Continuous functional calculus
Theorem. Let x be a normal element of a C*-algebra A with an identity element e; then there is a unique mapping π : f → f(x) defined for f a continuous function on the spectrum Sp(x) of x such that π is a unit-preserving morphism of C*-algebras such that π(1) = e and π(id) = x, where id denotes the function z → z on Sp(x).
The proof of this fact is almost immediate from the Gelfand representation: it suffices to assume A is the C*-algebra of continuous functions on some compact space X and define
Uniqueness follows from application of the Stone-Weierstrass theorem.
In particular, this implies that bounded normal operators on a Hilbert space have a continuous functional calculus.