# 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*).[1]

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.

## References

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