Continuous functional calculus

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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.

Theorem[edit]

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 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(id) = x, where id denotes the function zz on Sp(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[edit]

References[edit]

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