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

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

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.

See also[]

• Borel functional calculus
• Holomorphic functional calculus

References[]

External links[]

• Continuous functional calculus on PlanetMath