PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.CStarAlgebra.Compatibility

Commuting observables are Jordan compatible

i. Overview

If two self-adjoint elements of a C⋆-algebra commute, their Jordan multiplication operators commute. Expanding `a ∘ (b ∘ x)` and `b ∘ (a ∘ x)` in the associative product, the terms agree using `a b = b a`.

ii. Key results

`JB.isJordanCompatible_of_commute` : commuting observables are Jordan compatible.

iii. Table of contents

A. Commuting observables

A. Commuting observables

1 declaration

theorem

Commutativity of self-adjoint elements implies Jordan-compatibility

#isJordanCompatible_of_commute

Let $A$ be a unital $C^*$ -algebra and let $a, b \in \text{selfAdjoint}(A)$ be self-adjoint elements. If $a$ and $b$ commute under the associative product of $A$ , such that $ab = ba$ , then they are Jordan-compatible. This compatibility means that their Jordan multiplication operators commute, satisfying $a \circ (b \circ x) = b \circ (a \circ x)$ for all self-adjoint elements $x \in \text{selfAdjoint}(A)$ , where the Jordan product is defined as $x \circ y = \frac{1}{2}(xy + yx)$ .