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State (functional analysis)

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In the mathematical field offunctional analysis, astate of anoperator system is apositive linear functional ofnorm 1. States in functional analysisgeneralize the notion ofdensity matrices in quantum mechanics, which representquantum states, bothmixed states andpure states. Density matrices in turn generalizestate vectors, which only represent pure states. ForM an operator system in aC*-algebraA with identity, the set of all states ofM, sometimes denoted by S(M), is convex, weak-* closed in the Banach dual spaceM^*. Thus the set of all states ofM with the weak-* topology forms a compact Hausdorff space, known as thestate space ofM.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

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States can be viewed as noncommutative generalizations ofprobability measures. ByGelfand representation, every commutative C*-algebraA is of the formC₀(X) for some locally compact HausdorffX. In this case,S(A) consists of positiveRadon measures onX, and thepure states are the evaluation functionals onX.

More generally, theGNS construction shows that every state is, after choosing a suitable representation, avector state.

A bounded linear functional on a C*-algebraA is said to beself-adjoint if it is real-valued on the self-adjoint elements ofA. Self-adjoint functionals are noncommutative analogues ofsigned measures.

TheJordan 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 $f {\displaystyle f}$ in $A^{\ast }$ ${\displaystyle }$ can be written as $f=f_{+}-f_{-}$ where $f_{+}$ and $f_{-}$ are positive functionals and $\Vert f\Vert =\Vert f_{+}\Vert +\Vert f_{-}\Vert$ .

Proof

A proof can be sketched as follows: Let $\Omega$ be the weak*-compact set of positive linear functionals on $A {\displaystyle A}$ with norm ≤ 1, and $C(\Omega )$ be the continuous functions on $\Omega$ .

$A {\displaystyle A}$ can be viewed as a closed linear subspace of $C(\Omega )$ (this isKadison's function representation). By Hahn–Banach, $f {\displaystyle f}$ extends to a $g {\displaystyle g}$ in $C(\Omega )^{\ast }$ with $\Vert g\Vert =\Vert f\Vert$ .

Using results from measure theory quoted above, one has:

$g(\cdot )=\int \cdot \;d\mu$

where, by the self-adjointness of $f {\displaystyle f}$ , $\mu$ can be taken to be a signed measure. Write:

$\mu =\mu _{+}-\mu _{-},\;$

a difference of positive measures. The restrictions of the functionals $\textstyle \int \cdot \;d\mu _{+}$ and $\textstyle \int \cdot \;d\mu _{-}$ to $A {\displaystyle A}$ has the required properties of $f_{+}$ and $f_{-}$ . This proves the theorem.

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

Some important classes of states

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Pure states

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By theKrein-Milman theorem, the state space ofM hasextreme points. The extreme points of the state space are termedpure states and other states are known asmixed states.

Vector states

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For a Hilbert spaceH and a vectorx inH, the formula ω_x(T) := ⟨Tx,x⟩ (forT inB(H)) defines a positive linear functional onB(H). Since ω_x(1)=||x||², ω_x is a state if ||x||=1. IfA is a C*-subalgebra ofB(H) andM anoperator system inA, then the restriction of ω_x toM defines a positive linear functional onM. The states ofM that arise in this manner, from unit vectors inH, are termedvector states ofM.

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

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

Factorial states

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Afactorial state of a C*-algebraA is a state such that the commutant of the corresponding GNS representation ofA is afactor.

References

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

Functional analysis (topics –glossary)

Spaces

Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev Topological vector
Properties	Barrelled Complete Dual (Algebraic /Topological) Locally convex Reflexive Separable

Theorems

Operators

Algebras

Open problems

Applications

Advanced topics

Category

v t e Hilbert spaces
Basic concepts	Adjoint Inner product andL-semi-inner product Hilbert space andPrehilbert 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) withK compact &n<∞ Segal–BargmannF

v t e 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