The Algebraic State Space

The state space of a unital C*-algebra collects all positive normalized expectation rules on its observables. Its weak-star topology describes convergence of each bounded observable’s expectation. Compactness in this topology makes algebraic states useful for limits even when density operators do not converge in trace norm or no common trace representation has been chosen.

Required background. Positive State Functionals supplies positivity, normalization, and the Cauchy–Schwarz estimate. C*-Algebras supplies the positive cone. The operational introduction is States as Positive Linear Functionals.

Positivity and normalization define a compact convex set

Let $\mathcal A$ be a nonzero unital C*-algebra. A state is a complex-linear functional $\omega$ such that $\omega(a^*a)\geq0$ and $\omega(1)=1$. Positivity implies $\|\omega\|=1$; the proof uses the same Cauchy–Schwarz inequality and order bound $a^*a\leq\|a\|^2 1$ as for bounded operators.

Write $S(\mathcal A)$ for all such states. Its natural weak-star topology is the restriction of $\sigma(\mathcal A^*,\mathcal A)$:

$$\omega_\alpha\longrightarrow\omega \quad\Longleftrightarrow\quad \omega_\alpha(a)\longrightarrow\omega(a) \quad(a\in\mathcal A).$$

Banach–Alaoglu makes the dual unit ball compact. The positivity conditions and $\omega(1)=1$ are weak-star closed conditions, so $S(\mathcal A)$ is compact. It is convex because convex combinations preserve those conditions.

This argument supplies convergent subnets, not necessarily convergent subsequences. If $\mathcal A$ is separable in norm, the weak-star topology on its dual unit ball is metrizable, and sequential compactness follows. The distinction matters for observable algebras with uncountably many independent norm-separated elements, such as the universal Weyl algebra.

Commutative states are probability measures

For compact Hausdorff $X$, the Riesz representation theorem identifies states of $C(X)$ with regular Borel probability measures:

$$\omega_\mu(f)=\int_X f\,d\mu.$$

Weak-star convergence is therefore convergence against every continuous test function. It does not demand convergence of probabilities for every measurable set. In particular, masses can approach a boundary whose indicator is not continuous.

Point evaluations $\delta_x(f)=f(x)$ are pure states, meaning extreme points of this convex set. More generally, purity is an algebraic extremality condition. The irreducible GNS theorem translates it into a statement about representation theory.

A state on a commutative algebra need not have a density relative to a chosen reference measure. Point mass at $x=0$ is a state of $C([0,1])$ but has no integrable density relative to Lebesgue measure. There is no contradiction with its normalized expectation rule.

Restriction is unique; extension generally is not

If $\mathcal B\subset\mathcal A$ is a unital C*-subalgebra with the same identity, every state of $\mathcal A$ restricts to a state of $\mathcal B$. Conversely, the state-extension theorem says that each state of $\mathcal B$ has at least one extension to $\mathcal A$.

The extension theorem follows from a norm-preserving Hahn–Banach extension and the C*-fact that a functional satisfying $\|\varphi\|=\varphi(1)=1$ is positive. These are existence results, not algorithms selecting a distinguished extension.

For $\mathcal A=C([0,1])$ and $\mathcal B=\mathbb C1$, the unique state of $\mathcal B$ extends to every probability measure on the interval. Even specifying the expectation of the coordinate function as $1/2$ leaves alternatives: $\delta_{1/2}$ and $(\delta_0+\delta_1)/2$ agree on affine functions but differ on $x^2$.

The affine functions are an operator system, rather than a subalgebra; this last example illustrates incomplete expectation data, not an application of the subalgebra theorem with affine functions as its input.

States detect the norm of a self-adjoint observable

For $a=a^*$,

$$\|a\|=\sup_{\omega\in S(\mathcal A)}|\omega(a)|.$$

The upper bound follows from $\|\omega\|=1$. For the reverse bound, choose $\lambda\in\sigma(a)$ with $|\lambda|=\|a\|$. Continuous functional calculus identifies $C^*(1,a)$ with $C(\sigma(a))$, where evaluation at $\lambda$ is a state. Extend it to $\mathcal A$ using the state-extension theorem. Its value on $a$ is $\lambda$.

This also shows that nonzero positive elements have nonzero expectation in some state. It is the separation fact needed when assembling faithful representations from GNS representations. It does not say one fixed state detects every nonzero positive element; that stronger property defines a faithful state.

Exercise: weak-star convergence can preserve a point mass

On $C([0,1])$, let $\omega_n(f)=n\int_0^{1/n}f(x)\,dx$. Find its weak-star limit and explain what happens to a density description.

Solution

Continuity at zero gives $|\omega_n(f)-f(0)|\leq\sup_{0\leq x\leq1/n}|f(x)-f(0)|\to0$. Thus $\omega_n\to\delta_0$. Every approximating state has a Lebesgue density, while the limit measure has none. Weak-star compactness holds for the full algebraic state space, not for the subclass with a density in this fixed measure representation.

References

• Bratteli, Ola, and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed., Springer, 1987, doi:10.1007/978-3-662-02520-8, Chapters 2 and 4 — states, extension, and convex decomposition.
• Naaijkens, Pieter, Quantum Spin Systems on Infinite Lattices, Lecture Notes in Physics 933, Springer, 2017, arXiv:1311.2717 — state spaces and representation-theoretic uses of positivity.