Continuous functional calculus

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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 : ff(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 zz 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

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

References

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

External links

  • Continuous functional calculus on PlanetMath
Retrieved from "https://en.wikipedia.org/w/index.php?title=Continuous_functional_calculus&oldid=839280521"
This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/Continuous_functional_calculus
This page is based on the copyrighted Wikipedia article "Continuous functional calculus"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA