I need help with the following problem:
Let $A$ be a unital $C^{*}$-algebra.
(a) If $r(a)<1$ and $b=(\sum_{n=0}^{\infty}a^{*n}a^{n})^{1/2}$, show that $b\geq 1$ and $||bab^{-1}||<1$.
(b) For all $a\in A$, show that $r(a)=\inf_{b\in\mathrm{Inv}(A)}||bab^{-1}||=\inf_{b\in A_{sa}}||e^{b}ae^{-b}||$.