Calkin algebra

In mathematics , the Calkin algebra (after John Williams Calkin ) is a special Banach algebra that is assigned to a Banach space . In Calkin's algebra one can consider properties of continuous linear operators in a simplified manner by identifying operators whose difference is compact . This is how you arrive at classification sentences for normal operators modulo compact operators.

definition

Be a Banach space. Then the Banach algebra of compact operators is on a two-sided, closed ideal in the algebra of all bounded linear operators on . Then the quotient algebra with the quotient norm is again a Banach algebra, the Calkin algebra of . be the quotient mapping . ${\displaystyle E}$ ${\displaystyle K(E)}$ ${\displaystyle E}$ ${\displaystyle B(E)}$ ${\displaystyle E}$ ${\displaystyle B(E)/K(E)}$ ${\displaystyle E}$ ${\displaystyle \pi \colon B(E)\rightarrow B(E)/K(E)}$

Fredholm operators

Fredholm operators can be characterized using Calkin's algebra. The set of FV Atkinson says that for a bounded linear operator the following are equivalent: ${\displaystyle T\in B(E)}$

• ${\displaystyle T}$ is a Fredholm operator.
• There is an operator such that and are compact. ${\displaystyle S\in B(E)}$ ${\displaystyle \mathrm {id} _{E}-ST}$ ${\displaystyle \mathrm {id} _{E}-TS}$
• ${\displaystyle \pi _{E}(T)}$is invertible in . ${\displaystyle B(E)/K(E)}$

An important consequence is that the set of Fredholm operators is an open set in , because according to this theorem it is the archetype of the open set of invertible elements in under the continuous mapping . ${\displaystyle B(E)}$ ${\displaystyle B(E)/K(E)}$ ${\displaystyle \pi }$

C * algebra

If a Hilbert space is, then the quotient of a C ^* -algebra is again a C ^* -algebra. For the rest of this section, let it be separable and infinitely dimensional. Then the Calkin algebra is simple , i. that is, it has no two-sided, closed ideals other than and itself, because it is a maximal two-sided ideal. Furthermore, the Calkin algebra (see continuum (mathematics) ) has pairwise orthogonal projections . The Calkin algebra has no non-separable representations other than 0 ; i.e. , if there is a * - homomorphism , then the Hilbert space is either the null vector space or it is not separable. ${\displaystyle H}$ ${\displaystyle B(H)/K(H)}$ ${\displaystyle H}$ ${\displaystyle B(H)/K(H)}$ ${\displaystyle \{0\}}$ ${\displaystyle B(H)/K(H)}$ ${\displaystyle K(H)\subset B(H)}$ ${\displaystyle 2^{\aleph _{0}}}$ ${\displaystyle \varphi :B(H)/K(H)\rightarrow B({\tilde {H}})}$ ${\displaystyle {\tilde {H}}}$

Applications

With regard to the classification of normal operators , there are considerable simplifications if one uses terms modulo compact operators; such terms usually have the addition essential . In the following, let H be a separable Hilbert space again.

The essential spectrum of an operator is defined as the spectrum without the isolated points of finite multiplicity ( multiplicity means dimension of the associated eigenspace ). The essential spectrum of a normal operator T is exactly the normal spectrum of calculated with respect to the Calkin algebra . ${\displaystyle \sigma _{e}(T)}$ ${\displaystyle T\in B(H)}$ ${\displaystyle \pi (T)}$

One calls two operators and unitary equivalent modulo K (H) , if there is a unitary operator such that is compact. This means that and are unitarily equivalent in Calkin algebra, whereby the unitary transformation can be chosen so that it has a unitary archetype in . ${\displaystyle T_{1}}$ ${\displaystyle T_{2}}$ ${\displaystyle U\in B(H)}$ ${\displaystyle U^{*}T_{1}U-T_{2}}$ ${\displaystyle \pi (T_{1})}$ ${\displaystyle \pi (T_{2})}$ ${\displaystyle B(H)}$

The following theorem from Hermann Weyl , John von Neumann and ID Berg now applies : For two normal operators are equivalent: ${\displaystyle T_{1},T_{2}\in B(H)}$

• ${\displaystyle T_{1}}$and are unitarily equivalent modulo K (H). ${\displaystyle T_{2}}$
• ${\displaystyle \sigma _{e}(T_{1})=\sigma _{e}(T_{2})}$.

Addition: If compact , there is a normal operator with . ${\displaystyle \emptyset \not =X\subset {\mathbb {C} }}$ ${\displaystyle T\in B(H)}$ ${\displaystyle \sigma _{e}(T)=X}$

The next step is to consider the concept of normality only as modulo compact operators. An operator is said to be essentially normal if its image in Calkin's algebra is normal. A classification modulo K (H) is also possible for these operators, as the following theorem by LG Brown , RG Douglas and PA Fillmore shows ( BDF theory ). The following are equivalent for two essentially normal operators : ${\displaystyle T\in B(H)}$ ${\displaystyle \pi (T)}$ ${\displaystyle T_{1},T_{2}\in B(H)}$

• ${\displaystyle T_{1}}$and are unitarily equivalent modulo K (H). ${\displaystyle T_{2}}$
• ${\displaystyle \sigma _{e}(T_{1})=\sigma _{e}(T_{2})=:X}$and for all true . ${\displaystyle \lambda \in {\mathbb {C} }\setminus X}$ ${\displaystyle \mathrm {index} (T_{1}-\lambda 1_{H})=\mathrm {index} (T_{2}-\lambda 1_{H})}$

Here index stands for the Fredholm index ; note that this is defined by Atkinson for the operators specified in the sentence according to the above theorem.

Automorphisms on the Calkin Algebra

In 1977, within the framework of the BDF theory mentioned above, the authors asked whether all * -automorphisms on the Calkin algebra are inner , i.e. whether there is a unitary operator for every such automorphism with for all . * -Automorphisms that are not of this form are called outer * -automorphisms . So the question is whether there are outer * -automorphisms on the Calkin algebra. ${\displaystyle \varphi }$ ${\displaystyle u\in B(H)/K(H)}$ ${\displaystyle \varphi (a)=uau^{*}}$ ${\displaystyle a\in B(H)/K(H)}$

It is known for that every * -automorphism is internal. The proof uses that an * -automorphism has to map operators with a one-dimensional image again to such and constructs a unitary operator from this, so it makes substantial use of compact operators. But it is precisely this that is no longer available in Calkin algebra, so that the proof cannot be transferred. The problem of the existence of external * -automorphisms remained open for a long time until it found a surprising solution in 2007 and 2011. This problem has proven to be independent, that is, the axioms of the Zermelo-Fraenkel set theory with axiom of choice , or ZFC for short, do not allow a decision on this question. ${\displaystyle B(H)}$

First of all, NC Phillips and N. Weaver showed that assuming the continuum hypothesis, the existence of external automorphisms follows. Since the continuum hypothesis is consistent with ZFC, as K. Gödel had already shown in 1938 with the model of constructible sets , the existence of external * -automorphisms is also consistent with ZFC.

A proof that all * -automorphisms are internal is no longer possible, but it was not excluded that there could be a proof of the existence of external * -automorphisms on the basis of the ZFC axioms, which does not use the continuum hypothesis. I. Farah showed in 2011 that this is not the case either . If one adds the open coloring axiom to ZFC , then all * -automorphisms on the Calkin algebra are inner. Since the open coloring axiom is also consistent with ZFC, as S. Todorcevic had shown in 1989, one cannot refute the existence of outer * -automorphisms on the Calkin algebra in ZFC, i.e. the existence of outer * -automorphisms on the Calkin algebra is entirely independent of ZFC.

Individual evidence

1. ^ JW Calkin: Two-sided ideals and congruences in the ring of bounded operators in Hilbert space. In: Annals of Mathematics . 42, 839-873 (1941)
2. ^ FV Atkinson: The normal solvability of linear equations in normed spaces. In: Mat. Sb. 28 (70), 3-14 (1951)
3. ^ ID Berg: An Extension of the Weyl-von Neumann theorem to normal operators. In: Trans. American Mathematical Society . 160, 365-371 (1971)
4. ^ RG Douglas: C * -Algebra Extensions and K-Homology. Princeton University Press 1980
5. ^ LG Brown, RG Douglas, PA Fillmore: Extensions of C * -algebras and K-Homology , Annals of Mathematics (1977), Volume 105, pages 265-324, page 270 before Def. 1.7
6. ^ NC Phillips, N. Weaver: The Calkin algebra has outer automorphisms , Duke Mathematical Journal (2007), Volume 139, pages 185-202
7. ^ I. Farah: All automorphisms of the Calkin algebra are inner , Annals of Mathematics (2011), Volume 173, pp. 619-661