Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 10 / 5 / a / 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 $\lambda \mapsto \lambda 1$. With the usual normalization $\Vert 1\Vert =1$, that isomorphism is isometric. Commutativity need not be assumed in the theorem; it follows from the conclusion.

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.