Let $A$ be a unital $C^*$-algebra and $a\in A$ such that $r(a) < 1$. Define b = $(\sum_{n=0}^\infty (a^*)^n a^n)^{1/2}$. We can prove that $b\geq e$ and that $b$ is invertible. I want to show $\| b a b^{-1} \| < 1$.
From the definition of $b$ we see that $a^* b^2 a = b^2-e$ and we know $r(bab^{-1}) = r(a) <1$.
So it suffices to prove $r(b a b^{-1}) = \| b a b^{-1} \|$. It can follow from the fact $c = ba b^{-1}$ is a normal element... I don't know how to prove it (I have tried to compare $c^* c$ and $c c^*$...).
Context: The question appears in Murphy's book, page 74. I have managed to prove the first part. The second part of the question is to prove $r(a)= \inf_{c\in Inv(A)}\{\|cac^{-1} \| \}$ It's easy to see $r(a) \leq \inf_{c\in Inv(A)}\{\|cac^{-1} \| \}$ But I can't prove $r(a) \geq \inf_{c\in Inv(A)}\{\|cac^{-1} \| \}$ If we have had $r(a) = \|b a b^{-1} \|$ then it was obvious... But this is not true, so how we can prove this inequality?