# Calkin algebra

In functional analysis, the **Calkin algebra**, named after John Wilson 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]}

Since the compact operators is a (in fact, the only) maximal norm-closed ideal in *B*(*H*), the Calkin algebra is simple.

As a quotient of two C* algebras, the Calkin algebra is a C* algebra itself. There is a short exact sequence

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 are 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 remarkable because it is not isomorphic to an algebra of operators on a separable Hilbert space; instead, a larger Hilbert space has to be chosen (the GNS theorem says that every C* algebra is isomorphic to an algebra of operators on a Hilbert space; for many other simple C* algebras, there are explicit descriptions of such Hilbert spaces, but for the Calkin algebra, this is not the case).

The same name is now used for the analogous construction for a Banach space.

The Calkin algebra is the Corona algebra of the algebra of compact operators on a Hilbert space.

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]}