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Concept# Calkin algebra

Summary

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. Here the addition in B(H) is addition of operators and the multiplication in B(H) is composition of operators; it is easy to verify that these operations make B(H) into a ring. When scalar multiplication is also included, B(H) becomes in fact an algebra over the same field over which H is a Hilbert space.
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 a C*-algebra by a two-sided ideal, the Calkin algebra is a C*-algebra itself and there is a short exact sequence ::0 \to K(H) \to B(H) \to B(H)/K(H) \to 0 :which induces a six-term cyclic exact sequence in K-theory. Those operators in B(H) which are mapped to an invertibl

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