nLab
functional calculus

Contents

Idea

What is called functional calculus or function calculus are operations by which for ff a function (on the complex numbers , for instance) and aa a suitable linear operator (on a Hilbert space, for instance) one makes sense of the expression f(a)f(a) as a new operator. Usually one requires that the assignment ff(a)f \mapsto f(a) is an algebra homomorphism, but not always, namely in some contexts as quantization the ordering effects may not respect the homomorphism property, see Weyl functional calculus.

Statements

Let AA be a C-star algebra (possibly non-commutative) and aAa \in A a normal operator. With sp(a)sp(a) the operator spectrum of aa write C(sp(a))C(sp(a)) for the commutative C *C^\ast-algebra of contrinuous complex-valued functions on sp(a)sp(a). Finally write ι:C(sp(A))\iota : \in C(sp(A)) for the function ι:xx\iota : x \mapsto x.

Theorem

There is a unique star-algebra homomorphism

ϕ a:C(sp(a))A \phi_a : C(sp(a)) \to A

such that ϕ(ι)=a\phi(\iota) = a.

For all fC(sp(a))f \in C(sp(a)) we have that ϕ(f)A\phi(f) \in A is a normal operator.

This appears for instance as (KadisonRingrose, theorem 4.4.5).

Proof

Let aA\langle a \rangle \subset A be the C *C^\ast-algebra generated by aa, or in fact any commutative C *C^\ast-subalgebra of AA containing aa.

Then by Gelfand duality there is a compact topological space XX and an isomorphism ψ:aC(X)\psi : \langle a \rangle \stackrel{\simeq}{\to} C(X).

Define a morphism

()ψ(a):C(sp(a))C(X) (-) \circ \psi(a) : C(sp(a)) \to C(X)

by ffψ(a)f \mapsto f \circ \psi(a). This is a continuous *\ast-algebra homomorphism. Therefore so is the composite

ϕ:C(sp(a))()ψ(a)C(X)ψ 1aA. \phi : C(sp(a)) \stackrel{(-)\circ \psi(a)}{\to} C(X) \stackrel{\psi^{-1}}{\to} \langle a\rangle \hookrightarrow A \,.

And this satisfies ϕ(ι)=ψ 1(ιψ(a))=ψ 1ψ(a)=a\phi(\iota) = \psi^{-1}(\iota \circ \psi(a)) = \psi^{-1}\psi (a) = a.

This establishes the existence of ϕ\phi. To see uniqueness, notice that any other morphism with these properties coincides with ϕ\phi on all polynomials in ι\iota and ι¯\bar \iota. By the Stone-Weierstrass theorem such polynomials form an everywhere-dense subset of C(sp(a))C(sp(a)). Since moreover one can see that the two morphisms must be isometric (…) it follows that they in fact agree.

References

A standard textbook is for instance

  • Richard Kadison, John Ringrose, Fundamentals of the theory of operator algebra , Academic Press (1983)

See also

Revised on October 12, 2013 21:51:28 by Toby Bartels (98.19.41.253)