Construct the unitary from the given unitary operator conjugation operator

Question

Let $A$ be a C*-algebra (or just consider $B(H)$ for some Hilbert space $H$). Now for a given $T\in \text{Aut}(A)$ and $T$ is of this form: $$T(x)=uxu^*$$ for some unitary $u\in A$, $x\in A$. (We only know $T$ but not $u$.) Can we construct $u$ from $T$? In other word, is there a certain way to construct a unitary $u$ from the given $T$ and if $T(x)=vxv^*$ for some other unitary $v\in A$, then $v=\theta u$ for some $\theta\in\mathbb{C},|\theta|=1$?

Answer 1

In the case of $B(H)$, consider $x = e_j e_k^*$ where $\{e_i\}$ are an orthonormal basis of $H$. Then $u x u^* = (u e_j)(u e_k)^*$. Thus $u e_j$ is a unit vector that is a scalar multiple of $(u x u^*) w$ for any $w \in H$ such that $(u x u^*) w \ne 0$.

EDIT: First try $j=k$: $T(e_j e_j^*)$ must be of the form $v_j v_j^*$ for some unit vector $v_j$, which $v_j v_j^*$ determines up to multiplication by a scalar of absolute value $1$. Then $T(e_j e_k^*) = v_j v_k^*$. This determines the scalar for $v_k$ relative to that for $v_j$. Thus everything is determined up to one scalar.