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

So far I can only see one direction, namely $𝑟(𝑎)≤$ inf$_{𝑏∈𝐼𝑛𝑣(𝐴)}||𝑏𝑎𝑏^{−1}||$. I'm wondering how to prove the other direction.

Thank you very much for the help.