Exercise 26, § 6

Let A be a unital star algebra. Let $a$ and $b$ be normal elements of A and let $x\in A^*$ such that $b=xax^{-1}$. There exists a unitary element $u$ of A such that $b=uau^*$. (Use the previous exercise and the polar decomposition of $x$, noting that $|x|$ belongs to the bicommutant of $x$).