CALKIN ALGEBRA

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
:$0$
ightarrow K(H)
ightarrow B(H)
ightarrow B(H)/K(H)
ightarrow 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.

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Reference

Reference

★ Calkin, J.W. (1941).''Two-sided ideals and congruences in the ring of bounded operators in Hilbert space''. ''Annals of Mathematics'', '42', 839-873.