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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.

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    $\begingroup$ 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? $\endgroup$ – Yemon Choi Feb 14 at 5:25
  • $\begingroup$ 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. $\endgroup$ – user152346 Feb 14 at 5:44
  • $\begingroup$ This question was crossposted on MSE: math.stackexchange.com/questions/3546121/…. $\endgroup$ – MaoWao Feb 14 at 13:26
  • $\begingroup$ 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. $\endgroup$ – Yemon Choi Feb 14 at 13:49
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    $\begingroup$ I'm voting to close this question because it was subsequently asked on MSE and got feedback there which led to a resolution $\endgroup$ – Yemon Choi Feb 14 at 17:40