# Characterizing spectral radius using invertible elements in unital C* algebra [closed]

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.

• If I recall correctly, this is an exercise in Murphy's book. Where did you come across this, and what's the context in which you need to know the answer? – Yemon Choi Feb 14 at 5:25
• Yes this is a problem in Murphy's book. I'm a theoretical physicist and need some knowledge in C* algebra. I got stuck on this problem. – user152346 Feb 14 at 5:44
• This question was crossposted on MSE: math.stackexchange.com/questions/3546121/…. – MaoWao Feb 14 at 13:26
• By the way I should remark that I don't think this is a trivial question - without the hint given by the first part of the exercise, I remember being stuck on this for a while, a long time ago - but probably MSE is a better home for the question, now that it is getting feedback there. – Yemon Choi Feb 14 at 13:49
• I'm voting to close this question because it was subsequently asked on MSE and got feedback there which led to a resolution – Yemon Choi Feb 14 at 17:40