Continuity and uniqueness of holomorphic functional calculus (${\Theta }_x:\mathcal{O}(U)\to A$)

Equip holomorphic functions on an open spectral neighborhood with the compact-open topology. A fixed admissible contour gives a bound on $\Vert {\Theta }_x(f)\Vert$ by a constant times the supremum of $|f|$ on the contour, proving continuity. The Runge theorem makes rational functions with poles outside the open set dense. A unital homomorphism sending the coordinate function to $x$ must send each inverse coordinate difference to the corresponding resolvent, so its rational values and then all its holomorphic values are forced.