State (functional analysis)

A state is a mathematical term that is being studied in the functional analysis. There are certain linear functionals on a real or complex vector spaces, which are normalized in some way. The definitions are often designed so that the conditions with respect to an ordered structure are positive, that is, they represent the positive elements of this order on a non -negative real numbers. Further, the state-space forms, which is the set of states, a topological or geometric interesting area.

• 4.1 C *-algebras
• 4.2 Convex hull of the spectrum

• 5.1 Normal states
• 5.2 faithfulness states
• 5.3 Pure states

Involutive algebras

The most important applications is the case, a state on a involutive algebra, which is explained as follows. It is a standardized algebra, where one of the body or standing, in addition to the involution is defined.

A state is a continuous linear functional with

• For all
• .

The set of all states is called the state space, and is often denoted by ( S stands for the English word for state condition). If we replace the condition by, then one speaks of a quasi state; quasi state space is the set of quasi- states. If an identity, then we demand additionally.

Examples

Vector states

Be an involutive subalgebra of, the algebra of bounded linear operators on a Hilbert space with unit element. Then a vector of the standard 1, as defined by this

A state, the so-called defined by vector state, because it applies to any

This is true equality, because. Therefore a state.

If a number of magnitude 1, define a particular in physics so-called phase factor, and the same condition than for

In quantum mechanics, one identifies normalized to 1 Hilbert space vectors with quantum mechanical states, but really means that they define vector states, since the measured value of an observable in the state. It is thus clear that a Hilbert space vector, uniquely determines a state only up to a phase factor, which occurs in the form.

Spaces of dimensions

C * is the algebra of continuous functions, the involution is defined by the complex conjugate. Dual chamber is known, the spatial extent of the signed Borel, wherein the operation of such a measure for a continuous function by

Is given. because

Are the states in exactly the positive Borel measures with total variation. These considerations can be generalized to arbitrary algebras of C0 functions.

Locally Compact Groups

It is the group algebra of a locally compact group, which is the convolution algebra of respect to the left- hair dimension integrable functions. Dual chamber is known, that is, the space of the substantially limited capabilities. A function to operate by the definition

Where the Haar measure is. is exactly then to a state when

• Agrees almost everywhere consistent with a continuous, positive definite function.

In this case, a function is positive definite if the matrix for any finite number of elements is positive definite.

Importance, GNS - construction

A Hilbert space representation of an involutive Banach algebra is a * - homomorphism into the algebra of bounded linear operators on a Hilbert space. For simplicity, we assume, have an identity and that it was. ( Has no identity, then you can, if necessary, adjoin one or consider algebras with an approximation of one. ) Is now a vector state, and so is a condition on, because

The essential significance of conditions resulting from the fact that you can reverse this consideration, which means you can get from one state to a Hilbert space representation and a vector such that

For construction, the one by Gelfand, Neumark and Segal also called GNS - construction, one first forms the state, the left ideal

On the space factor is defined by the formula

Defines an inner product which turns into a pre-Hilbert space whose completion is a Hilbert space. By means of the left ideal property of, one can show that each defines a continuous linear mapping that clearly continues to a continuous linear mapping. The thus defined mapping is a Hilbert space representation and the definition

Follows the desired relationship, because is for.

Each state can thus be written by means of a Hilbert space representation as the vector state.

Properties

C *-algebras

For C *-algebras with identity statuses can be defined without reference to the involution. For the state space

The property follows automatically. It applies even more generally, for C *-algebras without identity:

Is a constant, linear, and is functional for any one - limited approximation of one of such a condition.

Convex hull of the spectrum

Since the state space of a C *-algebra is convex and weak - * - compact with unit element, and since for each, the mapping is linear and weak - * -continuous, is also

Convex and compact. It can be shown that the range is included in this set of always, that is to say is valid

Where stands for the convex hull of a set. For normal elements does equality, in general, the inclusion is true but, as the example shows. The spectrum of this nilpotent element, that agrees with the own convex hull, but for the unit vector is not contained in the convex hull of the spectrum.

Special conditions

Normal conditions

On von Neumann algebras one has more operator topologies in addition to the norm topology and hence it is of interest that states with respect to these topologies are continuous. The ultra - weak steady states are called normal, there are precisely those which can be written as a countable sum of multiples of vector states. They can be characterized in various ways and play an important role in the theory of von Neumann algebras, especially because the GNS construction leads to a homomorphism between von Neumann algebras.

Loyalty states

A state is called faithful if from already follows. In this case, the left ideal of the GNS construction is equal to the zero ideal and the construction is considerably simplified the constructed representation is faithful, ie injective. On separable C *-algebras, there is always faithful states. The existence of a faithful, normal states characterizes the σ - finite von Neumann algebras.

Pure states

The quasi- state space is convex and weak - * - compact, so has, by the theorem of Krein - Milman many extreme points. The different from 0 extremum points of the quasi- state space are states and are called pure states, as it blends, that is, convex combinations, can not be other states.

In the case of commutative C *-algebras, the pure states are precisely the * - homomorphisms. In the case of non- commutative C *-algebras are the pure states precisely those who lead their GNS - construction for irreducible representations.

Banach algebras

The characterization of states on a C *-algebra with unit element such as continuous, linear functionals, applies to the, can be transferred to any Banach algebras with identity. We define

Is called the state space, a numerical range of values. As in the case described above, the C *-algebras is a convex, compact subset of the complex plane containing the spectrum of. This conceptualization has many applications in the theory of Banach algebras, it leads to particular characterizations of C *-algebras ( set of Vidav - Palmer ).

Ordered vector spaces

If a minor vector space with an order unit, it is called a linear functional a state if and for all. The state space, ie the set of all states is convex, the extreme points of this set are called pure states. A state is then just pure if already follows for each linear functional with all that.

Taking as the space of self-adjoint elements of a C * - algebra with identity, then also acts as a structural unit. It is thus in the situation described above, the states on C * - algebras.

Parent groups

The concept of the state can even be generalized to ordered abelian groups. If such a group with positive semigroup and it is excellent, a scale, it means a mapping a state if

• Is a group homomorphism into the additive group of real numbers,
• ,
• .

The key for the C * - Algebrentheorie application is the K0 - group of a C * - algebra, in particular of AF C * - algebras. States of the K0 - group belong to tracks on the AF C * - algebras.