C*-algebra Totally Explained

Everything about C Algebra totally explained

C*-algebras are an important area of research in functional analysis, a branch of mathematics. The prototypical example of a C*-algebra is a complex algebra, A, of linear operators on a complex Hilbert space with two additional properties:

A is a topologically closed set in the norm topology of operators.

A is closed under the operation of taking adjoints of operators. It is generally believed that C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began in an extremely rudimentary form with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators. These papers considered a special class of C*-algebras which are now known as von Neumann algebras.
Around 1943, the work of Israel Gelfand, Mark Naimark and Irving Segal yielded an abstract characterisation of C*-algebras making no reference to operators.
C*-algebras are now an important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics.

Abstract characterization

We begin with the abstract characterization of C*-algebras given in the 1943 paper by Gel'fand and Naimark.
A C*-algebra, A, is a Banach algebra over the field of complex numbers, together with a map, * : A → A, called involution. The image of an element x of A under involution is written x*. Involution has the following properties:

For all x, y in A:
ť :

(x + y)^* = x^* + y^* quad

ť :

(x y)^* = y^* x^*. quad

For every λ in C and every x in A:
ť :

(lambda x)^* = overline lambda leq mu.

ť In case A is separable, A has a sequential approximate identity. More generally, A will have a sequential approximate identity if and only if A contains a strictly positive element, for example a positive element h such that hAh is dense in A.

Using approximate identities, one can show that the algebraic quotient of a C*-algebra by a closed proper two-sided ideal, with the natural norm, is a C*-algebra.

Similarly, a closed two-sided ideal of a C*-algebra is itself a C*-algebra.

Type for C*-algebras

A C*-algebra A is of type I if and only if for all non-degenerate representations π of A the von Neumann algebra π(A)′′ (that is, the bicommutant of π(A)) is a type I von Neumann algebra. In fact it's sufficient to consider only factor representations, for example representations π for which π(A)′′ is a factor.
A locally compact group is said to be of type I if and only if its group C*-algebra is type I.
However, if a C*-algebra has non-type I representations, then by results of James Glimm it also has representations of type II and type III. Thus for C*-algebras and locally compact groups, it's only meaningful to speak of type I and non type I properties.

C*-algebras and quantum field theory

In quantum field theory, one typically describes a physical system with a C*-algebra A with unit element; the self-adjoint elements of A (elements x with x* = x) are thought of as the observables, the measurable quantities, of the system. A state of the system is defined as a positive functional on A (a C-linear map φ : A → C with φ(u u*) > 0 for all u∈A) such that φ(1) = 1. The expected value of the observable x, if the system is in state φ, is then φ(x).
See Local quantum physics.

Further Information

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