Borel functional calculus

Let B(H) be the algebra ( of bounded operatorsMathworldPlanetmathPlanetmath over a complex Hilbert spaceMathworldPlanetmath H and TB(H) a normal operator.

The Borel functional calculus is a functional calculusMathworldPlanetmath which enables the expression


to make sense as a bounded operator in H, for a boundedPlanetmathPlanetmathPlanetmathPlanetmath ( Borel function f.

In particular, it allows the definition of operatorsMathworldPlanetmath χS(T) for any characteristic functionMathworldPlanetmathPlanetmathPlanetmath χS, which are of significant importance on the of the of T.

The Borel functional calculus will be constructed by extending the continuous functional calculus for arbitrary bounded Borel functions.

1 Preliminary Facts

Let us set some notation first:

  • σ(T) will denote the spectrum ( of T.

  • C(σ(T)) will denote the C*-algebra ( of continuous functionsMathworldPlanetmathPlanetmath σ(T).

  • B(σ(T)) will denote the C*-algebra of bounded Borel functions σ(T), endowed with the sup norm.

The continuous functional calculus for T allows the expression f(T) to make sense for continuous functions fC(σ(T)), by the assignment of a unital *-homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath


that sends the identity function to T. This unital *-homomorphism is in fact uniquely determined by this property (see the entry on the continuous functional calculus ( for more details).

The objective is to extend π to a unital *-homomorphism π~:B(σ(T))B(H).

Since B(σ(T)) is a much larger C*-algebra than C(σ(T)), there is no reson to presume that there is only one extensionPlanetmathPlanetmathPlanetmath of π. Which extension would be the most natural then? It turns out that there is a unique extension that satisfies a good continuity property.

It is known that *-homomorphisms between C*-algebras are continuous (see this entry (, so that whenever a net fiB(σ(T)) convergesPlanetmathPlanetmath in the sup norm to a function fB(σ(T)) we will have that fi(T)f(T) in the operator norm. All extensions of π will automaticaly satisfy this continuity property, but this can be improved in a satisfactory manner.

Notation - Let X be a compactPlanetmathPlanetmath Hausdorff space, M(X) the space of all finite regularPlanetmathPlanetmathPlanetmathPlanetmath ( Borel measures in X and B(X) the C*-algebra of all bounded Borel functions in X. The weakest topologyMathworldPlanetmath in B(X) for which integration against any measureMathworldPlanetmath ν is continuous will be reffered to as the μ-topology. This means that fif in the μ-topology if and only if fi𝑑νf𝑑ν for all νM(X).

Notice that we can identify each function fB(X) with a bounded linear functionalMathworldPlanetmath ( ωf in M(X), given by


and the μ-topology corresponds exactly to the weak-* topology under this identification.

We will see in the next that there is an unique extension of π that is continuous from the μ-topology to the weak operator topology.

Just like the Stone-Weierstrass theorem ( allowed the passage from the polynomial functional calculus to the continuous functional calculus, the Riesz representation theoremMathworldPlanetmath ( will allow the passage from the latter to the Borel functional calculus.

2 Definition

The following result is the key for the definition of the Borel functional calculus.

TheoremMathworldPlanetmath 1 - Let T be a normal operator in B(H) and π:C(σ(T))B(H) the unital *-homomorphism corresponding to the continuous functional calculus for T. Then, π extends uniquely to a *-homomorphism π~:B(σ(T))B(H) that is continuous from the μ-topology to the weak operator topology. Moreover, each operator π(f) lies in strong operator ( closureMathworldPlanetmathPlanetmathPlanetmath ( of the unital *-algebra generated by T.

: See this attached entry (

We are now able to define the Borel functional calculus:

Definition - Let T be a normal operator in B(H). Let π~:B(σ(T))B(H) be the unique *-homomorphism defined in Theorem 1. This *-homomorphism is denoted by


and it is called the Borel functional calculus for T.

Since this functional calculus extends the polynomial functional calculus, we have that for any polynomialPlanetmathPlanetmath p(z):=cn,mznz¯m,


Moreover, since f(T) lies in the strong operator closure of the unital *-algebra generated by T, for any function fB(σ(T)), we see that f(T) is the strong operator limit of polynomials cn,mTn(T*)m.

3 Borel Calculus in von Neumann Algebras

The Borel functional calculus is in fact applicable for any normal operator T in any von Neumann algebraMathworldPlanetmathPlanetmathPlanetmath .

That is due to the fact, expressed in Theorem 1, that for every fB(σ(T)) the operator f(T) belongs to the strong operator closure of the unital *-algebra generated by T. Being a von Neumann algebra, is closed ( in the strong operator topology, and therefore all operators f(T) belong to .

Thus, by restrictionPlanetmathPlanetmath, we have in fact a *-homomorphism


satisfying the properties of Theorem 1, i.e. we have a Borel functional calculus for normal operators of a von Neumann algebra.


  • 1 W. Arveson, A Short Course on Spectral Theory, Graduate Texts in Mathematics, 209, Springer, New York, 2002
  • 2 N. Weaver, Mathematical Quantization, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001
Title Borel functional calculus
Canonical name BorelFunctionalCalculus
Date of creation 2013-03-22 18:48:44
Last modified on 2013-03-22 18:48:44
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 10
Author asteroid (17536)
Entry type Feature
Classification msc 47A60
Classification msc 46L10
Classification msc 46H30
Related topic FunctionalCalculus
Related topic PolynomialFunctionalCalculus
Related topic ContinuousFunctionalCalculus2
Defines Borel functions of a normal operator