Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 106 / 2

The holomorphic functional calculus assigns to $a$ in a unital complex Banach algebra and every function holomorphic near $\sigma (a)$ the elementwhere $\Gamma$ winds once around the spectrum. It is a unital algebra homomorphism, extends polynomial evaluation, is independent of the admissible contour, and obeys the spectral mapping theorem

Let $z_K\in \mathcal{R}(K)$ be the coordinate function. For a character $\phi$, put $\lambda =\phi (z_K)$. If $\lambda \notin K$, then $(z_K-\lambda {)}^{-1}$ is one of the admitted rational functions, contradicting the fact that $z_K-\lambda$ lies in the kernel of $\phi$. Thus $\lambda \in K$. For every rational function $r$ without poles on $K$,Continuity of characters and uniform density extend this identity to every member of $\mathcal{R}(K)$. Conversely evaluation at every $\lambda \in K$ is a character. Therefore the character space of R(K) is naturally $K$.

Runge approximation theorem says that if $K\subset \mathbb{C}$ is compact, $f$ is holomorphic on a neighbourhood of $K$, and one chooses one point in every bounded component of $\mathbb{C}\setminus K$, then $f$ can be approximated uniformly on $K$ by rational functions whose finite poles belong only to the chosen points.

To prove it, surround $K$ by finitely many small rectangles contained in the domain of $f$ and apply the Cauchy integral formula on their oriented boundaries:Riemann sums approximate this integral uniformly on $K$ by rational functions with poles on $\Gamma$. If a pole $b$ lies in a component containing the selected point $a$, choose a polygonal path from $b$ to $a$ inside that component and subdivide it finely. The resolvent identityallows the pole to be moved step by step along the path, with arbitrarily small uniform error on $K$. Moving every pole proves the theorem.

For an open set $U$, choose a compact exhaustion $K_1\subset K_2\subset \cdots$ such that every component of $\mathbb{C}\setminus K_n$ meets $\mathbb{C}\setminus U$. Runge's theorem approximates a holomorphic function on $K_n$ by a rational function with all poles outside $U$. A diagonal choice gives convergence uniformly on every compact subset, proving that such rational functions are dense in the space of holomorphic functions $\mathcal{O}(U)$ with its compact-open topology.

Finally, equality of the Gelfand transforms givesLet this common compact spectrum be $K$. The divided differenceis holomorphic near $K\times K$. The two-variable holomorphic functional calculus for the commuting pair $(x_1,x_2)$ gives an element $u=g(x_1,x_2)$ satisfyingFor every character,An element of a commutative unital Banach algebra is invertible exactly when its Gelfand transform has no zero. Thus $u$ is invertible, and $f(x_1)=f(x_2)$ implies $x_1=x_2$. This is injectivity through a holomorphic functional calculus with nonvanishing derivative.