Calkin-Algebra

In mathematics, the Calkin algebra ( by John Williams Calkin ) a special Banach algebra that is associated with a Banach space. In the Calkin algebra can be considered properties of continuous linear operators simplified by operators whose difference is compact, can be identified. So we come to classification rates for normal operators modulo compact operators.

Definition

Be a Banach space. Then the Banach algebra of compact operators on a two-sided, closed ideal in the algebra of all bounded linear operators on. Then the quotient algebra is the quotient norm is a Banach algebra again, the Calkin algebra of. is the quotient map.

Fredholm operators

Fredholm operators can be characterized by means of the Calkin algebra. The set of FV Atkinson states that the following are equivalent for a bounded linear operator:

• Is a Fredholm operator.
• There is an operator, are so and compact.
• Is invertible in.

An important implication is that the set of Fredholm operators is an open set in, because it is after that set the archetype of open set of invertible elements in under the continuous map.

C * - algebra

Is a Hilbert space, then the quotient of a C *-algebra again a C *-algebra. For the remainder of this section is separable and infinite- dimensional. Then the Calkin algebra is simple, that is, it has no two-sided ideals except and completed itself, because is a maximal two-sided ideal. Next has the Calkin algebra (see continuum (mathematics) ) pairwise orthogonal projections. The Calkin algebra has no non- different from 0 separable representations, ie is a * - homomorphism, then the Hilbert space is either the zero vector space or non- separable.

Applications

Regarding the classification of normal operators have significant simplifications when terms modulo compact operators used such terms usually have the additional material. In the following let H be a separable Hilbert space again.

The main range of an operator is defined as the spectrum without the isolated points finite multiplicity ( multiplicity means dimension of the associated eigenspace ). The essential spectrum of a normal operator T is exactly regarding the Calkin algebra calculated spectrum of ordinary.

We call two operators and is unitarily equivalent modulo K (H ), if there is a unitary operator, so that compact. This means that and are in the Calkin algebra are unitarily equivalent, where the unitary transformation can be chosen so that it has a unitary prototype in.

It is now considered the following set of Hermann Weyl, John von Neumann and ID Berg: For two normal operators are equivalent:

• And unitarily equivalent modulo K (H).
• .

Extra: Is compact so it is with a normal operator.

The next step is to consider the notion of normality only modulo compact operators. This is called an operator much normal, if its image in the Calkin algebra is normal. Also for these operators manage a classification modulo K ( H) as the following set of LG Brown, RG Douglas and PA Fillmore shows (BDF - theory). For two essentially normal operators are equivalent:

• And unitarily equivalent modulo K (H).
• And applies to all.

This index represents the Fredholm index, note that this is defined for the specified set of operators in accordance with the above set of Atkinson.

Automorphisms on the Calkin algebra

As part of the above-mentioned BDF theory, the authors in 1977 the question of whether all *- automorphisms on the Calkin algebra are inner, ie, whether it each such automorphism a unitary operator are with all. * Automorphisms, which are not of this form, called external * automorphisms. So the question is whether there are external to the Calkin algebra * - automorphisms.

For it is known that every * - automorphism is inner. Used to prove that a * - automorphism operators with one-dimensional image again must reflect on such and constructed from a unitary operator, makes so much of compact operators use. But just this one has in the Calkin algebra yes no longer available, so that the evidence can not be transferred. The problem of the existence of external * - automorphisms has long remained open until it has been found in the years 2007 and 2011, a surprising solution. This problem has proved to be independent, that is, the axioms of Zermelo -Fraenkel set theory with the axiom of choice, ZFC short, leave no decision of this question to.

First, NC Phillips and N. Weaver have shown that under the additional assumption of the continuum hypothesis, the existence of external automorphisms follows. Since the continuum hypothesis to ZFC is consistent, as K. Gödel had shown with the model of constructible sets in 1938, including the existence of external * - automorphisms to ZFC is consistent.

This is a proof that all *- automorphisms are inner, no longer possible, but it was not impossible that there could be a proof of the existence of external * - automorphisms on the basis of ZFC axioms, which does not use the continuum hypothesis. This too is not the case, I. Farah has shown in 2011. Assuming ZFC to add the Open Coloring Axiom, so all *- automorphisms on the Calkin algebra are inner. Since the Open Coloring Axiom is consistent with ZFC as well, such as S. Todorcevic had shown in 1989, one can not disprove the existence of external * - automorphisms on the Calkin algebra in ZFC, that is, the existence of external * - automorphisms on the Calkin algebra is generally independent of ZFC.