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. 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'').

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
:$\pi(f) = f \circ x.$
Uniqueness follows from application of the Stone–Weierstrass theorem. Furthermore, the spectral mapping theorem holds:

:$\sigma(f(x)) = f(\sigma(x)).$Spectral mapping theorem on PlanetMath
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See also

* Borel functional calculus
* Holomorphic functional calculus

References

External links

Continuous functional calculus on PlanetMath

{{DEFAULTSORT:Continuous Functional Calculus
Theory of continuous functions
C*-algebras
Functional calculus