State (functional analysis)
Encyclopedia
In functional analysis
, a state on a C*-algebra is a positive linear functional
of norm
1. The set of states of a C*-algebra A, sometimes denoted by S(A), is always a convex set
. The extremal points of S(A) are called pure states. If A has a multiplicative identity, S(A) is compact in the weak*-topology.
In the C*-algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i.e. mappings from physical observables to their expected measurement outcome.
s. 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 measure
s on X, and the pure states are the evaluation functionals on X.
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 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−||.
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 ||g|| = ||f||.
Using results from measure theory quoted above, one has
where, by the self-adjointness of f, μ can be taken to be a signed measure. Write
a difference of positive measures. The restrictions of the functionals ∫ · dμ+ and ∫ · dμ− to A has the required properties of f+ and f−. This proves the theorem.
It follows from the above decomposition that A* is the linear span of states.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, a state on a C*-algebra is a positive linear functional
Positive linear functional
In mathematics, especially in functional analysis, a positive linear functional on an ordered vector space is a linear functional f on V so that for all positive elements v of V, that is v≥0, it holds thatf\geq 0....
of norm
Operator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
1. The set of states of a C*-algebra A, sometimes denoted by S(A), is always a convex set
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
. The extremal points of S(A) are called pure states. If A has a multiplicative identity, S(A) is compact in the weak*-topology.
In the C*-algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i.e. mappings from physical observables to their expected measurement outcome.
Jordan decomposition
States can be viewed as noncommutative generalizations of probability measureProbability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
s. By Gelfand representation
Gelfand representation
In mathematics, the Gelfand representation in functional analysis has two related meanings:* a way of representing commutative Banach algebras as algebras of continuous functions;...
, 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 measure
Radon measure
In mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...
s on X, and the pure states are the evaluation functionals on X.
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
Signed measure
In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Some authors may call it a charge, by analogy with electric charge, which is a familiar distribution that takes on positive and negative values.-Definition:There are two slightly...
s.
The Jordan decomposition
Hahn decomposition theorem
In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that given a measurable space and a signed measure μ defined on the σ-algebra Σ, there exist two sets P and N in Σ such that:...
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−||.
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 ||g|| = ||f||.
Using results from measure theory quoted above, one has
where, by the self-adjointness of f, μ can be taken to be a signed measure. Write
a difference of positive measures. The restrictions of the functionals ∫ · dμ+ and ∫ · dμ− to A has the required properties of f+ and f−. This proves the theorem.
It follows from the above decomposition that A* is the linear span of states.
Properties of states
- a tracial state is a state
fulfilling
- a state
is called normal, iff for every monotone, increasing netNet (mathematics)In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is...
of operators with upper bound 
, 
converges to 
. - a state is said to be faithful, if the image of strictly positive operators is itself strictly positive
See also
- Quantum state
- Gelfand–Naimark–Segal construction
- Quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
- Quantum state
- Density matrixDensity matrixIn quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...