Continuous functional calculus
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 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
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.
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 abstract 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.
- Theorem VII.1 p. 222 in Modern methods of mathematical physics, Vol. 1, Reed M., Simon B.
- A course in Functional analysis, Conway J., 4.7 p. 201