The Functional Calculus of a Self-Adjoint Element

Introduction

The spectral theorem for a self-adjoint element $a$ identifies the commutative $\mathrm{C}^*$-algebra $C^*(a,1)$ it generates with the algebra $C(\sigma(a))$ of continuous functions on its compact real spectrum, and under that identification the element $a$ is the identity function. Reading the identification backwards gives the continuous functional calculus: every continuous complex function $f$ on $\sigma(a)$ produces an element $f(a)$ of $A$, the assignment $f \mapsto f(a)$ is an isometric $*$-isomorphism of $C(\sigma(a))$ onto $C^*(a,1)$, and it intertwines the algebra of functions with the algebra of elements. The calculus is the analytic instrument of the whole theory: it produces the square roots, the absolute values and the positive elements, it proves the spectral mapping theorem $\sigma(f(a)) = f(\sigma(a))$ in one line, and it turns every property of a continuous function into a property of the element.

This article assumes the spectral theorem and the reality of the spectrum from The Spectrum of a Self-Adjoint Element; the commutative Gelfand–Naimark theorem from Involutive Banach Algebras and the Gelfand–Naimark Theorem; the positivity, the order, the square roots and the polar decomposition from Hermitian and Self-Adjoint Elements of a Banach Algebra and Self-Adjoint Elements and the Positive Cone; the spectrum and the holomorphic functional calculus from Topological Algebras and Banach Algebras and Holomorphic Functional Calculus; and the continuous calculus of the operator setting from Operator Algebras and Operators on a C-Algebra*. The grade involution $\alpha$ is not used.

Throughout, $A$ is a unital $\mathrm{C}^*$-algebra over $\mathbb{C}$ with involution $a \mapsto a^*$; $a$ is self-adjoint (or normal) with spectrum $\sigma(a)$ and $C^*(a,1)$ the closed unital $*$-subalgebra it generates; $C(\sigma(a))$ is the algebra of continuous complex functions on the compact set $\sigma(a)$ with the sup norm; and the functional calculus is the map $\Phi_a : C(\sigma(a)) \to A$, $f \mapsto f(a)$.

The Continuous Calculus

Theorem (the continuous functional calculus). Let $a$ be self-adjoint. There is a unique unital $*$-homomorphism

$$ \Phi_a : C(\sigma(a)) \to A , \qquad f \mapsto f(a) , $$

and it is an isometric $*$-isomorphism onto $C^*(a,1)$; it sends the identity function to $a$, the constant $1$ to the unit, and conjugation to the involution, $\overline{f}(a) = f(a)^*$. For a polynomial $p$, $p(a)$ is the element computed by the algebra, so the calculus extends the polynomial calculus.

Proof. By The Spectrum of a Self-Adjoint Element the Gelfand transform is an isometric $*$-isomorphism $C^*(a,1) \to C(\sigma(a))$ sending $a$ to the identity function; its inverse is $\Phi_a$, it is an isometric $*$-isomorphism, and on polynomials it agrees with evaluation because the Gelfand transform is multiplicative and unital. Uniqueness: two unital $*$-homomorphisms agreeing on the identity function agree on its continuous functions by continuity and the Stone–Weierstrass density of the polynomials. $\square$

Theorem (the spectral mapping theorem). For self-adjoint $a$ and continuous $f : \sigma(a) \to \mathbb{C}$,

$$ \sigma\bigl(f(a)\bigr) = f\bigl(\sigma(a)\bigr) . $$

Proof. Under the identification of $C^*(a,1)$ with $C(\sigma(a))$, the element $f(a)$ is the function $f$, whose spectrum as an element of the algebra $C(\sigma(a))$ is its range $f(\sigma(a))$; the spectrum is an invariant of the algebra $C^*(a,1)$ because invertibility in a unital $\mathrm{C}^*$-subalgebra with the same unit is the same as invertibility in $A$. $\square$

Corollary (positivity and the calculus). For self-adjoint $a$, $f(a) \geq 0$ exactly when $f \geq 0$ on $\sigma(a)$; the calculus is positive, and it produces the square root and the absolute value:

$$ f \geq 0 \Rightarrow \Phi_a(f) \geq 0 , \qquad a^{1/2} = \Phi_a(\lambda \mapsto \lambda^{1/2}) , \qquad \lvert a\rvert = \Phi_a(\lvert\lambda\rvert) , $$

with unique $a^{1/2} \geq 0$ and $(a^{1/2})^2 = a$.

Proof. If $f \geq 0$ then $f = g\bar g$ for the continuous $g = f^{1/2}$, so $\Phi_a(f) = \Phi_a(g)\Phi_a(g)^* \in P$; conversely $f(a) \geq 0$ implies $f \geq 0$ because $f$ is the Gelfand transform of $f(a)$ and the characters are positive on positive elements. The square root and the absolute value are the functions displayed; the uniqueness of the positive square root is Self-Adjoint Elements and the Positive Cone. $\square$

The Normal Case and the Comparison

Theorem (the calculus for a normal element). The continuous functional calculus holds for a normal element $a$ ($a^*a = aa^*$): $\Phi_a$ is an isometric $*$-isomorphism $C(\sigma(a)) \to C^*(a,1)$, and for two continuous functions $f,g$, $f(a)g(a) = (fg)(a)$. The spectrum $\sigma(a)$ is a compact subset of $\mathbb{C}$, and $\lVert f(a)\rVert = \lVert f\rVert_\infty$.

Proof. For normal $a$ the $\mathrm{C}^*$-algebra $C^*(a,1)$ is commutative, so the Gelfand transform identifies it with $C(\sigma(a))$ and the same argument applies; multiplicativity and isometry are inherited. $\square$

Remark (the two calculi). The continuous calculus works for self-adjoint and, more generally, normal elements, because the algebra generated is commutative; for a general element the polynomial calculus does not extend continuously, and one uses the holomorphic functional calculus, which for a function holomorphic on a neighbourhood of $\sigma(a)$ produces $f(a)$ by a contour integral. The holomorphic calculus is Holomorphic Functional Calculus; the continuous calculus is its boundary case for normal elements, and it is this calculus that is used in the spectral theory of a self-adjoint element. The operator-valued version, with the projection-valued measure, is Operator Algebras.

Example (the Hermitian matrix). For a Hermitian matrix $X = U\operatorname{diag}(\lambda_i)U^*$ the calculus is $f(X) = U\operatorname{diag}(f(\lambda_i))U^*$, the spectral mapping theorem reads that the eigenvalues of $f(X)$ are the values of $f$ on the eigenvalues of $X$, and the positive square root is the matrix with the square roots of the eigenvalues on the diagonal.

Example (the function algebra). For $A = C(X,\mathbb{C})$ and a real-valued $f$ self-adjoint, the calculus is $g(f) = g \circ f$; the spectral mapping theorem reads $\sigma(g\circ f) = (g\circ f)(X)$, and the square root is the pointwise square root.

Example (the multiplication operator). On $A = L^\infty(X,\mu)$ a real self-adjoint function acts on $L^2(X,\mu)$ as a multiplication operator; the calculus is $g(a) = g\circ a$, and the projection-valued measure of the spectral theorem is the family of indicator functions of the sublevel sets, the operator form of the spectral decomposition.

The Analytic and the Borel Calculi

Theorem (the analytic calculus). For a self-adjoint $a$ and a function $f$ holomorphic on an open neighbourhood of $\sigma(a)$,

$$ f(a) = \frac{1}{2\pi i}\oint_\Gamma f(\lambda)(\lambda - a)^{-1}\,d\lambda , $$

where $\Gamma$ is a finite cycle in the domain of $f$ surrounding $\sigma(a)$; the element is independent of $\Gamma$, and the assignment $f \mapsto f(a)$ is a unital algebra homomorphism extending the polynomial calculus, the holomorphic functional calculus. The contour integral and the general theory are Holomorphic Functional Calculus.

Theorem (the Borel calculus, named). For a self-adjoint operator $a \in B(H)$ the calculus extends to the bounded Borel functions on $\sigma(a)$ and the spectral theorem delivers a projection-valued measure $E$ on the Borel sets of $\sigma(a)$ with

$$ a = \int_{\sigma(a)} \lambda\,dE(\lambda) , \qquad f(a) = \int_{\sigma(a)} f\,dE . $$

The measure $E$, the measurable calculus and the spectral decomposition are Operator Algebras; here they are named and deferred.

Proposition (the polynomial core). The continuous calculus is the unique continuous extension of the polynomial calculus, and by the Stone–Weierstrass theorem the polynomials are dense in $C(\sigma(a))$; the spectral mapping theorem for a polynomial is the algebraic statement that the algebra homomorphism sends the identity function to $a$.

Summary

For a self-adjoint element $a$ of a unital $\mathrm{C}^*$-algebra the continuous functional calculus is the unique unital $*$-homomorphism $\Phi_a : C(\sigma(a)) \to A$, $f \mapsto f(a)$, and it is an isometric $*$-isomorphism onto the commutative $\mathrm{C}^*$-algebra $C^*(a,1)$; it sends the identity function to $a$ and conjugation to the involution, and it extends the polynomial calculus. The spectral mapping theorem reads $\sigma(f(a)) = f(\sigma(a))$, the calculus is positive, $f \geq 0$ implying $f(a) \geq 0$, and it produces the square root $a^{1/2}$ and the absolute value $\lvert a\rvert$. The calculus holds for a normal element, whose generated algebra is commutative, and its boundary case for general elements is the holomorphic functional calculus, which produces $f(a)$ by a contour integral; the operator-valued spectral theorem with the projection-valued measure is the form the calculus takes in Operator Algebras.

Summary of Notation

Symbol	Meaning
$a = a^*$	Self-adjoint element, $\sigma(a) \subseteq \mathbb{R}$
$\Phi_a : C(\sigma(a)) \to A$, $f \mapsto f(a)$	The continuous functional calculus
$C^*(a,1)$	The commutative unital $\mathrm{C}^*$-algebra generated by $a$
$\sigma(f(a)) = f(\sigma(a))$	Spectral mapping theorem
$f \geq 0 \Rightarrow f(a) \geq 0$	Positivity of the calculus
$a^{1/2} = \Phi_a(\lambda^{1/2})$, $\lvert a\rvert = \Phi_a(\lvert\cdot\rvert)$	Square root and absolute value
Normal $a$	The calculus extends, $C^*(a,1)$ commutative
Holomorphic calculus	Contour-integral calculus for a general element
Borel calculus, PV measure $E$	$a=\int\lambda\,dE$; deferred to Operator Algebras