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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
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can be described both using K-theory and directly. One can conclude, for instance, that the collection of unitary operators in the Calkin algebra consists of homotopy classes indexed by the integers
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Appell, Jürgen (2005). "Measures of noncompactness, condensing operators and fixed points: An application-oriented survey".
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Calkin, J. W. (1 October 1941). "Two-Sided Ideals and
Congruences in the Ring of Bounded Operators in Hilbert Space".
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As a C*-algebra, the Calkin algebra is not isomorphic to an algebra of operators on a separable
Hilbert space. The
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One can define a Calkin algebra for any infinite-dimensional complex
Hilbert space, not just separable ones.
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Phillips, N. Christopher; Weaver, Nik (1 July 2007). "The Calkin algebra has outer automorphisms".
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The existence of an outer automorphism of the Calkin algebra is shown to be independent of
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Farah, Ilijas (1 March 2011). "All automorphisms of the Calkin algebra are inner".
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by a two-sided ideal, the Calkin algebra is a C*-algebra itself and there is a
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90:) is composition of operators; it is easy to verify that these operations make
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281:) which are mapped to an invertible element of the Calkin algebra are called
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of the algebra of compact operators on a
Hilbert space.
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335:An analogous construction can be made by replacing
254:{\displaystyle 0\to K(H)\to B(H)\to B(H)/K(H)\to 0}
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320:, by work of Phillips and Weaver, and Farah.
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343:, which is also called a Calkin algebra.
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308:Gelfand-Naimark-Segal construction
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348:The Calkin algebra is the
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406:The Annals of Mathematics
293:. This is in contrast to
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46:bounded linear operators
74:. Here the addition in
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470:Annals of Mathematics
273:. Those operators in
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52:infinite-dimensional
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166:short exact sequence
110:is a Hilbert space.
26:John Williams Calkin
160:As a quotient of a
18:functional analysis
507:Fixed Point Theory
283:Fredholm operators
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72:compact operators
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265:which induces a
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388:. Retrieved
381:the original
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341:Banach space
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535:C*-algebras
139:. In fact,
524:Categories
412:(4): 839.
390:2020-01-17
357:References
162:C*-algebra
114:Properties
483:0705.3085
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