= Continuity and uniqueness of holomorphic functional calculus
{title2=$\Theta_x:\mathcal O(U)\to A$}
Equip <holomorphic functions> on an open spectral neighborhood with the compact-open topology. A fixed admissible contour gives a bound on $\|\Theta_x(f)\|$ 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.
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