Cuntz-Algebra

In the functional analysis, the so-called Cuntz algebras are ( according to Joachim Cuntz ) a special class of C *-algebras generated by n pairwise orthogonal isometries on a separable Hilbert space.

Definition

Be a separable infinite-dimensional Hilbert space. For a natural number are isometries on H, ie it applies to. In addition, they are the property

Meet the projectors are thus pairwise orthogonal. In the event one calls a sequence of isometries with the property

We define now

Than that of generated C * - subalgebra in. In order to maintain a uniform notation, we retain this notation also in the case.

Properties

The Cuntz algebra has a number of remarkable properties, it is an example of a separable, unital, and simple C *-algebra.

Unambiguity

Are more isometrics with, it follows

The isomorphism class does not depend on the choice of generators. The notation, which does not rely on the producers is justified.

A special role in the investigation of sub- passes, the C * algebra generated by elements of the mold. One can show that this is isomorphic to the UHF algebra supernatural number. Substituting a producer firm, for example, and writes, so there are pictures, so that each can be represented as

An important step in the proof above uniqueness property is to use these to interpret analogous to Fourier coefficients in a Laurent series. This makes it possible to show that can exist of only one C * - norm on the purely algebraic product, so the assertion is shown.

Simplicity

A C * - algebra is called simple if it has no non-trivial closed two -sided ideals. is easy even in the algebraic sense.

Theorem: Let. Then there exist with.

In addition, Cuntz algebras are related in the following sense with simple, unital, infinite C * - algebras.

Theorem: Let a simple, infinite, unital C * - algebra. Then a C * - subalgebra exists of which is isomorphic to. For finite, there exists a C * - subalgebra containing an ideal, so.

Classification

It should be as above. If we define, as also with isometrics and it is obvious.

Is obtained in this way, the inclusions

With K- theoretical methods, one can show that not and are isomorphic if. If finite, then the group is calculated from to. In the event arises. Since the group is an isomorphism - invariant, it follows immediately the assertion.

View as cross product

On there is a * - automorphism so. Because as a UHF - algebra is nuclear, it follows from this representation as cross product, that is nuclear.