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> I am trying to see that if we have a complex Banach algebra with unity, we will have that $$r\_\sigma(x)=\|x\| \iff \|x^2\|=\|x\|^2.$$
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I was able to do the first implication: Since we know that $\|x^2||\leq \|x||^2$ always and we have that $r\_\sigma(x^2)=r\_\sigma(x)^2=\|x\|^2$ so we will have that $\|x^2\|\geq r\_\sigma(x^2)=\|x\|^2$.

Now for the other one, I haven't figured out how to do it yet I just can't seem to see why we have $r\_\sigma(x)\geq \|x\|$ if $\|x^2\|=\|x\|^2$, so any tips are appreciated. Thanks.