Calkin algebra

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In functional analysis, the Calkin algebra 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.

Since the compact operators is the norm-closed minimal 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

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

which induces an 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.

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.