Approximately finite-dimensional C*-algebra

AF-C *-algebras, or shorter AF - algebras form a considered in the mathematical branch of functional analysis class of C * - algebras that can be constructed from finite dimensional C *-algebras, AF stands for Approximately finite (almost finally). This C *-algebras can be set to specific groups in relationship and fully describe in this way by means of K- theory.

Definition

An AF - algebra is a C *-algebra for which there exists a sequence of finite dimensional C *-algebras, so that

  • ,
  • Is dense in.

Examples

  • Finite dimensional C *-algebras are AF - algebras.
  • The C * - algebra of compact operators on the Hilbert space is an AF - algebra. If the canonical basis of, so is the sub- algebra of linear operators, which map the linear hull of in and disappear on the orthogonal complement thereof. They are obviously isomorphic to the matrix algebra and satisfy the above definition.
  • Be the Cantor set. Then the commutative C * - algebra of continuous functions is an AF - algebra. Be the subalgebra of constant functions on. Then a sequence -dimensional C *-algebras form and meet the above definition.
  • The C * - algebra of continuous functions on is not an AF - algebra, because the zero - algebra and the algebra of constant functions are the only finite-dimensional subalgebras.
  • UHF algebras are AF - algebras.

Properties

  • AF - algebras are separable.
  • A separable C * -algebra if and only an AF - algebra if it admits a finite number of elements and each a finite dimensional sub -C *-algebra, such that for all.
  • The AF - algebras are exactly the countable inductive limits of finite dimensional C *-algebras in the category of C * - algebras.
  • Completed, two -sided ideals and quotients of AF - algebras are AF - algebras again. Sub -C *-algebras of AF - algebras are in general not AF - algebras, such as the irrational Rotationsalgebren in AF - algebras are included.
  • Countable inductive limits of AF - algebras are AF - algebras again.
  • Tensor products of AF algebras are AF - algebras again.
  • Is an AF algebra and can be seen by the adjunction of a unit element, so is also an AF - algebra.
  • A commutative C * - algebra, X a compact Hausdorff space if and only an AF - algebra if is totally disconnected ( see example above = Cantor set ).
  • AF - algebras are nuclear.

K0 - group of an AF - algebra

Dimension Group

The functor assigns to each C * - algebra ( general each ring ), a scaled, ordered, abelian group. More is on the set of isomorphism classes of finitely generated projective modules. The direct sum makes this lot to a commutative semigroup. is defined as the Grothendieck group of and is the image of in. Finally, it can be shown that each projection is defined via a projective module; is the image of the set of projections from in and is called the scale. Instead Group is also said dimension group.

Simple examples are or.

Next, one can show that a * - homomorphism between C *-algebras is a homomorphism between the associated dimension groups induced, this is positive, that is, forms the positive semigroups from each other, and scaled, that is, it forms the scales from one another. This is clear, because induces a Halbgruppenhomomorphismus, and the Scales awareness, note that natural projections maps from those made of. Overall, defines a functor from the category of C *-algebras in the category of scaled, ordered, abelian groups.

Set of Elliott

  • Set of Elliott: Two AF - algebras are isomorphic if the associated dimension groups are isomorphic as scaled, ordered, abelian groups. Each group isomorphism between two dimension groups is induced by a * - isomorphism of the associated AF - algebras.

This can be succinctly be formulated as:

  • For AF - algebras, the associated dimension group is therefore a complete Isomorphieinvariante.

Isomorphieinvariante means that the dimension groups isomorphic AF - algebras are isomorphic. This is clearly because of the functorial properties described above and even applies to all C * - algebras. Completeness of Isomorphieinvariante now means that the non- isomorphic AF algebra can be distinguished by their dimensional groups, that is, that the associated dimension groups are not isomorphic. That's the hard part of the set of Elliot.

Set of Effros - Handelman - Shen

As an AF - algebra is determined up to isomorphism by its dimension group arises naturally the question of which groups can occur as a dimension groups of AF algebras. This question is answered completely by the

  • Set of Effros - Handelman - Shen: The countable, non-perforated, scaled, ordered, abelian groups with the Riesz interpolation property are precisely the dimension groups of AF algebras.

For the occurring here proper theoretical concepts to consult the article parent abelian groups.

Importance

With the above sets of Elliot and Effros - Handelman - Shen the study of AF - algebras can be attributed to the Riesz interpolation property on the study of countable, non-perforated, scaled, ordered, abelian groups.

So you can show that the closed two -sided ideals of an AF - algebra in one to one way the order ideals of the dimension group match, ie those subgroups, where, and the property that follows.

One can therefore construct a simple AF - algebras, ie those without true of various two -sided ideals by countable unperforated, scaled, is subordinate groups with the Riesz interpolation property that have no real order ideals. The dense subgroups of embracing with a scale, are examples (UHF - algebras ), as well as groups with and defined by the order unit scale.

Bratteli diagrams

Another important tool for the study of AF - algebras are Bratteli diagrams, certain infinite, directed graphs, which reflect the structure of the algebra.

C*-Algebra Sequence space Closed set Nuclear C*-algebra Isomorphism (disambiguation) Projective object George A. Elliott
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