Commutation theorem for traces: Difference between revisions

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{{shortShort description|Identifies the commutant of a specific von Neumann algebra acting on a Hilbert space}}

{{short description|none}}

In [[mathematics]], a '''commutation theorem for traces''' explicitly identifies the [[commutant]] of a specific [[von Neumann algebra]] acting on a [[Hilbert space]] in the presence of a [[Von Neumann algebra#Weights, states, and traces|trace]].

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Moreover if

:<math>M = \lambda(\mathfrak{A})^{\prime\prime},</math>

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The commutation theorem follows immediately from the last assertion. In particular

:<math display="block">M_0M = \lambda(\mathfrak{B})''.</math>

* ''M'' = λ(<math>\mathfrak{B}</math>)".

The space of all bounded elements <math>\mathfrak{B}</math> forms a Hilbert algebra containing <math>\mathfrak{A}</math> as a dense *-subalgebra. It is said to be '''completed''' or '''full''' because any element in ''H'' bounded relative to <math>\mathfrak{B}</math> must actually already lie in <math>\mathfrak{B}</math>. The functional τ on ''M''₊ defined by

:<math display="block"> \tau(x) = (a,a)</math>

if ''x'' = ''λ''(''a'')*''λ''(''a'') and ∞ otherwise, yields a faithful semifinite trace on ''M'' with

:<math> \tau(x) = (a,a)</math>

<math display="block">M_0 = \mathfrak{B}.</math>

if ''x'' = λ(a)*λ(a) and ∞ otherwise, yields a faithful semifinite trace on ''M'' with

:<math>M_0 = \mathfrak{B}.</math>

Thus:

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==References==

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*{{citation|first=A.|last=Connes|authorlink=Alain Connes|title=Sur la théorie non commutative de l’intégrationl'intégration|series=Lecture Notes in Mathematics|volume=(Algèbres d'Opérateurs)|publisher=Springer-Verlag|year=1979|pages=19–143|isbn=978-3-540-09512-5}}

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*{{citation|last=Pedersen|first=G.K.|title=C* algebras and their automorphism groups|series=London Mathematical Society Monographs|volume=14|year=1979|

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*{{citation|last=Segal|first=I.E.| authorlink=Irving Segal|title=A non-commutative extension of abstract integration|journal=Ann. of Math. |volume=57|year=1953|pages= 401–457|doi=10.2307/1969729|issue=3|publisher=Annals of Mathematics|jstor=1969729}} (Section 5)

*{{citation|last=Simon|first= B.|authorlink=Barry Simon|title=Trace ideals and their applications|series=London Mathematical Society Lecture Note Series|volume= 35|publisher= Cambridge University Press|year= 1979|isbn = 0-521-22286-9}}