Continuous functional calculus

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

Theorem. Let x be a normal element of C*-algebra A with an identity element 1; 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) = 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.

In particular, this implies that bounded self-adjoint operators on a Hilbert space have a continuous functional calculus.

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