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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

\( \pi (f)=f\circ x. \)

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

Theorem VII.1 p. 222 in Modern methods of mathematical physics, Vol. 1, Reed M., Simon B.

External links

Continuous functional calculus on PlanetMath

Functional analysis (topics – glossary)
Spaces

Hilbert space Banach space Fréchet space topological vector space

Theorems

Hahn–Banach theorem closed graph theorem uniform boundedness principle Kakutani fixed-point theorem Krein–Milman theorem min-max theorem Gelfand–Naimark theorem Banach–Alaoglu theorem

Operators

bounded operator compact operator adjoint operator unitary operator Hilbert–Schmidt operator trace class unbounded operator

Algebras

Banach algebra C*-algebra spectrum of a C*-algebra operator algebra group algebra of a locally compact group von Neumann algebra

Open problems

invariant subspace problem Mahler's conjecture

Applications

Besov space Hardy space spectral theory of ordinary differential equations heat kernel index theorem calculus of variation functional calculus integral operator Jones polynomial topological quantum field theory noncommutative geometry Riemann hypothesis

Advanced topics

locally convex space approximation property balanced set Schwartz space weak topology barrelled space Banach–Mazur distance Tomita–Takesaki theory

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

Hellenica World - Scientific Library

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