Operator Algebras and Positive Functionals: a Bridge

Before entering algebraic QFT, a reader needs one bounded, representation-independent package. A complex unital C*-algebra records observables through multiplication, involution, norm, and a positive cone. A state is a positive normalized linear functional, a representation realizes the abstract algebra as bounded operators, and the GNS construction turns each state into a cyclic Hilbert-space representation that reproduces its expectation values.

That package does not by itself make a state normal, tracial, faithful, pure, symmetry-invariant, or a density-matrix state in a previously chosen representation. It also supplies no locality, covariance, dynamics, spectral condition, or preferred vacuum. This page gives the bounded prerequisite map and the GNS construction pattern only; the primary Mathematical QFT treatment develops the theorem and the representation theory.

Required background. Banach and Hilbert Spaces, Completion, and Riesz Representation supplies completion, inner products, and continuous duality.

Helpful background. Bounded, Compact, and Integral Operators supplies operator norms, bounded adjoints, and $\mathcal B(\mathcal H)$.

Bounded star-algebras, states, and representations

This page gives a bounded prerequisite map for normed star-algebras, positivity, states, and the GNS construction statement. It uses only the minimum representation vocabulary needed to state GNS; primary operator-algebraic definitions and representation theory remain with Mathematical QFT.

The finite-dimensional regulated diagnostic below is the page’s one controlled QFT-facing application. Primary definitions of local nets, quasilocal completion, representation comparison, folia, and AQFT theorems remain downstream.

Keep algebra, state, and representation separate

The basic data play different roles:

$$\mathcal A \quad\text{is an abstract algebra},\qquad \omega:\mathcal A\longrightarrow\mathbb C \quad\text{is a state},$$

while

$$\pi:\mathcal A\longrightarrow\mathcal B(\mathcal H)$$

is a concrete realization. Neither a Hilbert space nor a basis is part of the abstract algebra or state.

A complex unital star-algebra has an identity $1$, associative multiplication, and a conjugate-linear involution satisfying

$$\begin{aligned} (ab)^*&=b^*a^*,& (\alpha a+\beta b)^*&=\bar\alpha a^*+\bar\beta b^*,& (a^*)^*&=a. \end{aligned}$$

A normed star-algebra also has a submultiplicative norm, $\|ab\|\leq\|a\|\,\|b\|$, and a continuous involution. It is a Banach star-algebra when this norm is complete. A C-algebra* is a Banach star-algebra satisfying

$$\|a^*a\|=\|a\|^2.$$

The C*-identity forces $\|a^*\|=\|a\|$ and tightly links the algebraic involution, norm, and spectrum. These definitions and consequences are developed in Blackadar 2006, II.1.1, PDF and summarized for AQFT in Fewster and Rejzner 2020, § 2.2, PDF.

Representative examples are:

• $M_n(\mathbb C)$, with the operator norm and matrix adjoint;
• $C(X)$ for compact Hausdorff $X$, with pointwise operations, $f^*(x)=\overline{f(x)}$, and the supremum norm;
• $\mathcal B(\mathcal H)$, with composition, operator norm, and Hilbert adjoint.

In the unital setting used on this page, a concrete C*-algebra is a norm-closed unital star-subalgebra of $\mathcal B(\mathcal H)$. C*-algebras can also be nonunital, but approximate identities and unitization are outside this bridge. An abstract C*-algebra need not arrive with a preferred concrete realization.

The boundedness qualifier matters. Smeared quantum fields and Hamiltonians are generally unbounded and require domains; they are not automatically elements of a C*-algebra. Algebraic formulations instead use bounded observables or separately constructed bounded objects such as unitaries or resolvents. This bridge does not perform that construction.

Positive elements and order

For a C*-algebra $\mathcal A$, its positive cone is

$$\mathcal A_+ = \{b^*b:b\in\mathcal A\}.$$

For an element $a\in\mathcal A$, the following are equivalent:

1. $a\in\mathcal A_+$;
2. $a=c^2$ for a unique positive $c\in\mathcal A$;
3. $a=a^*$ and its spectrum lies in $[0,\infty)$.

These equivalences use the C*-algebra hypotheses and continuous functional calculus; they should not be transferred unchanged to a bare star-algebra. The order on self-adjoint elements is

$$a\leq b \quad\Longleftrightarrow\quad b-a\in\mathcal A_+.$$

In particular,

$$0\leq a^*a\leq\|a\|^2 1.$$

In a concrete faithful representation, intrinsic and operator positivity agree:

$$\begin{aligned} a\in\mathcal A_+ \quad\Longleftrightarrow\quad \langle\psi|\pi(a)\psi\rangle\geq0 \qquad \text{for every }\psi\in\mathcal H. \end{aligned}$$

A bounded observable is represented by a self-adjoint element; a projection models a sharp yes-or-no event, while an effect satisfies $0\leq e\leq1$. Positive elements and their equivalent C*-algebra characterizations are treated in Blackadar 2006, II.3.1, PDF.

Positive functionals and states

A positive linear functional is a linear map $\omega:\mathcal A\to\mathbb C$ such that

$$\omega(a^*a)\geq0 \qquad \text{for every }a\in\mathcal A.$$

On a unital C*-algebra, positivity has strong automatic consequences:

$$\omega(a^*)=\overline{\omega(a)},$$

and the functional Cauchy–Schwarz inequality is

$$|\omega(a^*b)|^2 \leq \omega(a^*a)\,\omega(b^*b).$$

The functional is continuous and

$$\|\omega\|=\omega(1).$$

A state is a positive functional normalized by $\omega(1)=1$. Hence

$$|\omega(a)|\leq\|a\|, \qquad \omega(a^*a)\leq\|a\|^2.$$

A state is faithful when

$$\omega(a^*a)=0 \quad\Longrightarrow\quad a=0.$$

For any $a\in\mathcal A$, Cauchy–Schwarz also gives the nonnegative fluctuation

$$\begin{aligned} \omega(a^*a)-|\omega(a)|^2 &= \omega\!\left( (a-\omega(a)1)^*(a-\omega(a)1) \right)\\ &\geq0. \end{aligned}$$

For self-adjoint $a$, this is its variance. These results, including the Cauchy–Schwarz argument and automatic boundedness, are given in Blackadar 2006, II.6.2, PDF and Fewster and Rejzner 2020, § 2.2, PDF.

Positivity and normalization are independent requirements. On $M_2(\mathbb C)$, let

$$\eta(a)=2a_{11}-a_{22}.$$

Then $\eta(1)=1$, but the positive matrix $E_{22}=\operatorname{diag}(0,1)$ obeys $\eta(E_{22})=-1$. Thus a normalized linear functional need not be a state.

It is equally important not to add properties that the definition does not contain. A tracial state additionally satisfies $\omega(ab)=\omega(ba)$. A vector state and a density-matrix state are defined only after choosing a unital representation. They also require

$$\|\xi\|=1, \qquad \rho\geq0, \qquad \rho\text{ trace-class}, \qquad \operatorname{Tr}_{\mathcal H}\rho=1.$$

Then

$$\omega_\xi(a) = \langle\xi|\pi(a)\xi\rangle, \qquad \omega_\rho(a) = \operatorname{Tr}_{\mathcal H}\!\bigl(\rho\,\pi(a)\bigr).$$

Neither form is the definition of an abstract state. Every state becomes a vector state in its own GNS representation, but it need not be a vector or density-matrix state in an arbitrary representation chosen beforehand.

The caveat is genuine even for a commutative algebra. Evaluation $\delta_x(f)=f(x)$ is a state on $C([0,1])$. Represent this algebra faithfully by multiplication operators $M_f$ on $L^2([0,1],dt)$. Any positive trace-class density matrix $\rho$ would produce

$$\operatorname{Tr}(\rho M_f) = \int_0^1 f(t)r(t)\,dt$$

for some nonnegative $r\in L^1([0,1],dt)$, so the resulting measure is absolutely continuous. It cannot equal the point mass $\delta_x$. The GNS representation of $\delta_x$ is instead the one-dimensional evaluation representation. This does not make either representation defective; it shows that density-matrix language is representation-relative.

Representations and concrete adjoints

A unital star-representation is a unital star-homomorphism

$$\pi:\mathcal A\longrightarrow\mathcal B(\mathcal H).$$

The abstract involution and concrete Hilbert adjoint are related by

$$\pi(a^*)=\pi(a)^\dagger.$$

The notation matters: $a^*$ exists before a representation is selected, whereas $\pi(a)^\dagger$ is the adjoint of a bounded operator on $\mathcal H$. Every star-homomorphism between C*-algebras is contractive; a faithful one, meaning $\ker\pi=\{0\}$, is isometric.

A vector $\Omega\in\mathcal H$ is cyclic when

$$\overline{\pi(\mathcal A)\Omega}=\mathcal H.$$

Cyclicity says that the represented algebra acting on one vector generates a dense subspace. It does not say that $\Omega$ is separating, that $\pi$ is faithful, or that the representation is irreducible. The basic representation facts used here are collected in Blackadar 2006, II.6.1, PDF.

The GNS construction pattern

Let $\omega$ be a state on a unital C*-algebra $\mathcal A$. Define its null space

$$N_\omega = \{a\in\mathcal A:\omega(a^*a)=0\}.$$

Functional Cauchy–Schwarz shows that null vectors have zero pairing with every element. Together with $0\leq c^*c\leq\|c\|^2 1$, it also gives

$$\omega(a^*c^*ca) \leq \|c\|^2\omega(a^*a).$$

Consequently $N_\omega$ is a left ideal. It need not be $\ker\omega$, a right ideal, or the kernel of the resulting representation.

On the quotient $\mathcal A/N_\omega$, set

$$\langle[a]|[b]\rangle_\omega = \omega(a^*b).$$

This follows the site’s convention: the bra slot is conjugate-linear and the ket slot is linear. Quotienting removes exactly the zero-norm vectors. Completing gives a Hilbert space $\mathcal H_\omega$.

Left multiplication defines

$$\pi_\omega(c)[a]=[ca].$$

It is well-defined and bounded because

$$\begin{aligned} \|\pi_\omega(c)[a]\|_\omega^2 &= \omega(a^*c^*ca)\\ &\leq \|c\|^2\omega(a^*a) = \|c\|^2\|[a]\|_\omega^2. \end{aligned}$$

Multiplication and the identity law descend directly from $\mathcal A$, and

$$\begin{aligned} \langle\pi_\omega(c)[a]|[b]\rangle_\omega &= \omega(a^*c^*b)\\ &= \langle[a]|\pi_\omega(c^*)[b]\rangle_\omega. \end{aligned}$$

Therefore $\pi_\omega(c^*)=\pi_\omega(c)^\dagger$, so $\pi_\omega$ is a unital star-representation.

Finally, let $\Omega_\omega=[1]$. Then

$$\overline{ \pi_\omega(\mathcal A)\Omega_\omega } = \mathcal H_\omega,$$

and

$$\omega(c) = \langle\Omega_\omega| \pi_\omega(c)\Omega_\omega\rangle.$$

Thus a state determines a cyclic triple $(\mathcal H_\omega,\pi_\omega,\Omega_\omega)$. Any other cyclic triple that reproduces the same state is related to it by a unitary intertwiner carrying one cyclic vector to the other. The quotient, bounded action, cyclicity, and uniqueness statement are the GNS construction; see Blackadar 2006, II.6.4, PDF and Fewster and Rejzner 2020, § 2.3, PDF.

The construction does not guarantee faithfulness. For $\mathcal A=\mathbb C\oplus\mathbb C$ and $\omega(z,w)=z$,

$$N_\omega=\{0\}\oplus\mathbb C,$$

so $\mathcal H_\omega\cong\mathbb C$ and $\pi_\omega(z,w)=z$. The nonzero element $(0,1)$ is lost. A GNS representation therefore need not be faithful: elements of $\ker\pi_\omega$ act trivially on the entire state-generated cyclic space.

Controlled application: a regulated observable algebra

Observables and states described without a preferred basis. Suppose a finite-dimensional regulator or truncation supplies

$$\mathcal A=M_n(\mathbb C),$$

and let $\rho\geq0$ with $\operatorname{Tr}\rho=1$. Define

$$\omega_\rho(a)=\operatorname{Tr}(\rho a).$$

This is a state because

$$\omega_\rho(a^*a) = \operatorname{Tr} \!\left( \rho^{1/2}a^*a\rho^{1/2} \right) = \|a\rho^{1/2}\|_{\mathrm{HS}}^2 \geq0.$$

The GNS map has the concrete form

$$[a]\longmapsto a\rho^{1/2}.$$

Indeed,

$$\langle[a]|[b]\rangle_{\omega_\rho} = \operatorname{Tr}(\rho a^*b) = \left\langle a\rho^{1/2}\middle|b\rho^{1/2} \right\rangle_{\mathrm{HS}}.$$

Thus $\mathcal H_{\omega_\rho}$ identifies with the Hilbert–Schmidt subspace $\{a\rho^{1/2}:a\in M_n(\mathbb C)\}$, observables act by left multiplication, and $\rho^{1/2}$ is the cyclic vector corresponding to the identity. If $\rho$ has rank $k$, this GNS space has dimension $nk$: a full-rank state gives dimension $n^2$, while a rank-one state gives dimension $n$.

Nothing in this description privileges a basis. Under a simultaneous unitary change

$$\rho\longmapsto U\rho U^\dagger, \qquad a\longmapsto UaU^\dagger,$$

cyclicity of the trace gives

$$\operatorname{Tr} \!\left( U\rho U^\dagger UaU^\dagger \right) = \operatorname{Tr}(\rho a).$$

The algebra, state, and expectation values therefore have a basis-free meaning even though matrices are convenient coordinates.

This is a regulated diagnostic, not a construction of continuum AQFT. It does not show that continuum fields are bounded, that every algebraic state is a density matrix, or that a regulator-independent limit exists. Those questions require additional physical and analytic input.

What positivity and GNS do not supply

Algebraic state positivity,

$$\omega(a^*a)\geq0,$$

is not by itself Wightman positivity, Euclidean reflection positivity, or positive-energy spectral support. Those conditions concern different objects and require additional spacetime, distributional, or dynamical structure. Covariance, locality, clustering, and a choice of physical state also do not follow.

A state need not be tracial, faithful, pure, or invariant under a symmetry. After a representation and von Neumann closure have been selected, normality is a further condition. A GNS representation need not be faithful or irreducible, and its cyclic vector need not be separating. The C*-norm closure is intrinsic to the chosen C*-algebra, whereas weak operator closures are formed only after selecting a representation and may depend on that selection; see Blackadar 2006, I.3.1 and I.9.1, PDF.

The prerequisite vocabulary here tells us how to obtain one cyclic representation from one state. It does not compare different representations or determine which states are physically admissible.

Common pitfalls

Starting with matrices instead of the abstract algebra. Matrices are one representation, not the definition of an observable algebra. A basis change must not change intrinsic algebraic statements.

Confusing three kinds of positivity. A positive element $a\in\mathcal A_+$, a positive concrete operator $\pi(a)$, and a positive functional $\omega$ are different objects. Their relations require the C*-algebra and representation hypotheses stated above.

Calling every state a density matrix. Density matrices describe certain states relative to a chosen Hilbert-space representation. An abstract state is defined by positivity and normalization, and its own GNS representation may differ from the chosen one.

Treating a state as a trace. Positivity does not imply $\omega(ab)=\omega(ba)$. Traciality is an additional symmetry condition.

Using GNS to infer physical axioms. GNS supplies a cyclic representation, not locality, covariance, positive energy, dynamics, or a preferred folium.

Putting unbounded fields into a C-algebra without construction.* The domain problem does not disappear. Bounded functions, Weyl operators, or resolvents must be introduced under their own hypotheses.

Exercises

Positivity check. Let $\omega$ be a state and $e$ an effect, $0\leq e\leq1$. Show that $0\leq\omega(e)\leq1$.

Solution

Positivity gives $\omega(e)\geq0$. Because $1-e\geq0$,

$$1-\omega(e) = \omega(1-e) \geq0.$$

Hence $\omega(e)\in[0,1]$.

GNS ideal check. Why is $N_\omega$ a left ideal, and why is that the side needed for $\pi_\omega(c)[a]=[ca]$?

Solution

If $a\in N_\omega$, then

$$\omega(a^*c^*ca) \leq \|c\|^2\omega(a^*a) =0,$$

so $ca\in N_\omega$. Therefore changing $a$ by a null element also changes $ca$ by a null element, making left multiplication well-defined on equivalence classes. No right-ideal property is needed.

Representation check. For $\mathcal A=\mathbb C\oplus\mathbb C$ and $\omega(z,w)=z$, identify the GNS Hilbert space and decide whether $\pi_\omega$ is faithful.

Solution

Because

$$\omega\bigl((z,w)^*(z,w)\bigr)=|z|^2,$$

the null ideal is $\{0\}\oplus\mathbb C$. The quotient is $\mathbb C$, with $\pi_\omega(z,w)$ acting by multiplication by $z$ and $\Omega_\omega=1$. Since every $(0,w)$ lies in the representation kernel, $\pi_\omega$ is not faithful.

Entering algebraic QFT

States, GNS Representations, and Folia is the primary Mathematical QFT treatment. It develops the GNS quotient/completion and uniqueness theorem in context, representation kernels, purity and irreducibility, normal states, folia, disjointness, and representation-dependent von Neumann closure.

That target also relies on the Mathematical QFT treatments of positivity and locality hypotheses and of quasilocal C*-algebras and inductive limits. This bridge does not replace those prerequisites, define local nets, compare representations, or prove an AQFT theorem.

References

• Bruce Blackadar, Operator Algebras: Theory of C-Algebras and von Neumann Algebras*, revised and corrected author version of the 2006 Springer text, Open PDF, DOI 10.1007/3-540-28517-2. §§ II.1.1 and II.3.1 develop C*-algebras and positive elements; §§ II.6.1, II.6.2, and II.6.4 treat representations, states, and GNS. §§ I.3.1 and I.9.1 locate the representation-dependent weak-closure boundary. The book uses the opposite inner-product slot convention, so its formulas have been translated to the site’s conjugate-linear bra convention.
• Christopher J. Fewster and Kasia Rejzner (2020), “Algebraic Quantum Field Theory—An Introduction”, PDF, §§ 2.2–2.3, published in Progress and Visions in Quantum Theory in View of Gravity (Birkhäuser, 2020), pp. 1–61, DOI 10.1007/978-3-030-38941-3_1. This is the specialist anchor for the AQFT-facing distinctions among algebraic, vector, and density-matrix states and for the GNS bridge. Later local-net material in the chapter is not imported here.