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. 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).
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.
- Theorem VII.1 p. 222 in Modern methods of mathematical physics, Vol. 1, Reed M., Simon B.