Brans–Dicke theory: Difference between revisions

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In [[functional analysis]], the '''Calkin algebra''', named after [[John Williams Calkin]], 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|year=1941|month=October|volume=42|issue=4|pages=839|doi=10.2307/1968771}}</ref>
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Since the compact operators is a (in fact, the only) maximal norm-closed ideal in ''B''(''H''), the Calkin algebra is [[simple algebra|simple]].
 
As a quotient of two [[C* algebra]]s, the Calkin algebra is a C* algebra itself. There is a [[short exact sequence]]
:<math>0 \rightarrow K(H) \rightarrow B(H) \rightarrow B(H)/K(H) \rightarrow 0</math>
which induces a six-term cyclic exact sequence in [[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 are 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 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]].
 
The Calkin algebra is the [[Corona algebra]] of the algebra of compact operators on a Hilbert space.
 
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|coauthors=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}}</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}}</ref>
 
==References==
<references />
 
[[Category:Operator theory]]
[[Category:C*-algebras]]
[[Category:K-theory]]

Revision as of 08:57, 1 March 2014

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