View source for State (functional analysis)

{{Refimprove|date=July 2025}}
In the mathematical field of [[functional analysis]], a '''state''' of an [[operator system]] is a [[positive linear functional]] of [[operator norm|norm]] 1. States in functional analysis [[generalization|generalize]] the notion of [[density matrix|density matrices]] in quantum mechanics, which represent [[quantum state]]s, both [[quantum state#Mixed states|mixed states]] and [[quantum state#Pure states as rays in a complex Hilbert space|pure states]]. Density matrices in turn generalize [[quantum state|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''<sup>*</sup>. 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 measure]]s. By [[Gelfand representation]], every commutative C*-algebra ''A'' is of the form ''C''<sub>0</sub>(''X'') for some locally compact Hausdorff ''X''. In this case, ''S''(''A'') consists of positive [[Radon measure]]s on ''X'', and the [[#Pure states|pure states]] are the evaluation functionals on ''X''.

More generally, the [[GNS construction]] shows that every state is, after choosing a suitable representation, a [[State (functional analysis)#Vector states|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 measure]]s.

The [[Hahn decomposition theorem|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.

{{math theorem|math_statement= Every self-adjoint <math>f</math> in <math>A^{\ast}</math> <math></math> can be written as <math>f = f_{+} - f_{-}</math> where <math>f_{+}</math> and <math>f_{-}</math> are positive functionals and <math>\Vert f \Vert = \Vert f_{+}\Vert + \Vert f_{-}\Vert</math>.}}

{{Math proof|drop=hidden|proof= 
A proof can be sketched as follows: 
Let <math>\Omega</math> be the weak*-compact set of positive linear functionals on <math>A</math> with norm ≤ 1, and <math>C(\Omega)</math> be the continuous functions on <math>\Omega</math>. 

<math>A</math> can be viewed as a closed linear subspace of <math>C(\Omega)</math> (this is ''[[Richard V. Kadison|Kadison]]'s function representation''). By Hahn–Banach, <math>f</math> extends to a <math>g</math> in <math>C(\Omega)^{\ast}</math> with <math>\Vert g \Vert = \Vert f \Vert</math>.

Using results from measure theory quoted above, one has:

<math>g(\cdot) = \int \cdot \; d \mu</math>

where, by the self-adjointness of <math>f</math>, <math>\mu</math> can be taken to be a signed measure. Write:

<math>\mu = \mu_+ - \mu_-, \;</math>

a difference of positive measures. 
The restrictions of the functionals <math>\textstyle\int\cdot\;d\mu_{+}</math>  and <math>\textstyle\int\cdot\;d\mu_{-}</math> to <math>A</math> has the required properties of <math>f_{+}</math> and <math>f_{-}</math>. 
This proves the theorem.
}}

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 point]]s. 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 formula &omega;<sub>''x''</sub>(''T'') := ⟨''Tx'',''x''⟩ (for ''T'' in ''B(H)'') defines a positive linear functional on ''B(H)''. Since &omega;<sub>''x''</sub>(''1'')=||''x''||<sup>2</sup>, &omega;<sub>''x''</sub> 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 &omega;<sub>''x''</sub> 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 <math>\tau</math> is '''faithful''', if it is injective on the positive elements, that is, <math>\tau(a^* a) = 0</math> implies <math>a = 0</math>.

===Normal states===
A state <math>\tau</math> is called '''normal''', iff for every monotone, increasing [[net (mathematics)|net]] <math>H_\alpha</math> of operators with least upper bound <math>H</math>,  <math>\tau(H_\alpha)\;</math> converges to <math>\tau(H)\;</math>.

===Tracial states===
A '''tracial state''' is a state <math>\tau</math> such that

:<math>\tau(AB)=\tau (BA)\;.</math>

For any separable C*-algebra, the set of tracial states is a [[Choquet theory|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 [[Von Neumann algebra#Factors|factor]].

==See also==
*[[Quantum state]]
*[[Gelfand–Naimark–Segal construction]]
*[[Quantum mechanics]]
**[[Quantum state]]
**[[Density matrix]]

==References==

* {{citation|first=H.|last= Lin|title=An Introduction to the Classification of Amenable C*-algebras|publisher=World Scientific|year=2001}}

{{Functional analysis}}
{{Hilbert space}}
{{Ordered topological vector spaces}}

[[Category:Functional analysis]]
[[Category:C*-algebras]]