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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 operator|normal]] element of a C*-algebra ''A'' with an identity element e; 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) = e 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
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:<math> \pi(f) = f \circ x. </math>
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 [[operator (mathematics)|operator]]s on a Hilbert space of more interest is the [[Borel functional calculus]].
 
{{Functional Analysis}}
 
{{DEFAULTSORT:Continuous Functional Calculus}}
[[Category:Functional calculus]]
[[Category:Continuous mappings]]
[[Category:C*-algebras]]

Revision as of 01:36, 4 March 2014


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