Continuous functional calculus - Imported Knowlpedia Content

Definition

Let $A$ be a unital $C^*$ -algebra and let $a\in A$ be normal, meaning $a^*a=aa^*$ . On the spectrum $\sigma_A(a)$ , the continuous functional calculus is the unique unital isometric $*$ -homomorphism

\Phi_a\colon C(\sigma_A(a))\longrightarrow A

that sends the coordinate function $z\mapsto z$ to $a$ . Its image is the commutative $C^*$ -algebra $C^*(1,a)$ , and one writes $f(a)=\Phi_a(f)$ . Thus algebraic operations, involution, and uniform limits of functions become the corresponding operations on elements of $A$ .

Fundamental consequences

For every $f\in C(\sigma_A(a))$ ,

\|f(a)\|=\max_{\lambda\in\sigma_A(a)}|f(\lambda)| \quad\text{and}\quad \sigma_A(f(a))=f(\sigma_A(a)).

The first identity is the norm formula; the second is spectral mapping. For self-adjoint $a$ , real-valued functions give self-adjoint elements. In particular, nonnegative continuous functions construct positive square roots and absolute values Murphy, Theorem 2.1.10.

Nonunital algebras

If $A$ is nonunital, one applies the unital calculus in its unitization $\widetilde A$ . For $a\in A$ , the element $f(a)$ lies in $A$ whenever $f(0)=0$ . Equivalently, the nonunital algebra generated by $a$ corresponds to the continuous functions on $\sigma_{\widetilde A}(a)$ that vanish at $0$ . This condition must not be omitted when $0$ belongs to the unitized spectrum.

References

G. J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: Theorem 2.1.10 and Section 2.2.
G. K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: Section 1.3 on continuous functional calculus.