Spectral radii and norms of similar elements in a C*-algebra: $\|bab^{-1}\|<1$ if $b=(\sum_{n=0}^\infty (a^*)^n a^n)^{1/2}$
The first part can be solved by simplifying $(bab^{-1})^*bab^{-1}$ to show that it has norm less than $1$, using the fact that $b\geq 1$. (In particular, $a^*b^2a$ simplifies nicely.)
The second part can be solved, in the case when $r(a)>0$, by applying the first part to $\frac{1-\varepsilon}{r(a)}a$, yielding for $0<\varepsilon<1$ an invertible $b$ such that $\|bab^{-1}\|\leq\frac{r(a)}{1-\varepsilon}$. The case when $r(a)=0$ can then be solved by applying the previous case to $a+\varepsilon e$ and using the inequality $\|x+y\|\geq \|x\|-\|y\|$.