I reently proved the theorem for unital $C^\ast$-algebras that for $a\in A$ normal there exists a unique unital isometric $\ast$-homomorphism $\varphi : C(\sigma(a))\to A$ with $\varphi(i) = a$ where $i$ is the inclusion map.
It is not clear yet to me why but this is an important result (or so I believe). What is an example where this used?
The basic idea is that $\varphi(f)$ behaves like evaluation of $f$ at $a$ for any continuous function. The immediate application is that we can "evaluate" well-known functions with elements of a $C^{*}$-algebra, e.g. $f(x) = e^x$, logarithms, square roots, etc. They have analogous uses that these functions have when dealing with numbers. For example, if we have $\sigma(a)\subseteq [0,\infty)$, we can show that $A$ has a unique element $b$ so that $\sigma(b)\subseteq [0,\infty)$ and $b^2 = a$ or that if $a$ is positive, then $a$ has a unique positive square root.
Also since $f$ can be any continuous function, we can use piece-wise linear or other continuous functions to "evaluate" at $a$. For example, if $\sigma(a)$ is disconnected, there is a continuous function that takes the values $0$ and $1$ (without being constant) and so there exists a nontrivial projection $p = \phi(f)$ in $A$ so that $pa = ap$.