*-algebra problem

dray

Member
Let $X$ be a *-algebra with identity $e$, and let $e\in{X}$, $\lambda\in\mathbb{C}$. Can somebody show me how $\sigma(\lambda{e}-x)=\lambda-\sigma(x)$, where $\sigma(x)$ is the spectrum of an element.

Thanks in advance.

girdav

Member
$v\in\sigma(\lambda e-x)$ if and only if $(\lambda-v)e-x$ is invertible, that is if and only if $\lambda-v\in\sigma(x)$, which gives the result.