# Contents

## Idea

What is called functional calculus or function calculus are operations by which for $f$ a function (on the complex numbers , for instance) and $a$ a suitable operator (on a Hilbert space, for instance) one makes sense of the expression $f\left(a\right)$ as a new operator. Usually one requires that the assignment $f↦f\left(a\right)$ 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 $A$ be a C-star algebra (possibly non-commutative) and $a\in A$ a normal operator. With $\mathrm{sp}\left(a\right)$ the operator spectrum of $a$ write $C\left(\mathrm{sp}\left(a\right)\right)$ for the commutative ${C}^{*}$-algebra of contrinuous complex-valued functions on $\mathrm{sp}\left(a\right)$. Finally write $\iota :\in C\left(\mathrm{sp}\left(A\right)\right)$ for the function $\iota :x↦x$.

###### Theorem

There is a unique star-algebra homomorphism

${\varphi }_{a}:C\left(\mathrm{sp}\left(a\right)\right)\to A$\phi_a : C(sp(a)) \to A

such that $\varphi \left(\iota \right)=a$.

For all $f\in C\left(\mathrm{sp}\left(a\right)\right)$ we have that $\varphi \left(f\right)\in A$ is a normal operator.

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

###### Proof

Let $⟨a⟩\subset A$ be the ${C}^{*}$-algebra generated by $a$, or in fact any commutative ${C}^{*}$-subalgebra of $A$ containing $a$.

Then by Gelfand duality there is a compact topological space $X$ and an isomorphism $\psi :⟨a⟩\stackrel{\simeq }{\to }C\left(X\right)$.

Define a morphism

$\left(-\right)\circ \psi \left(a\right):C\left(\mathrm{sp}\left(a\right)\right)\to C\left(X\right)$(-) \circ \psi(a) : C(sp(a)) \to C(X)

by $f↦f\circ \psi \left(a\right)$. This is a continuous $*$-algebra homomorphism. Therefore so is the composite

$\varphi :C\left(\mathrm{sp}\left(a\right)\right)\stackrel{\left(-\right)\circ \psi \left(a\right)}{\to }C\left(X\right)\stackrel{{\psi }^{-1}}{\to }⟨a⟩↪A\phantom{\rule{thinmathspace}{0ex}}.$\phi : C(sp(a)) \stackrel{(-)\circ \psi(a)}{\to} C(X) \stackrel{\psi^{-1}}{\to} \langle a\rangle \hookrightarrow A \,.

And this satisfies $\varphi \left(\iota \right)={\psi }^{-1}\left(\iota \circ \psi \left(a\right)\right)={\psi }^{-1}\psi \left(a\right)=a$.

This establishes the existence of $\varphi$. To see uniqueness, notice that any other morphism with these properties coincides with $\varphi$ on all polynomials in $\iota$ and $\overline{\iota }$. By the Stone-Weierstrass theorem such polynomials form an everywhere-dense subset of $C\left(\mathrm{sp}\left(a\right)\right)$. 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)