**Operators on a C*-Algebra**

Introduction

A $\mathrm{C}^*$-algebra is a Banach algebra whose norm is tied to an involution by the identity $\lVert a^*a\rVert = \lVert a\rVert^2$, and that single axiom forces the algebra to be, up to isometric ${}^*$-isomorphism, an algebra of operators on a Hilbert space. The operator theory of a $\mathrm{C}^*$-algebra is therefore the theory of its ${}^*$-representations: the ways the algebra acts by bounded operators on a Hilbert space. The representations are indexed by the positive functionals and the states, and every state produces one representation through the GNS construction. This article develops that operator layer. It fixes the $\mathrm{C}^*$-structure as it constrains the operator norm, defines the positive elements and the states, proves the Cauchy–Schwarz inequality that makes a state a bounded functional, constructs the representation a state determines on a Hilbert space, and states the Gelfand–Naimark theorem that a $\mathrm{C}^*$-algebra is the direct sum of its GNS representations, so that its norm is recovered from the operators it defines.

The article assumes the Banach algebra, its norm, its spectrum, its spectral radius and the Neumann series from Topological Algebras and Banach Algebras; the $\mathrm{C}^*$-algebra, the involutive element $a^*$, the $\mathrm{C}^*$-identity and the isometry of the involution from Topological Algebras and Banach Algebras and Involutive Topological Linear Algebras; the bounded operators, the operator norm, the bounded operators of an algebra and the characters from Operators on a Banach Algebra; the Hilbert space, the inner product, the bounded operators $B(H)$, the adjoint of an operator and the ${}^*$-algebra $B(H)$ from Banach and Hilbert Spaces and The Adjoint of a Bounded Operator; and the completeness of $B(H)$ and the closed graph from The Operator Algebra of a Banach Space. The full theory of $\mathrm{C}^*$-algebras — the GNS construction in detail, the $\mathrm{C}^*$-modules, the von Neumann algebras and the modular theory — is Operator Algebras, later in this category, and the present article states the construction and its operator consequences and cites that article for the development. The element-level spectral theory of self-adjoint elements and the positive cone, the involution and the spectral radius, and the functional calculus are the - * Theory articles of this category, marked there; no measure, no integral and no spectral measure occurs.

Throughout, $A$ is a $\mathrm{C}^*$-algebra over $\mathbb{C}$ with norm $\lVert\cdot\rVert$ and involution $a \mapsto a^*$, unital with unit $1$ where a statement names $1$; $H$ is a complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$ linear in the first variable, $B(H)$ the $\mathrm{C}^*$-algebra of bounded operators, and $K(H)$ the compact operators. A ${}^*$-representation is a ${}^*$-homomorphism $\pi : A \to B(H)$, and a state is a positive functional of norm one.

The C*-Structure and its Norm

Definition. A $\mathrm{C}^*$-algebra is a complex Banach algebra $A$ with an involution $a \mapsto a^*$ satisfying

$$ (ab)^* = b^*a^* , \qquad (a^*)^* = a , \qquad (\lambda a + \mu b)^* = \bar\lambda a^* + \bar\mu b^* , \qquad \lVert a^*a\rVert = \lVert a\rVert^2 . $$

Proposition (the involution is isometric and the norm is algebraic). In a $\mathrm{C}^*$-algebra,

$$ \lVert a^*\rVert = \lVert a\rVert , \qquad \lVert a^*a\rVert = \lVert a\rVert^2 , \qquad \lVert a\rVert = r(a^*a)^{1/2} , $$

where $r$ is the spectral radius; hence the norm is determined by the algebraic structure, and a $\mathrm{C}^*$-algebra carries at most one $\mathrm{C}^*$-norm.

Proof. $\lVert a\rVert^2 = \lVert a^*a\rVert \leq \lVert a^*\rVert\lVert a\rVert$, so $\lVert a\rVert \leq \lVert a^*\rVert$, and replacing $a$ by $a^*$ gives equality. For the spectral formula, $a^*a$ is self-adjoint and $\lVert a^*a\rVert = r(a^*a)$ holds for every self-adjoint element of a $\mathrm{C}^*$-algebra by the spectral theory of self-adjoint elements; combining with the $\mathrm{C}^*$-identity gives the third display. Two $\mathrm{C}^*$-norms agree on self-adjoint elements, each being the spectral radius there, and then on all elements by the $\mathrm{C}^*$-identity applied to $a^*a$. $\square$

Theorem (${}^*$-homomorphisms are contractive). Every ${}^*$-homomorphism $\pi : A \to B$ between $\mathrm{C}^*$-algebras is bounded with $\lVert\pi\rVert \leq 1$, and an injective ${}^*$-homomorphism is isometric.

Proof. $\pi$ sends self-adjoint elements to self-adjoint elements and $\sigma(\pi(a)) \subseteq \sigma(a)$, so for $a$ self-adjoint $\lVert\pi(a)\rVert = r(\pi(a)) \leq r(a) = \lVert a\rVert$; for general $a$, $\lVert\pi(a)\rVert^2 = \lVert\pi(a)^*\pi(a)\rVert = \lVert\pi(a^*a)\rVert \leq r(a^*a) = \lVert a\rVert^2$. If $\pi$ is injective then $\pi(a^*a)$ and $a^*a$ have the same spectrum, so the inequalities are equalities. $\square$

Remark (the involution belongs to the involutive layer). The involution $a \mapsto a^*$ on the elements is the structure of the - * Theory group of this category: the positivity, the positive cone and the states are developed there in States and Positive Functionals on an Involutive Algebra, and the spectral law $\lVert a\rVert = r(a^*a)^{1/2}$ is The Involution and the Spectral Radius. The present article uses them for the operator theory they index and marks those articles as their owners.

Positive Elements

Definition. An element $a \in A$ is self-adjoint if $a^* = a$, positive, written $a \geq 0$, if $a = b^*b$ for some $b \in A$, and unitary if $a^*a = aa^* = 1$. The set of positive elements is the positive cone $A^+$.

Proposition (the positive cone). Every element $b^*b$ is positive and self-adjoint; the positive elements form a convex cone, $A^+ + A^+ \subseteq A^+$ and $\mathbb{R}_{\geq0}A^+ \subseteq A^+$, closed in the norm; the sum of positive elements is positive, and $a^*a \geq 0$ for every $a$.

Proof. $b^*b$ is self-adjoint, $(b^*b)^* = b^*b$, and it is the square of the self-adjoint $b^*b$ only in special cases; positivity is by definition $a = c^*c$, and $c_1^*c_1 + c_2^*c_2 = c^*c$ for a suitable $c$ requires the polarisation identity and is standard for $\mathrm{C}^*$-algebras. The cone is closed because the map $b \mapsto b^*b$ is continuous and the image of a closed bounded set is closed in the relevant range; the elementary case is that of $A = \mathbb{C}$, and the general statement is the positivity of a $\mathrm{C}^*$-algebra, quoted from Operator Algebras. $\square$

Proposition (the order). The relation $a \leq b$ defined by $b - a \geq 0$ is a partial order on the self-adjoint elements, compatible with addition and with multiplication by positive scalars; a self-adjoint $a$ is positive exactly when $\sigma(a) \subseteq [0,+\infty)$, and then $\lVert a\rVert = \max\sigma(a)$.

Proof. The relation is reflexive, antisymmetric and transitive because the cone is a cone. A self-adjoint $a$ with $\sigma(a) \subseteq [0,\infty)$ has $\lVert \lVert a\rVert - a\rVert < \lVert a\rVert$ when $a \neq 0$, so $\lVert a\rVert - a$ is not invertible and the standard positivity criterion applies; the converse is standard. The norm of a positive element is the largest spectral value, by the spectral radius formula for a self-adjoint element. $\square$

Remark (the boundary). The positivity criterion, the square root of a positive element, the polar decomposition and the order structure on the self-adjoint elements are developed in Hermitian and Self-Adjoint Elements of a Banach Algebra and Operator Algebras; the present article uses only the definitions and the two propositions above, which are the operator-theoretic minimum for the states and the GNS construction.

States and Positive Functionals

Definition. A linear functional $\omega : A \to \mathbb{C}$ is positive if $\omega(a^*a) \geq 0$ for all $a$, and it is a state if it is positive and, when $A$ is unital, $\omega(1) = 1$. A weight is a positive functional not normalised.

Proposition (Cauchy–Schwarz and boundedness). For a positive functional $\omega$,

$$ \lvert\omega(b^*a)\rvert^2 \leq \omega(a^*a)\,\omega(b^*b) \qquad \text{for all } a,b , $$

so $\omega$ is bounded with $\lVert\omega\rVert \leq \omega(1)$ on a unital algebra, and $\lVert\omega\rVert = \omega(1)$; a state has norm one. Moreover $\omega(a^*) = \overline{\omega(a)}$, and $\omega$ is Hermitian.

Proof. The form $[a,b] = \omega(b^*a)$ is sesquilinear and positive semidefinite, $\omega(a^*a) \geq 0$; the Cauchy–Schwarz inequality for a positive semidefinite form gives the display. For the norm, the $\mathrm{C}^*$-computation $\lvert\omega(a)\rvert^2 \leq \omega(1)\omega(a^*a)$ and the bound $\lVert a^*a\rVert \leq \lVert a\rVert^2$ give $\lvert\omega(a)\rvert \leq \omega(1)^{1/2}\lVert a\rVert$ when $\omega(1) > 0$, so $\lVert\omega\rVert \leq \omega(1)$; conversely a normalised state has $\lVert\omega\rVert \geq \omega(1) = 1$. Hermitianity follows by expanding $\omega((a + \lambda b)^*(a + \lambda b)) \geq 0$ over $\lambda \in \mathbb{C}$ and comparing. $\square$

Proposition (the state space is convex and weak-$*$ compact). The states of a unital $\mathrm{C}^*$-algebra form a convex subset of the unit sphere of the dual, closed in the weak-$*$ topology, hence compact; the extreme points are the pure states, and every state is a weak-$*$ integral of pure states.

Proof. Convexity is the convexity of the positive functionals and the normalisation, and closedness in the weak-$*$ topology is the pointwise closedness of the two conditions $\omega(a^*a) \geq 0$ and $\omega(1) = 1$; the weak-$*$ compactness follows from Banach–Alaoglu applied to the unit ball. The extreme-point statement is the Krein–Milman theorem, quoted. $\square$

The GNS Construction

Theorem (Gelfand–Naimark–Segal). Let $A$ be a unital $\mathrm{C}^*$-algebra and let $\omega$ be a state. Then there are a Hilbert space $H_\omega$, a ${}^*$-representation $\pi_\omega : A \to B(H_\omega)$ and a cyclic unit vector $x_\omega \in H_\omega$ with

$$ \omega(a) = \langle \pi_\omega(a)x_\omega, x_\omega\rangle \qquad \text{for all } a \in A . $$

The triple $(\pi_\omega, H_\omega, x_\omega)$ is the GNS triple of $\omega$, and it is unique up to unitary equivalence.

Proof (sketch). On $A$ define the sesquilinear form $[a,b] = \omega(b^*a)$, which is positive semidefinite by the positivity of $\omega$. The set $N_\omega = \{a : \omega(a^*a) = 0\}$ is a left ideal, by the Cauchy–Schwarz inequality $\lvert\omega((ca)^*(ca))\rvert = \lvert\omega(a^*c^*ca)\rvert \leq \lVert c^*c\rVert\omega(a^*a)$; the quotient $A/N_\omega$ carries the induced positive definite inner product $[a + N_\omega, b + N_\omega] = \omega(b^*a)$, and its completion is the Hilbert space $H_\omega$. For fixed $a$, the operator $\pi_\omega(a)$ defined on the dense image of $A$ by $b + N_\omega \mapsto ab + N_\omega$ is well defined because $N_\omega$ is a left ideal, and it is bounded with $\lVert\pi_\omega(a)\rVert \leq \lVert a\rVert$ by the bound $[ab,ab] = \omega(b^*a^*ab) \leq \lVert a^*a\rVert[b,b]$; it therefore extends to a bounded operator, and $\pi_\omega(ab) = \pi_\omega(a)\pi_\omega(b)$, $\pi_\omega(a^*) = \pi_\omega(a)^*$ are the multiplicativity and the involution of $\omega$. The class of the unit, $x_\omega = 1 + N_\omega$, is cyclic and satisfies $\langle\pi_\omega(a)x_\omega,x_\omega\rangle = \omega(a)$. $\square$

Proposition (the representation is a ${}^*$-representation). The map $\pi_\omega$ is a ${}^*$-homomorphism $A \to B(H_\omega)$ with $\lVert\pi_\omega\rVert \leq 1$, and $\pi_\omega$ is irreducible exactly when $\omega$ is a pure state.

Proof. Multiplicativity, the involution law and the norm bound are in the sketch; the irreducibility criterion is the standard equivalence of a pure state and an irreducible GNS representation, quoted. $\square$

Corollary (the left multiplication bound). For every $a$ and every state $\omega$, the operator $\pi_\omega(a)$ is bounded with $\lVert\pi_\omega(a)\rVert \leq \lVert a\rVert$, so the norm of $a$ dominates the norm of every operator representing it through a state.

Proof. The bound is the estimate of the proof, $[ab,ab] \leq \lVert a\rVert^2[b,b]$. $\square$

Gelfand–Naimark

Theorem (Gelfand–Naimark). Let $A$ be a $\mathrm{C}^*$-algebra. Then the direct sum of the GNS representations over all states,

$$ \pi = \bigoplus_{\omega \text{ state}} \pi_\omega : A \longrightarrow B\Bigl(\bigoplus_\omega H_\omega\Bigr) , $$

is an isometric ${}^*$-isomorphism of $A$ onto a $\mathrm{C}^*$-subalgebra of $B(H)$ for a Hilbert space $H$; in particular

$$ \lVert a\rVert = \sup_{\omega \text{ state}} \lVert\pi_\omega(a)\rVert \qquad \text{for all } a \in A . $$

Proof (sketch). Each $\pi_\omega$ is a ${}^*$-homomorphism, so the direct sum is a ${}^*$-homomorphism; each $\pi_\omega$ is contractive, so $\lVert\pi(a)\rVert \leq \lVert a\rVert$. For the reverse inequality, let $a \neq 0$. The element $a^*a$ is positive, and on the commutative $\mathrm{C}^*$-subalgebra it generates there is a state $\omega_0$ with $\omega_0(a^*a) = \lVert a^*a\rVert = \lVert a\rVert^2$, by the Gelfand theory of a commutative $\mathrm{C}^*$-algebra; extending $\omega_0$ to a state $\omega$ of $A$ and forming the vector state $\omega_x(b) = \omega(x^*bx)/\omega(x^*x)$ with $x = a$ gives a state whose GNS representation satisfies $\lVert\pi_{\omega_x}(a)\rVert = \lVert a\rVert$, because $\omega_x(a^*a) = \lVert a\rVert^2$ realises the top of the spectrum of $\pi_{\omega_x}(a^*a)$. Hence the supremum attains the norm, $\pi$ is isometric, and an isometric ${}^*$-homomorphism is injective. When $A$ has no unit the construction is applied to the unitisation. $\square$

Corollary (the abstract is the concrete). Every $\mathrm{C}^*$-algebra is isometrically ${}^*$-isomorphic to a norm-closed, adjoint-closed subalgebra of $B(H)$; the equality of the two notions is the content of the theorem, and the operator norm of the representation is the given norm.

Proof. The theorem exhibits the isomorphism, the image is norm-closed and adjoint-closed because $\pi$ is isometric and preserves the involution. $\square$

Examples

Example (the matrix algebra). For $A = M_n(\mathbb{C})$ with the conjugate transpose, the positive elements are the positive semidefinite matrices, the states are $\omega(a) = \mathrm{tr}(\rho a)$ with $\rho \geq 0$ a density matrix of trace one, and the GNS space of a faithful state is the Hilbert–Schmidt space with $\pi$ the defining action; the pure states are the vector states $\omega(a) = \langle a\xi,\xi\rangle$.

Example (the commutative algebra $C(X)$). For $A = C(X)$ with $X$ compact Hausdorff and $f^*(x) = \overline{f(x)}$, the positive elements are the nonnegative functions, the states are the probability measures on $X$ by the Riesz representation theorem, and the GNS construction of a point state $\omega(f) = f(x_0)$ produces the one-dimensional representation $f \mapsto f(x_0)$; the direct sum over point states is faithful and isometric, recovering the sup norm.

Example ($B(H)$ and the vector states). For $A = B(H)$, every unit vector $\xi$ defines a pure state $\omega_\xi(T) = \langle T\xi,\xi\rangle$ and the GNS representation of $\omega_\xi$ is the identity of $B(H)$ with cyclic vector $\xi$; the normal states are the trace-class ones, whose theory is Operator Algebras.

Example (the compact operators). For $A = K(H)$ in infinite dimension the algebra has no unit, the states are the functionals $\omega(T) = \mathrm{tr}(\rho T)$ with $\rho$ trace-class positive of trace one, and the GNS construction applies to the unitisation; the identity is not in $K(H)$ and the norm is recovered from the same supremum over states, as it must be by the theorem.

Summary

A $\mathrm{C}^*$-algebra $A$ is a complex Banach algebra with an involution for which $\lVert a^*a\rVert = \lVert a\rVert^2$; the involution is isometric, the norm is algebraic, $\lVert a\rVert = r(a^*a)^{1/2}$, and every ${}^*$-homomorphism is contractive and isometric when injective, so the $\mathrm{C}^*$-norm is unique. The positive elements are those of the form $b^*b$, they form a closed convex cone carrying a partial order on the self-adjoint elements, and a self-adjoint element is positive exactly when its spectrum lies in $[0,\infty)$. A positive functional satisfies the Cauchy–Schwarz inequality $\lvert\omega(b^*a)\rvert^2 \leq \omega(a^*a)\omega(b^*b)$, hence is bounded with $\lVert\omega\rVert = \omega(1)$ on a unital algebra; the states, the positive functionals of norm one, form a convex weak-$*$ compact set with the pure states as its extreme points. Every state $\omega$ carries a GNS triple $(\pi_\omega,H_\omega,x_\omega)$ with $\omega(a) = \langle\pi_\omega(a)x_\omega,x_\omega\rangle$, built from the form $[a,b] = \omega(b^*a)$ by quotienting the null ideal and completing; $\pi_\omega$ is a ${}^*$-representation with $\lVert\pi_\omega(a)\rVert \leq \lVert a\rVert$, irreducible exactly for a pure state. The direct sum of the GNS representations over all states is an isometric ${}^*$-isomorphism of $A$ onto a $\mathrm{C}^*$-subalgebra of $B(H)$, the Gelfand–Naimark theorem, so $\lVert a\rVert = \sup_\omega\lVert\pi_\omega(a)\rVert$ and every abstract $\mathrm{C}^*$-algebra is concrete. The element-level positivity, the involution and the spectral radius are the - * Theory articles, and the full development of the $\mathrm{C}^*$-theory, the $\mathrm{C}^*$-modules, the von Neumann algebras and the modular theory is Operator Algebras.

Summary of Notation

Symbol	Meaning
$A$, $a^$, $\lVert a^a\rVert = \lVert a\rVert^2$	$\mathrm{C}^$-algebra, involution, $\mathrm{C}^$-identity
$A^+$, $a \geq 0$	The positive cone, $a = b^*b$; order on self-adjoint elements
$r(a^*a)$, $\lVert a\rVert$	Norm is the square root of the spectral radius of $a^*a$
$\omega$, $\omega(a^*a)\geq0$	A positive functional
State, pure state	Positive functional with $\omega(1)=1$; an extreme point
$\lvert\omega(b^a)\rvert^2 \leq \omega(a^a)\omega(b^*b)$	The Cauchy–Schwarz inequality for a positive functional
$[a,b] = \omega(b^*a)$	The positive semidefinite form of a state
$N_\omega = \{a : \omega(a^*a)=0\}$	The null left ideal
$(\pi_\omega,H_\omega,x_\omega)$	The GNS triple, $x_\omega$ cyclic, $\omega(a)=\langle\pi_\omega(a)x_\omega,x_\omega\rangle$
$\pi = \bigoplus_\omega\pi_\omega$	The isometric faithful $^*$-representation (Gelfand–Naimark)
$H$, $B(H)$, $K(H)$	Hilbert space, bounded and compact operators
${}^*$-representation	A $^*$-homomorphism $\pi : A \to B(H)$

Operators on a C*-Algebra