> [!theorem] > > Let $\alg$ be a unital complex [[Banach Algebra|Banach algebra]] on which every non-zero element is invertible, then $\alg \iso \complex$. > > *Proof*. Suppose that $x \not\in \complex e$, then $\lambda e - x \ne 0$ for all $\lambda \in \complex$, so the [[Spectrum|spectrum]] would be empty, which is impossible.