Continuous functional calculus

Theorem. Let x be a normal element of a C*-algebra A with an identity element e. Let C be the C*-algebra of the bounded continuous functions on the spectrum σ(x) of x. Then there exists a unique mapping π : C → A, where π(f) is denoted f(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]

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

${\displaystyle \pi (f)=f\circ x.}$

Uniqueness follows from application of the Stone–Weierstrass theorem. Furthermore, the spectral mapping theorem holds:

${\displaystyle \sigma (f(x))=f(\sigma (x)).}$

• Borel functional calculus
• Holomorphic functional calculus

• Continuous functional calculus on PlanetMath