Calkin algebra

Properties

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

  Generalizations

• One can define a Calkin algebra for any infinite-dimensional complex Hilbert space, not just separable ones.
• 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.