Consider $A$ a unital $C^*$ algebra, I want to show that the spectral radius $r(a)$ satisfies the following: $$r(a) = \inf_{b\in \operatorname{Inv}(A)}\|bab^{-1}\|$$ where $\operatorname{Inv}(A)$ is the set of invertible elements in $A$.

So far I can only see one direction, namely $r(a) \leq \inf_{b\in \operatorname{Inv}(A)}\|bab^{-1}\|$. I wondering how to prove the other direction.

Thank you very much for the help.


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