Physlib.QuantumMechanics.OperatorAlgebra.Dynamics.Automorphism

Let $H$ be a complex Hilbert space and $\mathcal{B}(H)$ be the algebra of bounded linear operators on $H$. Every $\mathbb{C}$-algebra $\star$-automorphism $\Phi: \mathcal{B}(H) \to \mathcal{B}(H)$ is implemented by unitary conjugation. That is, for any such $\star$-automorphism $\Phi$, there exists a unitary operator $U \in \mathcal{B}(H)$ such that $\Phi(A) = U A U^*$ for all $A \in \mathcal{B}(H)$.

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ denote the algebra of bounded linear operators on $\mathcal{H}$. For any two unitary operators $u, v \in \mathcal{B}(\mathcal{H})$, the $\star$-automorphisms of $\mathcal{B}(\mathcal{H})$ implemented by conjugation—defined by $A \mapsto uAu^*$ and $A \mapsto vAv^*$ respectively—are equal if and only if there exists a scalar phase $c \in U(1)$ (i.e., a complex number $c$ with $|c| = 1$) such that $u = cv$.

Let $\mathcal{H}$ be a complex Hilbert space. The **projective unitary group** $PU(\mathcal{H})$ is defined as the quotient of the group of unitary operators $U(\mathcal{H})$ by the kernel of the conjugation homomorphism $\text{Ad}: U(\mathcal{H}) \to \text{Aut}_{\star}(\mathcal{B}(\mathcal{H}))$, which maps a unitary $U$ to the $\star$-automorphism $A \mapsto U A U^*$. This quotient identifies unitary operators that differ only by a scalar phase $c \in \mathbb{C}$ (where $|c| = 1$).

Let $\mathcal{H}$ be a complex Hilbert space. The projective unitary group $PU(\mathcal{H})$ is equipped with a group structure. This structure is the quotient group structure derived from the unitary group $U(\mathcal{H})$ and the kernel of the conjugation homomorphism $\text{Ad}: U(\mathcal{H}) \to \text{Aut}_{\star}(\mathcal{B}(\mathcal{H}))$.

Let $H$ be a complex Hilbert space and $\mathcal{B}(H)$ be the algebra of bounded linear operators on $H$. There is a group isomorphism between the projective unitary group $PU(H)$ and the group of $\mathbb{C}$-algebra $\star$-automorphisms of $\mathcal{B}(H)$, denoted by $\text{Aut}_{\star}(\mathcal{B}(H))$. This isomorphism identifies each projective unitary (an equivalence class of unitaries $[U]$) with the unique $\star$-automorphism it implements via conjugation, $A \mapsto UAU^*$.