Uniformly hyperfinite algebra

UHF algebras are studied in the mathematical branch of functional analysis. This is a class of C *-algebras, which are named after their discoverer, James Glimm also Glimm algebras. The UHF - algebras are simple, that is, they possess other than 0 and itself no two -sided ideals, and they can be used for the construction of certain von Neumann algebras.

Construction

Denote the C * - algebra of complex matrices. Is a divisor of, as was the one - homomorphism *, which maps to that of a matrix of matrix consisting of copies of the initial matrix along the diagonal, for example,

.

This * - homomorphism is injective and is the identity element to the identity element from. Since injective * - homomorphisms between C *-algebras are automatically isometric, can be conceived in this sense as Unterlagebra of, and instead we simply write.

Is now a sequence of natural numbers, we obtain a chain of inclusions:

.

On the association it is then a clear standard, which continues, each of the C * of standards, and hence until the completeness has all the properties of a C * standard. The completion is therefore a C *-algebra, called UHF algebra or glow- algebra of rank.

Properties

Isomorphisms

The UHF - algebras naturally depend on the defining sequence. For every prime number is the supremum of all, so that is a divisor of, which runs to infinity. This is assigned to the result of the defining sequence for one prime factorization of natural numbers as an analogy and writes in a supernatural number is called, which is of course a purely symbolic meaning; this runs through all prime numbers. It is

• Two UHF algebras of rank or are isomorphic if the assigned over natural numbers are equal, ie if for all primes.

This sentence is already in. In particular, there are uncountably many pairwise non- isomorphic UHF algebras.

UHF - algebras as AF - algebras

Back to top of specified construction UHF algebras are special AF - algebras; the latter, however, have been introduced later. If the rank of the UHF - algebra, then the corresponding Bratteli diagram is given by

.

It reads directly that all UHF algebras are simple, but also can be demonstrated without the use of the Bratteli diagrams. When AF - algebras UHF algebras are also classified by their parent scaled group, this is isomorphic to

With the given by [0,1] scale.

Representations

UHF algebras are antiliminal. Each irreducible representation is faithful and its image contains, besides 0 no further compact operator. UHF algebras have überabzähbar many pairwise non- equivalent irreducible representations.

Construction of factors

Each UHF - algebra has a unique trace state, that is a continuous linear functional with, and for all elements. The corresponding GNS construction provides a representation on a Hilbert space. It can be shown that the image is a type of Bikommutant II1 factor.

It's called factors hyperfinit, though they are sub -von Neumann algebras generated by Neumann algebras by an increasing sequence of finite-dimensional. Hence the name of the UHF algebras derives, because they are located in such a hyper- finite factors, UHF stands for Uniformly hyper - finite.

A special role is played by the CAR- algebra, which is equal to the UHF - algebra with supernatural number. In representations of this algebra are constructed whose images have type III factors as Bikommutanten.