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MessagePosté le: Sam 3 Sep - 04:24 (2016)    Sujet du message: Von Neumann Algebra Pdf Download Répondre en citant




Von Neumann Algebra Pdf Download > shorl.com/stevasebymani









Von Neumann Algebra Pdf Download

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This...paper...gives...their...basic...properties...and...the...division...into...types...I,...II,...and...III,...and...in...particular...finds...factors...not...of...type...I....A..bimodule..(or..correspondence)..is..a..Hilbert..space..H..with..module..actions..of..two..commuting..von..Neumann..algebras...More...precisely,...if...the...Connes...spectrum...(of...its...modular...group)...is...1...then...the...factor...is...of...type...III0,...if...the...Connes...spectrum...is...all...integral...powers...of......for...0<<1,...then...the...type...is...III,...and...if...the...Connes...spectrum...is...all...positive...reals...then...the...type...is...III1....Equivalent..subrepresentations..correspond..to..equivalent..projections..in..G...Another...way...of...stating...this...is...that...E...is...equivalent...to...F...if...E=uu*...and...F=u*u...for...some...partial...isometry...u...in...M....For..example,..all..finite-dimensional..projections..(or..subspaces)..are..finite..(since..isometries..between..Hilbert..spaces..leave..the..dimension..fixed),..but..the..identity..operator..on..an..infinite-dimensional..Hilbert..space..is..not..finite..in..the..von..Neumann..algebra..of..all..bounded..operators..on..it,..since..it..is..isometrically..isomorphic..to..a..proper..subset..of..itself...The...equivalence...relation......thus...defined...is...additive...in...the...following...sense:...Suppose...E1......F1...and...E2......F2....

Any....factor....has....a....trace....such....that....the....trace....of....a....non-zero....projection....is....non-zero....and....the....trace....of....a....projection....is....infinite....if....and....only....if....the....projection....is....infinite.....Murray,....F.J.;....von....Neumann,....J.....For...many...years...it...was...an...open...problem...to...find...a...type...II...factor...whose...fundamental...group...was...not...the...group...of...positive...reals,...but...Connes...then...showed...that...the...von...Neumann...group...algebra...of...a...countable...discrete...group...with...Kazhdan's...property...T...(the...trivial...representation...is...isolated...in...the...dual...space),...such...as...SL(3,Z),...has...a...countable...fundamental...group....It....is....a....factor....(of....type....II1)....if....every....non-trivial....conjugacy....class....of....G....is....infinite....(for....example,....a....non-abelian....free....group),....and....is....the....hyperfinite....factor....of....type....II1....if....in....addition....G....is....a....union....of....finite....subgroups....(for....example,....the....group....of....all....permutations....of....the....integers....fixing....all....but....a....finite....number....of....elements).....Application....Error.....Since..the..identity..operator..is..always..infinite..in..those..factors,..they..were..sometimes..called..type..III..in..the..past,..but..recently..that..notation..has..been..superseded..by..the..notation..III,..where....is..a..real..number..in..the..interval..[0,1]...The...tensor...product...of...a...factor...of...type...II1...and...an...infinite...type...I...factor...has...type...II,...and...conversely...any...factor...of...type...II...can...be...constructed...like...this....x...&#x21A6;...T...r...(...1...&#x03BB;...+...1...0...0...&#x03BB;...&#x03BB;...+...1...)...x.......The....relationship....between....commutative....von....Neumann....algebras....and....measure....spaces....is....analogous....to....that....between....commutative....C*-algebras....and....locally....compact....Hausdorff....spaces.....

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