A is unital $C^*$ algebra and $a \in A$. I need to prove that set of elements {$1,a,a^*$} is linearly dependent if and only if $a$ is normal element and $\sigma(a)$ lies in direction in the complex plane.
I am done with first direction, but I have a problem with the other one, to prove that set of {$1,a,a^*$} is linearly dependent if $a$ is normal and $\sigma(a)$ lies in direction in the complex plane.
Thank you for help and suggestions.
If $a$ is normal, then $\sigma(f(a))=f(\sigma(a))$ for any function $f$ which is continuous on $\sigma(A)$. Let $\lambda_{0}\in\sigma(A)$. If $a$ is normal and has spectrum in a particular direction, then $e^{i\theta}(a-\lambda_{0}1)$ is normal and has only real spectrum for some $\theta$. Thus $e^{i\theta}(a-\lambda_{0}1)$ is selfadjoint. So $e^{i\theta}a-\lambda_{0}1=e^{-i\theta}a^{\star}-\overline{\lambda_{0}}1$.