Continuous functional calculus

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

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[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(ι) = x, where ι denotes the function zz 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

 \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.

Related theorems[edit]

For the case of self-adjoint operators on a Hilbert space (incl. unbounded operators) the Borel functional calculus is of greater interest. The latter has various formulations, and is also known as Spectral theorem. If one wants to stick to an abtract algebraic formation as opposed to operators on a given Hilbert space, the Borel functional calculus holds in the context of von Neumann algebras.

One may also cite the holomorphic functional calculous that holds for an arbitrary element of a C*-algebra or Riesz functional calculus for elements of an unital Banach algebra (A course in Functional analysis, Conway J., 4.7 p. 201).

References[edit]

Theorem VII.1 p. 222 in

Modern methods of mathematical physics, Vol. 1, Reed M., Simon B.