Continuous functional calculus

In operator theory and C*-algebra theory the continuous functional calculus allows applications of continuous functions to normal elements of a associates to a normal element C*-algebra. More precisely,

Theorem. Let x be a normal element of C*-algebra A with an identity element 1; the 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) = 1 and π(ι) = x, where ι 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.

This implies in particular, that bounded self-adjoint operators on Hilbert spaces have a continuous functional calculus.

For the case of normal operators on a Hilbert space of more interest is the Borel functional calculus.