Continuous functional calculus
In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. Theorem. 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 a continuous function "f" on the spectrum σ("x") of "x", such that π is a unit-preserving morphism of C*-algebras and π(1) = e and π(id) = "x", where id denotes the function "z" → "z  on σ(x"). In particular, this theorem implies that bounded normal operators on a Hilbert space have a continuous functional calculus. Its proof 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. Furthermore, the spectral mapping theorem holds: