PhyslibAlpha.AlgebraicFramework.CStarAlgebra.ConjugationSymmetry

For a unitary element $u$ in a unital star-algebra $A$, this definition represents the $\mathbb{R}$-linear map from the space of self-adjoint elements $\text{selfAdjoint} A$ to itself, defined by the conjugation $a \mapsto u a u^*$. This map is well-defined because the conjugation of a self-adjoint element $a$ by a unitary $u$ remains self-adjoint.

Let $A$ be a unital star-algebra. For any unitary element $u \in A$ and any self-adjoint element $a \in \text{selfAdjoint } A$, the value of the conjugation linear map $\text{conjugationLinearMap}(u)$ applied to $a$ is given by $u a u^*$ when viewed as an element of $A$.

Let $A$ be a unital star-algebra and $\text{selfAdjoint } A$ be the set of elements $a \in A$ such that $a^* = a$. For any unitary element $u$ in $A$, let $C_u: \text{selfAdjoint } A \to \text{selfAdjoint } A$ be the $\mathbb{R}$-linear map defined by conjugation $C_u(a) = u a u^*$. For any two unitaries $u, v \in \text{unitary } A$ and any self-adjoint element $a \in \text{selfAdjoint } A$, the composition of conjugation by $v$ followed by conjugation by $u$ is equal to the conjugation by the product $uv$: $$C_u(C_v(a)) = C_{uv}(a)$$

For any self-adjoint element $a$ in a unital star-algebra $A$, the $\mathbb{R}$-linear map defined by conjugation with the identity unitary $1 \in \text{unitary } A$ acts as the identity map on the space of self-adjoint elements, satisfying $\text{conjugationLinearMap}(1)(a) = a$.

For a unitary element $u$ in a unital star-algebra $A$ and a self-adjoint element $a \in \text{selfAdjoint } A$, let $\Phi_u$ denote the $\mathbb{R}$-linear map of conjugation by $u$ defined by $\Phi_u(a) = uau^*$. Then applying the map for $u$ followed by the map for its adjoint $u^*$ returns the original element: $\Phi_{u^*}(\Phi_u(a)) = a$.

Let $A$ be a unital star-algebra and $u$ be a unitary element of $A$. For any self-adjoint element $a \in \text{selfAdjoint } A$, let the conjugation map by an element $x$ be defined as $C_x(a) = xax^*$. Then, applying the conjugation map by $u^*$ and then by $u$ returns the original element $a$: $$u(u^* a u)u^* = a$$

For a unitary element $u$ in a unital star-algebra $A$, this definition represents the unital positive $\mathbb{R}$-linear map from the space of self-adjoint elements $\text{selfAdjoint } A$ to itself, defined by the conjugation $a \mapsto u a u^*$. The map is unital because it maps the identity element to itself ($u 1 u^* = 1$) and is positive because it maps the cone of non-negative elements to itself.

Let $A$ be a unital star-algebra. For any unitary element $u \in A$ and any self-adjoint element $a$ of $A$, the result of applying the unital positive linear map $\text{conjugationUPLM } u$ to $a$ is the conjugation $u a u^*$, i.e., $$(\text{conjugationUPLM } u)(a) = u a u^*$$

Let $A$ be a unital star-algebra and let $\Phi_u: \text{selfAdjoint } A \to \text{selfAdjoint } A$ denote the unital positive $\mathbb{R}$-linear map defined by conjugation by a unitary element $u \in A$, specifically $\Phi_u(a) = uau^*$. For any two unitary elements $u, v \in \text{unitary } A$, the composition of their respective conjugation maps satisfies $\Phi_u \circ \Phi_v = \Phi_{uv}$.

For any unital star-algebra $A$, the unital positive $\mathbb{R}$-linear map on the space of self-adjoint elements $\text{selfAdjoint } A$ defined by conjugation with the identity element $1$ of the unitary group (mapping $a \mapsto 1 a 1^*$) is the identity map on $\text{selfAdjoint } A$.

For a unitary element $u$ in a unital star-algebra $A$, this definition constructs a symmetry (an order-automorphism) of the space of self-adjoint elements $\text{selfAdjoint } A$. The symmetry is defined by the conjugation map $a \mapsto u a u^*$, with its two-sided inverse given by conjugation by the adjoint $u^*$, i.e., $a \mapsto u^* a u$. This map preserves the identity, the linear structure, and the ordering of the self-adjoint part of the algebra.

Let $A$ be a unital star-algebra. For any unitary element $u \in \text{unitary } A$, the symmetry (order-automorphism) of the self-adjoint elements $\text{selfAdjoint } A$ defined by $a \mapsto uau^*$, denoted as $\text{conjugationSymmetry } u$, is equal to the unital positive $\mathbb{R}$-linear map $\text{conjugationUPLM } u$.

For a unital star-algebra $A$, let $U(A)$ be the group of unitary elements and $\text{selfAdjoint}(A)$ be the set of self-adjoint elements. The map $u \mapsto \Phi_u$, where $\Phi_u(a) = uau^*$ for $a \in \text{selfAdjoint}(A)$, defines a group homomorphism from $U(A)$ to the group of symmetries (order-automorphisms) of $\text{selfAdjoint}(A)$. This homomorphism satisfies $\Phi_{uv} = \Phi_u \circ \Phi_v$, where the composition follows the convention that the right-hand factor is applied first.

Let $A$ be a unital star-algebra. For any unitary element $u \in \text{unitary } A$, the value of the group homomorphism $\text{conjugationSymmetryHom}$ at $u$ is equal to the symmetry $\text{conjugationSymmetry } u$, which is the order-automorphism of the self-adjoint elements $\text{selfAdjoint } A$ defined by the conjugation map $a \mapsto uau^*$.

Given a group homomorphism $U : G \to \mathcal{U}(A)$ from a group $G$ to the group of unitary elements of a unital star-algebra $A$, this definition constructs a group homomorphism from $G$ to the group of symmetries (order-automorphisms) of the self-adjoint part $\text{sa}(A)$. For each $g \in G$, the resulting symmetry $\alpha_g$ is defined by the conjugation $\alpha_g(a) = U_g a U_g^*$ for any self-adjoint element $a \in \text{sa}(A)$. This represents the algebraic core of a symmetry action in a quantum system induced by a unitary representation.

Let $G$ be a group and $A$ be a unital star-algebra. Given a unitary representation $U : G \to \mathcal{U}(A)$ (where $\mathcal{U}(A)$ is the group of unitary elements in $A$), for any $g \in G$, the symmetry of the self-adjoint elements $\text{sa}(A)$ induced by the representation $U$ at $g$ is equal to the symmetry defined by the conjugation map $a \mapsto U_g a U_g^*$.

Let $G$ be a group and $A$ be a unital star-algebra. Given a unitary representation $U : G \to \mathcal{U}(A)$ (a group homomorphism into the unitary group of $A$), for any $g \in G$ and any self-adjoint element $a \in \text{sa}(A)$, the induced symmetry $\alpha_g$ acts on $a$ via conjugation: $$\alpha_g(a) = U_g a U_g^*$$ where $U_g$ is the image of $g$ under $U$ and $U_g^*$ is its adjoint.