= Solution
The <Gelfand-Mazur theorem> says that a nonzero complex unital <Banach algebra> in which every nonzero element is invertible is isomorphic to $\mathbb C$ by the scalar map $\lambda\mapsto\lambda1$. With the usual normalization $\|1\|=1$, that isomorphism is isometric. Commutativity need not be assumed in the theorem; it follows from the conclusion.
\b[Every complex Banach division <algebra> is the scalar <field>.] We use complex scalars throughout this question, including for $C^1[0,1]$; the complex-<algebra character> and spectrum conclusions depend on that convention.
Back to article page
