Calkin algebra

In functional analysis, the Calkin algebra, named after John Williams Calkin, is the quotient of B(H), the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K(H) of compact operators.^[1]

Properties

• Since K(H) is a maximal norm-closed ideal in B(H), the Calkin algebra is simple. In fact, K(H) is the only closed ideal in B(H).
• As a quotient of two C* algebras, the Calkin algebra is a C* algebra itself and there is a short exact sequence

$0 \rightarrow K(H) \rightarrow B(H) \rightarrow B(H)/K(H) \rightarrow 0$

which induces a six-term cyclic exact sequence in K-theory. Those operators in B(H) which are mapped to an invertible element of the Calkin algebra are called Fredholm operators, and their index can be described both using K-theory and directly. One can conclude, for instance, that the collection of unitary operators in the Calkin algebra consists of homotopy classes indexed by the integers Z. This is in contrast to B(H), where the unitary operators are path connected.

• As a C*-algebra, the Calkin algebra is not isomorphic to an algebra of operators on a separable Hilbert space. The Gelfand-Naimark-Segal construction implies that the Calkin algebra isomorphic to an algebra of operators on a nonseparable Hilbert space, but while for many other C*-algebras there are explicit descriptions of such Hilbert spaces, the Calkin algebra does not have an explicit representation.
• The existence of an outer automorphism of the Calkin algebra is shown to be independent of ZFC, by work of Phillips and Weaver, and Farah.^[2][3]

Generalizations

• An analogous construction can be made by replacing H with a Banach space, which is also called a Calkin algebra.
• The Calkin algebra is the Corona algebra of the algebra of compact operators on a Hilbert space.

References

1. Calkin, J. W. (1 October 1941). "Two-Sided Ideals and Congruences in the Ring of Bounded Operators in Hilbert Space". The Annals of Mathematics 42 (4): 839. doi:10.2307/1968771.
2. Phillips, N. Christopher; Weaver, Nik (1 July 2007). "The Calkin algebra has outer automorphisms". Duke Mathematical Journal 139 (1): 185–202. doi:10.1215/S0012-7094-07-13915-2.
3. Farah, Ilijas (1 March 2011). "All automorphisms of the Calkin algebra are inner". Annals of Mathematics 173 (2): 619–661. doi:10.4007/annals.2011.173.2.1.