continuous functional calculus

Let B⁒(H) be the algebraMathworldPlanetmathPlanetmath of bounded operatorsMathworldPlanetmathPlanetmath over a complex Hilbert spaceMathworldPlanetmath H. Let T∈B⁒(H) be a normal operator.

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


to make sense as a bounded operator in H, for continuous functionsPlanetmathPlanetmath f.

More generally, when π’œ is a C*-algebra ( with identity elementMathworldPlanetmath e, and x is a normal elementMathworldPlanetmath of π’œ, the continuous functional calculus allows one to define f⁒(x)βˆˆπ’œ when f is a continuous function.

More precisely, if σ⁒(x) denotes the spectrum of x and C⁒(σ⁒(x)) denotes the C*-algebra of complex valued continuous functions on σ⁒(x), we will define a continuous homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath



that the functional calculus ( .

There are several reasons to require the continuity of f on the spectrum σ⁒(x).

For example, suppose Ξ»0βˆˆΟƒβ’(x). The function f⁒(Ξ»)=1Ξ»-Ξ»0 is clearly not continuous in Ξ»0. By the functional calculus we would obtain


but x-Ξ»0⁒e is not invertiblePlanetmathPlanetmath since Ξ»0βˆˆΟƒβ’(x).

The abstraction towards C*-algebras is almost . Indeed, C*-algebras are the appropriate where to and prove the continuous functional calculus. The conclusionsMathworldPlanetmath towards B⁒(H) then follow as a particular case.

1 Preliminary construction

Let π’œ be a unital C*-algebra and x a normal element in π’œ. Let β„¬βŠ†π’œ be the C*-subalgebra generated by x and the identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath e of π’œ.

Thus, ℬ is the norm closureMathworldPlanetmathPlanetmathPlanetmath of the algebra generated by x, x* and e.

Moreover, since x is , x*⁒x=x⁒x*, it follows that ℬ is commutativePlanetmathPlanetmathPlanetmath and ℬ consists of those elements yβˆˆπ’œ that can be approximated by polynomialsMathworldPlanetmathPlanetmath p⁒(x,x*) in x and x*.

Recall the following facts:

The following result is perhaps the for the definition of the continuous functional calculus.

Theorem 1 - β–³ and σ⁒(x) are homeomorphicMathworldPlanetmath topological spacesMathworldPlanetmath. Moreover, the mapping S:β–³β†’Οƒ(x) defined by


is such an homeomorphism.

: We need to check that S is well defined, i.e. ϕ⁒(x)βˆˆΟƒβ’(x) for all Ο•βˆˆβ–³.

From the identity


follows that x-ϕ⁒(x)⁒e cannot be invertible in ℬ (recall that Ο• is a multiplicative linear functional on ℬ).

Thus, ϕ⁒(x)βˆˆΟƒβ„¬β’(x). By the spectral invariance theorem, we see that ϕ⁒(x)βˆˆΟƒβ’(x)=σℬ⁒(x), and so S is well defined.

  • β€’

    S is continuous - Suppose ϕα is a net in β–³ such that Ο•Ξ±βŸΆΟ•. Recall that the topology in β–³ is the weak-* topology, so ϕα⁒(x)βŸΆΟ•β’(x).

    Thus, S⁒(ϕα)⟢S⁒(Ο•) and so S is continuous.

  • β€’

    S is - Suppose S⁒(Ο•1)=S⁒(Ο•2). Then, Ο•1⁒(x)=Ο•2⁒(x). Since


    we must also have Ο•1⁒(x*)=Ο•2⁒(x*).

    This clearly implies that


    for every polynomial in two variables p.

    Recall that the ”polynomials” p⁒(x,x*) are dense in ℬ. So we must have Ο•1⁒(y)=Ο•2⁒(y) for every yβˆˆβ„¬, i.e. Ο•1=Ο•2.

  • β€’

    S is - Let Ξ»βˆˆΟƒβ’(x)=σℬ⁒(x). Then x-λ⁒e is not invertible.

    Since ℬ is commutative, x-λ⁒e is contained in a maximal idealMathworldPlanetmathPlanetmath β„³.

    As β„³ is maximal ideal, the quotient ℬ/β„³ is a division algebraMathworldPlanetmath, and so by the Gelfand-Mazur theorem, ℬ/β„³ must ne isomorphic to β„‚.

    Therefore the quotient homomorphism


    is a multiplicative linear functional such that ϕ⁒(x-λ⁒e)=0, i.e. ϕ⁒(x)=Ξ», i.e. S⁒(Ο•)=Ξ».

    Therefore, S is surjective.

Since S is a continuous bijective function from the compact Hausdorff space β–³ to σ⁒(x), it follows that it must be a homeomorphism. β–‘

2 Definition of the continuous functional calculus

If S is the homeomorphism between β–³ and σ⁒(x) defined as above, then the mapping f↦f∘S is a *-isomorphism between the algebras C⁒(σ⁒(x)) and C⁒(β–³). Since the Gelfand transform G:β„¬βŸΆC⁒(β–³) is a also a *-isomorphism, we obtain a *-isomorphism


by setting Γ⁒(f):=G-1⁒(f∘S).

Definition - Suppose x is a normal element in a unital C*-algebra A. For every f∈C⁒(σ⁒(x)) we define


The mapping Ξ“, such that f↦f⁒(x), is called the continuous functional calculus for x.

We now prove the functional calculus ( for the continuous functional calculus and show its uniqueness:

Theorem 2 - Let A be a unital C*-algebra, x∈A a normal element and id the identity function in C. The continuous functional calculus for x is the unique unital *-homomorphism between C⁒(σ⁒(x)) and A which sends id to x. In particular, for every polynomial p in C of the form p⁒(Ξ»):=βˆ‘cn,m⁒λn⁒λ¯m, we have p⁒(x)=βˆ‘cn,m⁒xn⁒(x*)m.

: We have seen that the continuous functional calculus Ξ“ for x is a *-homomorphism between C⁒(σ⁒(x)) and π’œ. Recall that Ξ“ was defined by Γ⁒(f):=G-1⁒(f∘S). It is clear by the definition that Ξ“ is unital. Also, Gβˆ˜Ξ“β’(f)=f∘S for every f∈C⁒(β–³). Taking the identity function id we obtain that for every Ο•βˆˆβ–³

Gβˆ˜Ξ“β’(id)⁒(Ο•) = id⁒(S⁒(Ο•))
= ϕ⁒(x)
= G⁒(x)⁒(Ο•)

Since the Gelfand transform is a *-isomorphism, we must have Γ⁒(id)=x.

Now, let p:β„‚β†’β„‚ be a polynomial of the form p⁒(Ξ»):=βˆ‘cn,m⁒λn⁒λ¯m. Notice that p=βˆ‘cn,m⁒idn⁒idΒ―m. If F is any unital *-homomorphism such that F⁒(id)=x, then one must have F⁒(p)=βˆ‘cn,m⁒xn⁒(x*)m. Thus all such unital *-homomorphisms coincide on the subspaceMathworldPlanetmath of polynomials of the above form. By the Stone-Weierstrass theorem (, this subspace is dense in C⁒(σ⁒(x)). Thus, all such unital *-homomorphisms coincide in C⁒(σ⁒(x)), and uniqueness is proven. β–‘

3 Properties

  • β€’

    The spectral mapping theorem assures that for f∈C⁒(σ⁒(x))

  • β€’

    When σ⁒(x)βŠ‚β„+ the continuous functional calculus assures the existence of a square root x of x, since Ξ» is defined and continuous on Ξ»βˆˆΟƒβ’(x).

Title continuous functional calculus
Canonical name ContinuousFunctionalCalculus
Date of creation 2013-03-22 17:30:02
Last modified on 2013-03-22 17:30:02
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 11
Author asteroid (17536)
Entry type Feature
Classification msc 47A60
Classification msc 46L05
Classification msc 46H30
Related topic FunctionalCalculus
Related topic PolynomialFunctionalCalculus
Related topic BorelFunctionalCalculus
Defines continuous functions of normal operators