponds to the Jordan square, which for self-adjoint elements in a $C^*$ -algebra coincides with the standard power.

theorem

$\|a^2\| \le \|a^2 + b^2\|$ for self-adjoint elements $a$ and $b$

#norm_mul_self_le_add'

Let $A$ be a unital $C^*$ -algebra and let $a$ and $b$ be self-adjoint elements of $A$ . Then the norm of the square of $a$ is less than or equal to the norm of the sum of the squares of $a$ and $b$ : $\|a^2\| \le \|a^2 + b^2\|$ where $\|\cdot\|$ is the $C^*$ -norm and the squares are defined via the Jordan product (which, for self-adjoint elements in this context, coincides with the standard algebraic square).

instance

The self-adjoint part of a unital $C^*$ -algebra is a JB-algebra

#instJBAlgebra

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 Jordan product $a \circ b = \frac{1}{2}(ab + ba)$ and the norm inherited from $A$ , forms a JB-algebra.

theorem

Quadratic Representation for Self-Adjoint Elements is $U_a b = aba$

#quadRep_eq_conj

Let $A$ be a unital $C^*$ -algebra and let $\text{selfAdjoint}(A)$ be the set of its self-adjoint elements. For any $a, b \in \text{selfAdjoint}(A)$ , the quadratic representation $U_a$ in the Jordan algebra of self-adjoint elements (defined by $U_a b = 2a \circ (a \circ b) - a^2 \circ b$ , where $x \circ y = \frac{1}{2}(xy + yx)$ ) satisfies the identity $U_a b = aba$ where the product on the right-hand side is the associative product in $A$ .

theorem

Self-adjoint Jordan projection iff $p^2 = p$

#isJordanProjection_iff_isIdempotentElem

Let $A$ be a $C^*$ -algebra and let $\text{selfAdjoint}(A)$ be the set of its self-adjoint elements. For any $p \in \text{selfAdjoint}(A)$ , $p$ is a Jordan projection with respect to the normalized Jordan product $a \circ b = \frac{1}{2}(ab + ba)$ if and only if $p$ is an idempotent element in $A$ , satisfying $p^2 = p$ .

theorem

Jordan Orthogonality $\iff pq + qp = 0$ for Self-Adjoint Elements

#jordanOrthogonal_iff

Let $A$ be a unital $C^*$ -algebra. For any two self-adjoint elements $p, q \in \text{selfAdjoint}(A)$ , they are Jordan-orthogonal if and only if their anticommutator vanishes, i.e., $pq + qp = 0$ .

PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.CStarAlgebra.Basic

The self-adjoint part of a C⋆-algebra is a JB-algebra

i. Overview

ii. Key results

iii. Table of contents

A. C⋆ order compatibility

B. The JB-algebra instance

C. Squares, `U_a = aba` and projections

1∘a=a1 \circ a = a1∘a=a for self-adjoint aaa

The Jordan square a∘aa \circ aa∘a equals the associative square a2a^2a2

The Jordan square of a self-adjoint element is non-negative (0≤a∘a0 \le a \circ a0≤a∘a)

The self-adjoint part of a unital C∗C^*C∗-algebra is a Jordan order-unit space

∥a∘b∥≤∥a∥∥b∥\|a \circ b\| \leq \|a\| \|b\|∥a∘b∥≤∥a∥∥b∥ for self-adjoint elements a,ba, ba,b

selfAdjoint(A)\text{selfAdjoint}(A)selfAdjoint(A) is a normed Jordan algebra

The self-adjoint part of a C∗C^*C∗-algebra is complete

∥a2∥=∥a∥2\|a^2\| = \|a\|^2∥a2∥=∥a∥2 for self-adjoint elements

∥a2∥≤∥a2+b2∥\|a^2\| \le \|a^2 + b^2\|∥a2∥≤∥a2+b2∥ for self-adjoint elements aaa and bbb

The self-adjoint part of a unital C∗C^*C∗-algebra is a JB-algebra

Quadratic Representation for Self-Adjoint Elements is Uab=abaU_a b = abaUa​b=aba

Self-adjoint Jordan projection iff p2=pp^2 = pp2=p

Jordan Orthogonality ⟺ pq+qp=0\iff pq + qp = 0⟺pq+qp=0 for Self-Adjoint Elements

$1 \circ a = a$ for self-adjoint $a$

The Jordan square $a \circ a$ equals the associative square $a^2$

The Jordan square of a self-adjoint element is non-negative ( $0 \le a \circ a$ )

The self-adjoint part of a unital $C^*$ -algebra is a Jordan order-unit space

$\|a \circ b\| \leq \|a\| \|b\|$ for self-adjoint elements $a, b$

$\text{selfAdjoint}(A)$ is a normed Jordan algebra

The self-adjoint part of a $C^*$ -algebra is complete

$\|a^2\| = \|a\|^2$ for self-adjoint elements

$\|a^2\| \le \|a^2 + b^2\|$ for self-adjoint elements $a$ and $b$

The self-adjoint part of a unital $C^*$ -algebra is a JB-algebra

Quadratic Representation for Self-Adjoint Elements is $U_a b = aba$

Self-adjoint Jordan projection iff $p^2 = p$

Jordan Orthogonality $\iff pq + qp = 0$ for Self-Adjoint Elements