PhyslibAlpha.ProbabilisticTheory.HilbertSpace.Dynamics.Automorphism

Let $H$ be a complex Hilbert space and let $\mathcal{B}(H)$ (denoted as $H \to L[\mathbb{C}] H$) be the algebra of bounded linear operators on $H$. The map that sends a unitary operator $U \in \mathcal{B}(H)$ to the $\star$-automorphism defined by unitary conjugation $A \mapsto U A U^*$ is surjective. In other words, every $\star$-automorphism of $\mathcal{B}(H)$ is implemented by a unitary conjugation.

Let $H$ be a complex Hilbert space and let $B(H)$ (denoted as `H →L[ℂ] H`) be the algebra of bounded linear operators on $H$. For any two unitary operators $u, v \in B(H)$, the $\star$-automorphisms of $B(H)$ induced by conjugation (where $u$ acts as $a \mapsto u a u^*$) are equal if and only if there exists a scalar phase $c \in \mathbb{C}$ (where $|c| = 1$) such that $u = cv$.

Given a complex Hilbert space $H$, the projective unitary group $PU(H)$ is defined as the quotient of the group of unitary operators $U(H)$ on $H$ by the subgroup of scalar phases, which are operators of the form $c I$ for $c \in \mathbb{C}$ with $|c| = 1$. This group is isomorphic to the group of $\star$-automorphisms of the algebra of bounded linear operators $\mathcal{B}(H)$.

For a complex Hilbert space $H$, the projective unitary group $PU(H)$ is equipped with a group structure. This group is formed by the quotient of the group of unitary operators $U(H)$ on $H$ by its center, which consists of scalar phases $\{c I \mid c \in \mathbb{C}, |c| = 1\}$.

Let $H$ be a complex Hilbert space, and let $\mathcal{B}(H)$ (denoted as $H \to L[\mathbb{C}] H$) be the algebra of bounded linear operators on $H$. There exists a group isomorphism between the projective unitary group $PU(H)$ and the group of $\star$-algebra automorphisms of $\mathcal{B}(H)$. This isomorphism maps a projective unitary $[U] \in PU(H)$ to the automorphism defined by the conjugation map $A \mapsto U A U^*$ for $A \in \mathcal{B}(H)$.

Let $H$ be a complex Hilbert space and let $\mathcal{B}(H)$ denote the space of bounded linear operators on $H$. Let $f : \mathbb{R} \to \mathcal{B}(H)$ be a function. If for every $s \in \mathbb{R}$, the derivative of $f$ at $s$ is the zero operator, then $f(t) = f(0)$ for all $t \in \mathbb{R}$.

Let $H$ be a complex Hilbert space and $\mathcal{B}(H)$ be the algebra of bounded linear operators on $H$. Let $U: \mathbb{R} \to \mathcal{B}(H)$ be a norm-continuous unitary one-parameter group with generator $G$. Suppose $\alpha: \mathbb{R} \to (\mathcal{B}(H) \to \mathcal{B}(H))$ is a flow such that for a fixed operator $a \in \mathcal{B}(H)$, the path $t \mapsto \alpha_t(a)$ satisfies the Heisenberg equation: $$\frac{d}{dt} \alpha_t(a) = [\alpha_t(a), iG]$$ where $[\cdot, \cdot]$ denotes the commutator. Then for any $s \in \mathbb{R}$, the derivative of the conjugated path $r \mapsto U(r)^* \alpha_r(a) U(r)$ at $r = s$ is zero: $$\frac{d}{dr} \left( U(r)^* \alpha_r(a) U(r) \right) \bigg|_{r=s} = 0$$