Editing Calkin algebra

In [[functional analysis]], the '''Calkin algebra''', named after [[John Williams Calkin]],<ref>{{cite web|url=https://library.ias.edu/files/pdfs/hs/cos.pdf|website=ias.edu|title=A Community of Scholars, the Institute for Advanced Study, Faculty and Members 1930–1980|access-date=2020-01-17|archive-date=2011-11-24|archive-url=https://web.archive.org/web/20111124234612/http://library.ias.edu/files/pdfs/hs/cos.pdf|url-status=dead}}</ref> is the [[quotient space (linear algebra)|quotient]] of ''B''(''H''), the [[ring (algebra)|ring]] of [[bounded linear operator]]s on a [[separable space|separable]] infinite-dimensional [[Hilbert space]] ''H'', by the [[ideal (ring theory)|ideal]] ''K''(''H'') of [[Compact operator on Hilbert space|compact operator]]s.<ref>{{cite journal|last=Calkin|first=J. W.|title=Two-Sided Ideals and Congruences in the Ring of Bounded Operators in Hilbert Space|journal=The Annals of Mathematics|date=1 October 1941|volume=42|issue=4|pages=839–873|doi=10.2307/1968771|jstor=1968771 }}</ref> 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 algebra|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]]
::<math>0 \to K(H) \to B(H) \to B(H)/K(H) \to 0</math>
:which induces a [[Bott periodicity|six-term cyclic exact sequence]] in [[operator K-theory|K-theory]]. Those operators in ''B''(''H'') which are mapped to an invertible element of the Calkin algebra are called [[Fredholm operator]]s, and their [[linear map#Index|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 consists of 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 not isomorphic to an algebra of operators on a separable Hilbert space. The [[Gelfand-Naimark-Segal construction]] 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 representation.{{citation needed|date=January 2016}}

* 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.<ref>{{cite journal|last=Phillips|first=N. Christopher|author2=Weaver, Nik|title=The Calkin algebra has outer automorphisms|journal=Duke Mathematical Journal|date=1 July 2007|volume=139|issue=1|pages=185–202|doi=10.1215/S0012-7094-07-13915-2|arxiv=math/0606594}}</ref><ref>{{cite journal|last=Farah|first=Ilijas|title=All automorphisms of the Calkin algebra are inner|journal=Annals of Mathematics|date=1 March 2011|volume=173|issue=2|pages=619–661|doi=10.4007/annals.2011.173.2.1|arxiv=0705.3085}}</ref>

== Generalizations ==

* One can define a Calkin algebra for any infinite-dimensional complex Hilbert space, not just separable ones.

* An analogous construction can be made by replacing ''H'' with a [[Banach space]], which is also called a Calkin algebra.<ref>{{cite journal|last=Appell|first=Jürgen|title=Measures of noncompactness, condensing operators and fixed points: An application-oriented survey |journal=Fixed Point Theory|volume=6|issue=2|pages=157–229|year=2005}}</ref>

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

== References ==
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[[Category:Operator theory]]
[[Category:C*-algebras]]
[[Category:K-theory]]