Commutation theorem for traces: Difference between revisions

Revision as of 22:00, 28 July 2013

Revision as of 19:01, 8 March 2014

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:<math>Ja\Omega=a^*\Omega</math>

for ''a'' in ''M'' defines a conjugate-linear isometry of ''H'' with square the identity ''J''² = ''I''. The operator ''J'' is usually called the '''modular conjugation operator'''.

It is immediately verified that ''JMJ'' and ''M'' commute on the subspace ''M'' Ω, so that

:<math>JMJ\subseteq M^\prime.</math>

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an orthogonal direct sum for the real part of inner product. This is just the real orthogonal decomposition for the ±1 eigenspaces of ''J''.

On the other hand for ''a'' in ''M''_sa and ''b'' in ''M'''_sa, the inner product (''ab''Ω, Ω) is real, because ''ab'' is self-adjoint. Hence ''K'' is unaltered if ''M'' is replaced by ''M'' '.

In particular Ω is a trace vector for ''M''' and ''J'' is unaltered if ''M'' is replaced by ''M'' '. So the opposite inclusion

:<math>JM^\prime J\subseteq M</math>

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If in addition τ is non-zero on every non-zero projection, then τ is called a '''faithful trace'''.

If τ is a faithul trace on ''M'', let ''H'' = ''L''²(''M'', τ) be the Hilbert space completion of the inner product space

:<math>M_0=\{a\in M| \tau(a^*a) <\infty\}</math>

with respect to the inner product

:<math>(a,b)=\tau(b^*a).</math>

The von Neumann algebra ''M'' acts by left multiplication on ''H'' and can be identified with its image. Let

:<math>Ja=a^*</math>

for ''a'' in ''M''₀. The operator ''J'' is again called the ''modular conjugation operator'' and extends to a conjugate-linear isometry of ''H'' satisfying ''J''² = I. The commutation theorem of Murray and von Neumann

:{| border="1" cellspacing="0" cellpadding="5"

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These results were proved independently by {{harvtxt|Godement|1954}} and {{harvtxt|Segal|1953}}.

The proof relies on the notion of "bounded elements" in the Hilbert space completion ''H''.

An element of ''x'' in ''H'' is said to be '''bounded''' (relative to <math>\mathfrak{A}</math>) if the map ''a'' → ''xa'' of <math>\mathfrak{A}</math> into ''H'' extends to a

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

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

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

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==References==

*{{citation|first=O.|last=Bratteli|first2=D.W.|last2=Robinson|title=Operator Algebras and Quantum Statistical Mechanics 1, Second Edition|publisher=Springer-Verlag|year=1987|idisbn=ISBN 3-540-17093-6}}

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

*{{citation|first=J.|last = Dieudonné|authorlink=Jean Dieudonné|title=Treatise on Analysis, Vol. II |year=1976|publisher=Academic Press|idisbn=ISBN 0-12-215502-2}}

*{{citation|first=J.|last= Dixmier|authorlink=Jacques Dixmier|title=Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann|publisher= Gauthier-Villars |year=1957}}

*{{citation|first=J.|last= Dixmier|authorlink=Jacques Dixmier|title=Von Neumann algebras|publisher=North Holland| isbn=0-444-86308-7 |year=1981}} (English translation)

*{{citation|first=J.|last= Dixmier|authorlink=Jacques Dixmier|title=Les C*-algèbres et leurs représentations|publisher= Gauthier-Villars|year=1969|idisbn= ISBN 0-7204-0762-1}}

*{{citation|first=J.|last=Dixmier|authorlink=Jacques Dixmier|title=C* algebras|publisher=North Holland|year=1977|idisbn=ISBN 0-7204-0762-1}} (English translation)

*{{citation|first=R.|last=Godement|authorlink=Roger Godement|title=Mémoire sur la théorie des caractères dans les groupes localement compacts unimodulaires|journal=J. Math. Pures Appl.|volume= 30|year=1951|pages=1–110}}

*{{citation|first=R.|last=Godement|authorlink=Roger Godement|title=Théorie des caractères. I. Algèbres unitaires|journal=Ann. Of Math.|volume= 59|year=1954|pages=47–62|doi=10.2307/1969832|issue=1|publisher=Annals of Mathematics|jstor=1969832}}

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|title=On rings of operators IV|journal= Ann. Of Math. (2) |volume= 44 |year=1943|pages= 716–808|doi=10.2307/1969107|jstor=1969107|issue=4|publisher=Annals of Mathematics}}

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

publisher=Academic Press|idisbn=ISBN 0-12-549450-5}}

*{{citation|last=Rieffel|first= M.A.|last2= van Daele|first2=A.|title=A bounded operator approach to Tomita–Takesaki theory|journal=Pacific J. Math.|volume= 69 |year=1977|pages= 187–221}}

*{{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|idisbn = ISBN 0-521-22286-9}}

*{{citation|first=M. |last=Takesaki |title=Theory of Operator Algebras II|publisher=Springer-Verlag|idisbn= ISBN 3-540-42914-X|year=2002}}

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