Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 6 / 3 / ii

A complex Banach algebra is an associative complex algebra with a complete norm satisfying submultiplicativity, $\Vert ab\Vert \le \Vert a\Vert \Vert b\Vert$. A field has a nonzero identity $e$, and every nonzero element is invertible. We derive the Gelfand-Mazur theorem from these facts.

If $\Vert z\Vert <1$, the series $e+z+z^2+\cdots$ converges in the complete algebra. Multiplying its partial sums by $e-z$ and passing to the limit proves the Neumann series inverse. In particular, for any $x\in B$ and $|\lambda |>\Vert x\Vert$,This resolvent of an element tends to zero in norm at infinity; the estimate follows from the geometric bound on $\Vert x^n\Vert$.

We next prove that the spectrum of an element cannot be empty. Suppose every $\lambda e-x$ were invertible. At each ${\lambda }_0$, writeThe Neumann series on a small disk about ${\lambda }_0$ proves that $R$ is analytic there. Consequently, for each continuous linear functional $\ell :B\to \mathbb{C}$, the function $\lambda \mapsto \ell (R(\lambda ))$ is entire. It is bounded outside a large disk by the estimate above and bounded on that disk by continuity. The Liouville theorem makes it constant; its zero limit at infinity makes the constant zero.

Continuous complex linear functionals separate points of $B$. Here is the needed consequence of the Hahn-Banach theorem: for $z\ne 0$, define $g(tz)=t\Vert z\Vert$ on the real span of $z$ and extend it as a bounded real linear functional of norm one. Then $\ell (w)=g(w)-ig(iw)$ is complex linear and continuous, and $\mathrm{Re}\ell (z)=\Vert z\Vert \ne 0$. The real extension theorem is proved explicitly in Question 4(i) below. Thus $\ell (R(\lambda ))=0$ for every $\ell$ forces $R(\lambda )=0$, contradicting $(\lambda e-x)R(\lambda )=e\ne 0$. This proves the nonemptiness of the Banach-algebra spectrum.

Choose $\lambda \in {\sigma }_B(x)$. Since $B$ is a field, the noninvertible element $x-\lambda e$ must be zero. Hence every $x\in B$ is a scalar multiple of $e$. The mapis therefore a bijective unital complex-algebra homomorphism. It and its inverse are bounded, since $\Vert \lambda e\Vert =|\lambda |\Vert e\Vert$. Thus it is an isomorphism of Banach algebras. If the usual normalization $\Vert e\Vert =1$ is included in the definition, it is an isometry; without that normalization, it is still the required continuous algebra isomorphism.