State (functional analysis)

In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both §§ Mixed states and pure states. Density matrices in turn generalize state vectors, which only represent pure states. For M an operator system in a C*-algebra A with identity, the set of all states of M, sometimes denoted by S(M), is convex, weak-* closed in the Banach dual space M^*. Thus the set of all states of M with the weak-* topology forms a compact Hausdorff space, known as the state space of M .

In the C*-algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i.e. mappings from physical observables (self-adjoint elements of the C*-algebra) to their expected measurement outcome (real number).

Jordan decomposition

States can be viewed as noncommutative generalizations of probability measures. By Gelfand representation, every commutative C*-algebra A is of the form C₀(X) for some locally compact Hausdorff X. In this case, S(A) consists of positive Radon measures on X, and the § pure states are the evaluation functionals on X.

More generally, the GNS construction shows that every state is, after choosing a suitable representation, a vector state.

A bounded linear functional on a C*-algebra A is said to be self-adjoint if it is real-valued on the self-adjoint elements of A. Self-adjoint functionals are noncommutative analogues of signed measures.

The Jordan decomposition in measure theory says that every signed measure can be expressed as the difference of two positive measures supported on disjoint sets. This can be extended to the noncommutative setting.

Theorem — Every self-adjoint f in A^* can be written as f = f₊ − f₋ where f₊ and f₋ are positive functionals and ||f|| = ||f₊|| + ||f₋||.

Proof

A proof can be sketched as follows: Let Ω be the weak*-compact set of positive linear functionals on A with norm ≤ 1, and C(Ω) be the continuous functions on Ω. A can be viewed as a closed linear subspace of C(Ω) (this is Kadison's function representation). By Hahn–Banach, f extends to a g in C(Ω)* with

It follows from the above decomposition that A* is the linear span of states.

Some important classes of states

Pure states

By the Krein-Milman theorem, the state space of M has extreme points. The extreme points of the state space are termed pure states and other states are known as mixed states.

Vector states

For a Hilbert space H and a vector x in H, the equation ω_x(A) := ⟨Ax,x⟩ (for A in B(H) ), defines a positive linear functional on B(H). Since ω_x(1)=||x||², ω_x is a state if ||x||=1. If A is a C*-subalgebra of B(H) and M an operator system in A, then the restriction of ω_x to M defines a positive linear functional on M. The states of M that arise in this manner, from unit vectors in H, are termed vector states of M.

Faithful states

A state $\tau$ is faithful, if it is injective on the positive elements, that is, $\tau (a^{*}a)=0$ implies $a=0$ .

Normal states

A state $\tau$ is called normal, iff for every monotone, increasing net $H_{\alpha }$ of operators with least upper bound $H$ , $\tau (H_{\alpha })\;$ converges to $\tau (H)\;$ .

Tracial states

A tracial state is a state $\tau$ such that

\tau (AB)=\tau (BA)\;.

For any separable C*-algebra, the set of tracial states is a Choquet simplex.

Factorial states

A factorial state of a C*-algebra A is a state such that the commutant of the corresponding GNS representation of A is a factor.

References

Lin, H. (2001), An Introduction to the Classification of Amenable C*-algebras, World Scientific

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Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F

Ordered topological vector spaces
Basic concepts	Ordered vector space Partially ordered space Riesz space Order topology Order unit Positive linear operator Topological vector lattice Vector lattice
Types of orders/spaces	AL-space AM-space Archimedean Banach lattice Fréchet lattice Locally convex vector lattice Normed lattice Order bound dual Order dual Order complete Regularly ordered
Types of elements/subsets	Band Cone-saturated Lattice disjoint Dual/Polar cone Normal cone Order complete Order summable Order unit Quasi-interior point Solid set Weak order unit
Topologies/Convergence	Order convergence Order topology
Operators	Positive State
Main results	Freudenthal spectral