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]
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
Uniqueness follows from application of the Stone–Weierstrass theorem. Furthermore, the spectral mapping theorem holds:
continuous, functional, calculus, mathematics, particularly, operator, theory, algebra, theory, continuous, functional, calculus, functional, calculus, which, allows, application, continuous, function, normal, elements, algebra, contents, theorem, also, refere. 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 Contents 1 Theorem 2 See also 3 References 4 External linksTheorem EditTheorem Let x be a normal element of a C algebra A with an identity element e Then there is a unique mapping p f f x defined for a continuous function f on the spectrum s x of x such that p is a unit preserving morphism of C algebras and p 1 e and p id x where id denotes the function z z on s x 1 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 p f f x displaystyle pi f f circ x Uniqueness follows from application of the Stone Weierstrass theorem Furthermore the spectral mapping theorem holds s f x f s x displaystyle sigma f x f sigma x 2 See also EditBorel functional calculus Holomorphic functional calculusReferences Edit Theorem VII 1 p 222 in Modern methods of mathematical physics Vol 1 Reed M Simon B Spectral mapping theorem on PlanetMathExternal links EditContinuous functional calculus on PlanetMath Retrieved from https en wikipedia org w index php title Continuous functional calculus amp oldid 1127647849, wikipedia, wiki, book, books, library,
