Measure Theory in Noncommutative Spaces

Sym metry comma I n tegra b i l i ty a nd Geometry : .... Methods a nd App l i ca t i ons .... S I GMA 6 open parenthesis 20 1 0 closing \noindent Sym metry , I n tegra b i l i ty a nd Geometry : \ h f i l l Methods a nd App l i ca t i ons \ h f i l l S IGMA6 ( 20 10 ) , 72 , 36pa ges parenthesisSym metry comma , I n72 tegra comma b i l i3 ty 6 a pa nd ges Geometry : Methods a nd App l i ca t i ons S I GMA 6 ( 20 1 0 ) , 72 , 3 6 pa ges Measure .. Theory .. in .. Noncommutative .. Spaces to the power of big star ? \noindentMeasureMeasure \quad TheoryTheory \quad inin \quad NoncommutativeNoncommutative \quad $ SpacesSpaces ˆ{\ star }$ StevenSteven LORD LORD dagger ..† andand Fedor Fedor SUKOCHEV SUKOCHEV ddagger ‡ dagger School of Mathematical Sciences comma .. University of Adelaide comma Adelaide comma .. 5005 comma Australia \noindent† SchoolSteven of Mathematical LORD $ \dagger Sciences$ ,\quad Universityand Fedor of Adelaide SUKOCHEV , Adelaide $ \ddagger , 5005$ , Australia E hyphen mailE -: ..mail s t e : v e ..s tn e period v e lo .. n r . .. lo d at a r d e l a d ide@ perioda d e e ldu a period ide . e.. du a u . a u ddagger School of Mathematics and Statistics comma .... University of New South Wales comma Sydney comma 2052 comma Australia \noindent‡ School$ of\dagger Mathematics$ School and Statistics of Mathematical , University Sciences of New , \quad SouthUniversity Wales , Sydney of Adelaide , 2052 , , Adelaide , \quad 5005 , Australia E hyphenAustralia mail : .. f period s .. u k .. o ch .. e v at u n s .. w period edu period .. a u Received March 25 comma 20 1 0 comma in final form August 4 comma 20 1 0 semicolon Published online September 1 6 comma 20 1 0 \ centerline {EE− -mail mail : : \quadf . ss t u e k v e o\quad chn e .v l o@ \uquad n sr \ wquad . edud . $@$ a u adelaide.edu. \quad a u } d oReceived i : 1 0 period March 3 8 25 42 , 20slash 1 0 SI , in .. Gfinal M form .. A period August 2 4 0 , 1 20 0 1period 0 ; Published 0 72 online September 1 6 , 20 1 0 Abstract period The integral in noncommutative geometry open parenthesis NCG closing parenthesis involves a non hyphen st andard trace \noindentd o i : 1$ 0 .\ddagger 3 8 42 / SI$ G School M A . of 2 0 Mathematics 1 0 . 0 72 and Statistics , \ h f i l l University of New South Wales , Sydney , 2052 , Australia called .. aAbstract Dixmier .. . traceThe periodintegral .. in The noncommutative geometric origins geometry .. of this ( NCG integral ) involves are well a knownnon - st period andard .. trace From a measure hyphen theoretic view comma .. however comma .. the formulation contains several difficulties period .. We re hyphen \ centerlinecalled{E − amail Dixmier : \quad tracef . . The s \ geometricquad u k origins\quad ofo this ch integral\quad aree v well $ known @ $ . u nFrom s \quad w . edu . \quad a u } view resultsa measure concerning - theoretic the technical view , features however of , the the integral formulation in NCG contains and some several outstanding difficulties . We re - problems in this area period The review is aimed for the general user of NCG period \noindent Receivedview results March concerning 25 , the 20 technical 1 0 , features in final of the form integral August in NCG 4 and , 20 some 1 0 outstanding ; Published online September 1 6 , 20 1 0 Key words : .. Dixmierproblems trace semicolon in this area .. singular . The review trace is semicolon aimed for .. the noncommutative general user of integration NCG . semicolon .. noncommutative geometry semicolon Lebesgue integral semicolon noncommutative residue \noindentKeydoi words : :10.3842/SIDixmier trace ; singular\quad traceGM ; noncommutative\quad A . 2 0 integration 1 0 . 0 ; 72 noncommutative 201 0 Mathematics Subject Classificationgeometry ; Lebesgue : 46 L 5 1 integral semicolon ; noncommutative 47 B 1 0 semicolon residue 58 B 34 Contents \ centerline { Abstract201 0. Mathematics The integral Subject in Classification noncommutative : 46 L geometry 5 1 ; 47 B 1 ( 0 NCG; 58 B ) 34 involves a non − st andard trace } 1 ....Contents I nt ro duction .... 2 2 .... Dix .... m i e r .... t r a c e s .... 5 \ centerline1{ c a l l e d \quad a Dixmier \quad I ntt r ro a c duction e . \quad The geometric origins \quad of 2 this integral are well known . \quad From } 2 period2 1 .. 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I d e n t if ica t i o n s .... i n .... open parenthesis 2 period 3 7 closing parenthesis .... 29 \noindentRe f e r e ..6 n\ ch e f is l .. l 34Charac te r i s i \ h f i l l ng t h e \ h f i l l n o \ h f i l l n c o \ h f i l l m \ h f i l l m \ h f i l l u t a t i v e \ h f i l l i n \ h f i l l te g r a l \ h f i l l 24 hline \ hspacebig star∗{\ subf i This l l }6 paper . 1 is\quad a contributionCh a r to a the\quad Specialcter Issue quotedblleft isationan Noncommutative\quad Spacesd s in and gu FieldsRow l a r \quad 1 quotedblrightt r a c Row e s 2\quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 24 period Fields full collection is \ hspaceavailable∗{\ atf hi l t l t}6 p : . slash 2 \ slashquad wA w\ wquad periodd e o m\ iquad s periodmi d n e slash\quad j o ..at u edconver n .. a l s slash S I rgence G .. 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1 Introduction The basic obj ect of noncommutative geometry ( NCG ) i s a sp ectral triple (A,H,D)[1], originally called a K - cycle [ 2 , p . 546 ] . Here A is a countably generated unital non - degenerate ∗− algebra of bounded linear operators on a separable complex Hilbert space H, and D : Dom D → H i s a selfadj oint linear operator with the properties k [D, a] k < ∞ for all a ∈ A and hDi−1 := (1 + D2)− 21 i s a compact operator . In Connes ’ quantised calculus the compact operator hDi−1 is the analogue of an inﬁnitesimal length element ds [2, p . 545 ] , [ 3 , p . 1 5 7 ] , and the integral of a ∈ A is represented by the operator - theoretic expression [ 2 , p . 545 ] , [ 3 , p . 1 58 ],

−n Trω(ahDi ), a ∈ A, (1.1) where n( assumed t o exist ) i s the smallest value such that hDi−p i s trace class for p > n, and the positive trace Trω ( nominally a ‘ Dixmier trace ’ ) measures the log divergence of the trace of ahDi−n. The link b etween integration in the traditional sense and sums of eigenvalues of the product ahDi−n i s not obvious . Segal , in [ 4 ] , formalised an operator - theoretic view of integration theory . The subsequently developed theory 1 [ 5 , 6 ] , through t o [ 7 , 8 , 9 ] , features the notion of a faithful normal trace τ on a ∗− algebra N of bounded linear operators on H that is closed in the weak operator t opology ( a von Neumann algebra ) . The trace τ is used t o build an Lp− theory with L∞(N , τ) = N and L−p(N , τ) b eing the completion of the set {x ∈ N | τ(| x |p) < ∞}. The trace ( and the integration theory ) is called ﬁnit e if τ(1N ) < ∞ where 1N i s the identity of N . The integration theory associated t o the pair (N , τ) has analogous machinery to measure theory , including Radon – Nikodym theorems , dominated and monotone convergence theorems2, etc . Without the condition of normality this machinery fails . A commutative example of the theory ( and the prototype ) is a regular Borel measure space (X, µ). For the von Neumann algebra of complex functions f ∈ L∞(X, µ) acting on the separable Hilb ert space L2(X, µ) by ( almost everywhere ) pointwise multiplication , the integral Z ∞ τµ(f) = fdµ, f ∈ L (X, µ) X i s a faithful normal trace . The noncommutative Lp− spaces are the usual Lp− spaces , i . e .

p ∞ p L (L (X, µ), τµ) ≡ L (X, µ). A noncommutative example is the von Neumann algebra L(H) of bounded linear operators on a separable Hilb ert space H. The canonical trace

∞ X Tr(T ) = hhm, T hmi,T ∈ L(H), m = 1

∞ where {hm}m=1 i s an orthonormal basis of H, i s a faithful normal trace . The resulting Lp− spaces

Lp(H) := {T ∈ L(H) | Tr(| T |p) < ∞} (1.2)

1The explanation here is for semifinite von Neumann algebras with separable pre - dual . 2The condition of normality of a linear functional preempts the monotone convergence theorem . If N is 2 .... S period Lord and F period Sukochev \noindent 2 \ h f i l l S . Lord and F . Sukochev hlinea closed ∗− subalgebra of L(H) in the weak operator topology then a linear functional ρ ∈ N ∗ is normal if 1 ..ρ Introduction(S) = supα ρ(Sα) for all increasing nets of positive operators {Sα} ⊂ N in the strong operator topology with l . \ [ The\ r u basic l e {3em obj}{ ect0.4 of pt noncommutative}\ ] geometryP open parenthesisP NCG closing parenthesis i s a sp ectral triple open parenthesis A comma u . b .S. This condition is equivalent to ρ( i Pi) = i ρ(Pi) for all sets {Pi} of pairwise orthogonal projections H commafrom DN closing[10, p parenthesis . 67 ] , i . e open . analogous square to bracket additivity 1 closing of a measure square on bracket a σ− commaalgebra of originally sets . called a K hyphen cycle open square bracket 2 comma p period 546 closing square bracket period .. Here A is a countably generated unital non\noindent hyphen degenerate1 \quad *Introduction hyphen algebra of bounded linear operators on a separable complex Hilbert space H comma .. and D : .. Dom D right arrow H \noindenti s a selfadjThe oint basic linear operator obj ect with of the noncommutative properties .. bar open geometry square bracket ( NCG D ) comma i s a a sp closing ectral square triple bracket bar $ ( less infinity A , for allH a in, A and D .. ) angbracketleft [ 1 D ] right ,$ angbracket originally to the power of minus 1 : = c aopen l l e d parenthesis a K − cycle 1 plus [D 2to the,p power . 546 of 2 ] closing . \quad parenthesisHere to $ Athe $ power is of a minus countably hline 2 togenerated the power ofunital 1 .. i s non a compact− degenerate operator period$ ∗ − $ algebra In Connes quoteright quantised calculus the compact operator angbracketleft D right angbracket to the power of minus 1 is the analogue of an\noindent infinitesimalof bounded linear operators on a separable complex Hilbert space $ H , $ \quad and $ D : $length\quad .. elementDom .. $ ds D open\rightarrow square bracket 2H comma $ .. p period .. 545 closing square bracket comma .. open square bracket 3 comma .. p periodi s a.. 1 selfadj 5 7 closing oint square linear bracket operator comma .. and with the ..the integral properties .. of a in A\quad is .. represented$ \ parallel by the [ D , a ] \ parallel < operator\ infty hyphen$ f theoretic o r a l l expression $ a \ openin squareA $ bracket and \ 2quad comma$ p period\ langle 545 closingD square\rangle bracketˆ{ comma − 1 open} square: = bracket $ 3 comma p period 1 58 closing square bracket comma \noindentEquation: open$ ( parenthesis 1 + 1 D period ˆ{ 2 1 closing} ) ˆ parenthesis{ − \ r u .. l e Tr{3em sub}{ omega0.4 pt open}} parenthesis2 ˆ{ 1 } a$ angbracketleft\quad i s D a right compact angbracket operator to the . power of minus n closing parenthesis comma a in A comma Inwhere Connes n open ’ quantised parenthesis assumed calculus t o exist theclosing compact parenthesis operator i s the $ smallest\ langle valueD such that\rangle .. angbracketleftˆ{ − 1 D} right$ is angbracket the analogue to the of an infinitesimal powerlength of minus\quad p i selement trace class\quad for p greater$ ds n comma [ 2 and , $ \quad p . \quad 545 ] , \quad [ 3 , \quad p . \quad 1 5 7 ] , \quad and the \quad i n t e g r a l \quad o f $ athe positive\ in A trace $ Tr i s sub\quad omegarepresented open parenthesis by nominally the a quoteleft Dixmier trace quoteright closing parenthesis measures the log divergenceoperator of− thetheoretic trace expression [ 2 , p . 545 ] , [ 3 , p . 1 58 ] , of a angbracketleft D right angbracket to the power of minus n period .. The link b etween integration in the traditional sense and sums of eigenvalues\ begin { a l of i g the n ∗} Trproduct{\omega a angbracketleft} ( Da right\ langle angbracketD to the\rangle power of minusˆ{ − n in s not} obvious) , period a \ in A, \ tag ∗{$ ( 1 . 1 ) $Segal} comma in open square bracket 4 closing square bracket comma formalised an operator hyphen theoretic view of integration theory period\end{ ..a l The i g n ∗} subsequently developed theory 1 open square bracket 5 comma .. 6 closing square bracket comma through t o open square bracket 7 comma .. 8 comma ..\noindent 9 closing squarewhere bracket $ n comma ( $ features assumed the notion t o of exist a faithful ) i normal s the trace smallest tau on value such that \quad $ \ langle D \ranglea * hyphenˆ{ − algebrap N}$ of bounded i s trace linear class operators for on H $p that is> closedn in the , $ weak and operator t opology theopen positive parenthesis trace a von Neumann $ Tr {\ algebraomega closing} parenthesis( $ nominally period .. The a ‘ trace Dixmier tau is used trace t o ’build ) measures an L to the thepower log of p divergence hyphen theory of the trace witho f L $ to a the power\ langle of infinityD open\rangle parenthesisˆ{ N − comman } tau. closing $ \quad parenthesisThe =link N b etween integration in the traditional sense and sums of eigenvalues of the productand .. L-p $ open a parenthesis\ langle N commaD \ taurangle closingˆ{ parenthesis − n }$ .. b ieing s the not .. obvious completion . .. of the .. set .. open brace x in N bar tau open parenthesis bar x bar to the power of p closing parenthesis less infinity closing brace period .. The .. trace .. open parenthesis and .. the Segalintegration , in theory [ 4 ] closing , formalised parenthesis an is called operator finit e if− tautheoretic open parenthesis view 1 of sub integration N closing parenthesis theory less . infinity\quad whereThe 1subsequently sub N i s the identitydeveloped of N period theory .. The 1 integration[ 5 , \quad 6 ] , through t o [ 7 , \quad 8 , \quad 9 ] , features the notion of a faithful normal trace $ \theorytau $ .. associated on t o .. the pair .. open parenthesis N comma tau closing parenthesis .. has .. analogous .. machinery .. to .. measure theorya $ comma∗ − ..$ including algebra $ N $ of bounded linear operators on $ H $ that is closed in the weak operator t opology (Radon a von endash Neumann Nikodym algebra theorems ) . comma\quad dominatedThe t r a and c e monotone $ \tau convergence$ is used theorems t o build to the power an $Lˆ of 2 comma{ p } etc − period$ theory.... Without with the$ L ˆ{\ infty } (N, \tau ) = N $ andcondition\quad of normality$ L−p this ( machinery N , fails\tau period ) $ \quad b eing the \quad completion \quad o f the \quad s e t \quad $ \{A .. commutativex \ in ..N example\mid .. of the\tau .. theory( ..\ openmid parenthesisx \mid andˆ ..{ thep ..} prototype) < closing\ infty parenthesis\} .. is. .. $ a ..\quad regularThe .. Borel\quad .. t r a c e \quad ( and \quad the measureintegration theory ) is called finit e if $ \tau ( 1 { N } ) < \ infty $ where $ 1 { N }$ i sspace the open identity parenthesis of X $N comma . mu $ closing\quad parenthesisThe integration period .. For the von Neumann algebra of complex functions f in L to the power oftheory infinity open\quad parenthesisassociated X comma t o mu\quad closingthe parenthesis p a i r \ actingquad on$ the ( N , \tau ) $ \quad has \quad analogous \quad machinery \quad to \quad measure theory , \quad i n c l u d i n g separable Hilb ert space L to the power of 2 open parenthesis X comma mu closing parenthesis by open parenthesis almost everywhere closing\noindent parenthesisRadon pointwise−− Nikodym multiplication theorems comma , the dominated integral and monotone convergence $ theorems ˆ{ 2 } , $ e t c . \ h f i l l Without the tau sub mu open parenthesis f closing parenthesis = integral sub X fd mu comma f in L to the power of infinity open parenthesis X comma mu\noindent closing parenthesiscondition of normality this machinery fails . i s a faithful normal trace period .. The noncommutative L to the power of p hyphen spaces are the usual L to the power of p hyphen spaces Acomma\quad i periodcommutative e period \quad example \quad o f the \quad theory \quad ( and \quad the \quad prototype ) \quad i s \quad a \quad r e g u l a r \quad Borel \quad measure spaceL to the $ power ( of X p open , parenthesis\mu ) L to . the $ power\quad of infinityFor the open von parenthesis Neumann X comma algebra mu closing of complex parenthesis functions comma tau $ sub f mu closing\ in Lparenthesis ˆ{\ infty equiv} L to(X, the power of p\mu open parenthesis) $ acting X comma on muthe closing parenthesis period A noncommutative example is the von Neumann algebra L open parenthesis H closing parenthesis of bounded linear operators \noindenton a separableseparable Hilb ert space Hilb H ert period space .. The $ canonical L ˆ{ 2 trace} (X, \mu ) $ by ( almost everywhere ) pointwise multiplication , the integral Line 1 infinity Line 2 Tr open parenthesis T closing parenthesis = sum angbracketleft h sub m comma Th sub m right angbracket comma T\ [ in\ Ltau open{\ parenthesismu } H( closing f parenthesis ) = comma\ int Line{ X 3} m =fd 1 \mu , f \ in L ˆ{\ infty } (X, \mu ) \where] open brace h sub m closing brace sub m = 1 to the power of infinity i s an orthonormal basis of H comma i s a faithful normal trace period .... The resulting L to the power of p hyphen spaces Line 1 L to the power of p open parenthesis H closing parenthesis : = open brace T in L open parenthesis H closing parenthesis bar Tr open\noindent parenthesisi s bar a T faithful bar to the normalpower of p trace closing . parenthesis\quad The less noncommutative infinity closing brace $open L ˆparenthesis{ p } − 1$ period spaces 2 closing are parenthesis the usual Line 2$ hline L ˆ{ p } − $ spaces , i . e . 1 sub The explanation here is for semifinite von Neumann algebras with separable pre hyphen dual period \ [2 L sub ˆ{ Thep } condition( L of ˆ normality{\ infty of} a linear(X, functional preempts\mu the), monotone\tau convergence{\mu theorem} ) period\equiv .. If NL is ˆ{ p } (X , a closed\mu *). hyphen subalgebra\ ] of L open parenthesis H closing parenthesis in the weak operator topology then a linear functional rho in N to the power of * .. is normal if rho open parenthesis S closing parenthesis = supremum sub alpha rho open parenthesis S sub alpha closing parenthesis for all increasing Anets noncommutative of positive operators example open brace is S the sub alphavon Neumann closing brace algebra subset N $ in L the strong ( H operator ) $ topology of bounded with linear operators onl period a separable u period b Hilb period ert S period space This $H condition . is$ equivalent\quad The to rho canonical open parenthesis trace sum sub i P sub i closing parenthesis = sum sub i rho open parenthesis P sub i closing parenthesis for all sets open brace P sub i closing brace of pairwise orthogonal projections \ [ \frombegin N{ opena l i g square n e d }\ bracketinfty 1 0\\ comma p period 67 closing square bracket comma i period e period analogous to additivity of a measure on a sigmaTr hyphen ( T algebra ) of = sets period\sum \ langle h { m } , Th { m }\rangle ,T \ in L(H) , \\ m = 1 \end{ a l i g n e d }\ ]

\noindent where $ \{ h { m }\} ˆ{\ infty } { m = 1 }$ i s an orthonormal basis of $H , $ i s a faithful normal trace . \ h f i l l The resulting $ L ˆ{ p } − $ spaces

\ [ \ begin { a l i g n e d } L ˆ{ p } ( H ) : = \{ T \ in L(H) \mid Tr ( \mid T \mid ˆ{ p } ) < \ infty \} ( 1 . 2 ) \\ \ r u l e {3em}{0.4 pt }\end{ a l i g n e d }\ ]

\ centerline { $ 1 { The }$ explanation here is for semifinite von Neumann algebras with separable pre − dual . }

\ hspace ∗{\ f i l l } $ 2 { The }$ condition of normality of a linear functional preempts the monotone convergence theorem . \quad I f $ N $ i s

\noindent a c l o s e d $ ∗ − $ subalgebra of $ L ( H ) $ in the weak operator topology then a linear functional $ \rho \ in N ˆ{ ∗ }$ \quad i s normal i f $ \rho ( S ) = \sup {\alpha }\rho (S {\alpha } ) $ for all increasing nets of positive operators $ \{ S {\alpha }\}\subset N $ in the strong operator topology with l . u . b $ . S . $ This condition is equivalent to $ \rho ( \sum { i } P { i } ) = \sum { i }\rho (P { i } )$ for all sets $ \{ P { i }\} $ of pairwise orthogonal projections from $N [ 1 0 , $ p . 67 ] , i . e . analogous to additivity of ameasureona $ \sigma − $ algebra of sets . Measure Theory in Noncommutative Spaces .... 3 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 3 3 are the Schatten endash von Neumann ideals of compact operators open square bracket 1 1 closing square bracket open parenthesis open square\ [ \ r ubracket l e {3em 1}{ 2 closing0.4 pt }\ square] bracket for finit e dimensional H closing parenthesis period A consequenceare the Schatten of the Radon – von endash Neumann Nikodym ideals theorem of in compact this context operators i s that for [ any1 1 normal ] ( [ 1 linear 2 ] for finit e functionaldimensional phi on a vonH). NeumannA consequence algebra N of subset the Radon L open parenthesis – Nikodym H closingtheorem parenthesis in this contextthere i s a i trace s that class for quoteleft density quoteright T\noindent in L to theare power the of 1 Schatten open parenthesis−− von H closing Neumann parenthesis ideals of compact operators [ 1 1 ] ( [ 1 2 ] for finit e dimensional $ Hany ) normal . $ linear functional φ on a von Neumann algebra N ⊂ L(H) there i s a trace class ‘ suchdensity that comma ’ T ∈ open L1(H square) such bracket that ,1 [ 0 1 comma 0 , p p. 55period ] , 55 closing square bracket comma AEquation: consequence open parenthesis of the Radon 1 period−− 3Nikodym closing parenthesis theorem .. in phi this open parenthesis context i a closing s that parenthesis for any = normal Tr open linear parenthesis aT closing parenthesisfunctional comma $ forall\phi a$ in N on period a von Neumann algebra $ N \subset L ( H )$ there i sa trace class ‘ density ’ $ TCombining\ in theL examples ˆ{ 1 } comma( H for every ) $ regularφ(a) =Borel Tr(aT measure), ∀a mu∈ N on. a space X there is a positive (1.3) such that , [ 10 ,p . 55 ] , trace classCombining operator Tthe sub examples mu on the ,for Hilb every ert space regular L toBorel the power measure of 2 openµ on parenthesis a space X commathere is mu apositive closing parenthesis such that 2 Equation:trace class open operatorparenthesisT 1µ periodon the 4 Hilb closing ert parenthesis space L ..(X, integral µ) such sub that X fd mu = Tr open parenthesis fT sub mu closing parenthesis comma\ begin f{ ina l L i g to n ∗} the power of infinity open parenthesis X comma mu closing parenthesis period \phi ( a )=Tr ( aT ) , \ f o r a l l a \ in N. \ tag ∗{$ ( 1 . 3 ) $} This formula exhibits the standard associationZ between integration and the eigenvalues of the \end{ a l i g n ∗} ∞ product fT sub mu comma a compact linearfdµ operator= Tr(fT periodµ), f ∈ L (X, µ). (1.4) What are the differences between open parenthesisX 1 period 1 closing parenthesis comma the integral in NCG comma and open parenthesis 1Combining periodThis 3 closing formula the parenthesis examples exhibits comma , the for standard the every integral regular association according Borel between measure integration $ \mu and$ the on eigenvaluesa space $X$ of the there is a positive trace class operator $ T {\mu }$ on the Hilb ert space $ L ˆ{ 2 } (X, \mu ) $ such that t o standardproduct noncommutativefTµ, a compact integration linear operator theory ? .. . Let us mention basic facts about open parenthesis 1 period 1 closing parenthesis period FirstlyWhat comma are since the angbracketleft differences D between right angbracket ( 1 . 1 ) to , the the powerintegral of minus in NCG n b , elongs and ( t 1 o .the 3 )ideal , the of integral compact linear operators \ beginEquation:according{ a l i g open n ∗} parenthesis t o standard 1 period noncommutative 5 closing parenthesis integration .. M sub theory1 comma? infinityLet open us mention parenthesis basic H closing facts parenthesis : = open brace T\ inint Labout to{ theX power (} 1 .fd of 1 infinity) . \mu open= parenthesis Tr ( H closing fT {\ parenthesismu } vextendsingle-vextendsingle-vextendsingle) , f \ in L ˆ{\ infty } supremum(X, n in N 1\ dividedmu ) by log. \ opentag ∗{ parenthesis$ ( 1Firstly 1 plus . n , closingsince 4 )h parenthesisD $}i−n b elongs sum from t o the k = ideal 1 to n of mu compact k open parenthesis linear operators T closing parenthesis less infinity closing brace \endopen{ a l parenthesis i g n ∗} where L to the power of infinity open parenthesis H closing parenthesis denotes the compact linear operators and mu n open parenthesis T closing parenthesis comma n in N comma are the eigenvaluesk=1 of \noindent This formula exhibits the standard∞ association1 X between integration and the eigenvalues of the bar T bar = open parenthesisM T1,∞ to(H the) := power{T ∈ Lof *(H T)| closingsup parenthesisµk to( theT ) < power∞} of 1 slash 2 arranged(1.5) in decreasing order closing log(1 + n) parenthesisproduct comma $ fT and{\ themu positive} , traces $ a Tr compact sub omega linear aren∈ linearN operator functionalsn . on M( where sub 1 commaL∞(H infinity) denotes comma the the compact formula linear operators and µn(T ), n ∈ , are the eigenvalues of What are the differences between ( 1 . 1 ) , the integral in NCGN , and ( 1 . 3 ) , the integral according Equation:| T |= (T open∗T )1 parenthesis/2 arranged 1 inperiod decreasing 6 closing order parenthesis ) , and .. the Tr subpositive omega traces open parenthesisTr are linear S angbracketleft functionals D right angbracket to the t o standard noncommutative integration theory $ ? $ \quad Letω us mention basic facts about ( 1 . 1 ) . power ofon minusM1,∞ n, closingthe formula parenthesis comma S in L open parenthesis H closing parenthesis defines a linear functional on L open parenthesis H closing parenthesis period .. In particular comma .. by restriction .. open parenthesis 1 \ centerline { Firstly , since $ \ langle D \rangle ˆ{ − n }$ b elongs t o the ideal of compact linear operators } period 6 closing parenthesis .. i s a linear functional −n on the * hyphen algebra A comma on the closureTrω(S AhD ofi A) in,S the∈ uniform L(H) operator t opology open parenthesis (1. a6) separable Case 1 * Case 2 \ begin { a l i g n ∗} hyphendefines a linear functional on L(H). In particular , by restriction ( 1 . 6 ) i s a linear M { 1 , \ infty } ( H ) : = \{ T \ in L ˆ{\ infty } (H) \arrowvert \sup { n subalgebrafunctional of L open on the parenthesis∗− algebra H closingA, on parenthesis the closure since AA iof s countablyA in the generated uniform closing operator parenthesis t opology comma ( and on the closure A to \ in N }\ f r a c { 1 }{\ log ( 1 + n ) }\sum ˆ{ k = 1 } { n }\mu k ( T ) < \ infty the power of prime prime of∗ A in the weak open parenthesis or \}\strongtaga separable∗{ closing$ ( parenthesis 1C .subalgebra operator 5 ) t $ opology} of L(H open) since parenthesisA i s countably the bicommutant generated comma ) a , vonand Neumann on the closure subalgebra of L open parenthesis − H\end closing{ a l i parenthesis g n ∗} open square bracket 1 3 comma S 2 period 4 closing square bracket closing parenthesis period A00 A Line 1 Secondlyof in comma the weak if 0 less ( or S in strong L open )parenthesis operator H t closing opology parenthesis ( the bicommutant comma Line 2 Tr , a sub von omega Neumann open parenthesis S angbracketleft L(H) [13, D\noindent rightsubalgebra angbracket( where to of the power$ L ˆ of{\ minus§infty2 . 4 n closing]} ) . ( parenthesis H ) =$ Tr denotes sub omega the parenleftbig compact square linear root ofoperators S angbracketleft and D $ right\mu angbracketn ( T ) , n \ in N , $ are the eigenvalues of to the power of minus n square root S parenrightbig greaterSecondly equal, if0 0< period S ∈ L(H), Hence open parenthesis 1 period 6 closing parenthesis defines√ a positive√ linear functional on A comma A or A to the power of prime prime \noindent $ \mid T \mid = (− Tn ˆ{ ∗ } T− )n ˆS){≥01. / 2 }$ arranged in decreasing order ) , and the positive traces period Trω(ShDi ) = Trω( ShDi $ Tr {\omega }$ are linear functionals ThirdlyHence comma ( 1 . 6 ) defines a positive linear functional on A,A or A00. Thirdly , onbar $ Tr M sub{ omega1 open , parenthesis\ infty } S angbracketleft, $ the formula D right angbracket to the power of minus n closing parenthesis bar less or equal C bar −n S bar comma forall S in L open parenthesis| Tr Hω( closingShDi parenthesis) |≤ C k S k, ∀S ∈ L(H) \ beginfor a{ constanta l i g n ∗} C greater 0 period Hence the same formula open parenthesis 1 period 6 closing parenthesis i s the unique uniformly continuous for a constant C > 0. Hence the same formula ( 1 . 6 ) i s the unique uniformly continuous extensionTr {\omega } (S \ langle D \rangle ˆ{ − n } ),S \ in L(H) \ tag ∗{$ ( 1 . 6extension ) $} t o A and A00 of the positive linear functional ( 1 . 1 ) on A. Denote t othis .. A and linear A to the functional power of by prime prime .. of the positive linear functional .. open parenthesis 1 period 1 closing parenthesis .. on A period\end{ ..a l Denote i g n ∗} this .. linear functional by 0 less Capital Phi sub omega in open parenthesis A to the power of prime prime Case 1 * Case 2 period \noindent defines a linear functional on $L (∗ H ) . $ \quad In particular , \quad by restriction \quad ( 1 . 6 ) \quad i s a linear functional Fourthly comma Capital Phi sub omega is a trace on A to the00 power of prime prime open square bracket 1 4 comma Theorem 1 period 3 0 < Φω ∈ (A ) closingon the square $ bracket∗ − comma$ algebra open square $A bracket , $ 2 comma on the p period closure. 3 1 3 $A$ closing square of $A$bracket comma in the open uniform square bracket operator 1 5 comma t opology p ( a separable $\ l e f t .C\ begin { a l i g n e d } & ∗ \\ period 280 closing square bracket comma00 open square bracket 1 6 comma p period 47 closing square bracket period Fourthly& − \,endΦω is{ a a l i traceg n e d }\ onrightA [1. $ 4, Theorem 1 . 3 ] , [ 2 , p . 3 1 3 ] , [ 1 5 , p . 280 ] , [ 1 6 , p . 47 ] . The fundamentalThe fundamental difference b difference etween the bfunctionals etween open the functionalsparenthesis 1 period ( 1 . 3 3 closing ) and parenthesis ( 1 . 6 and ) i open s that parenthesis 1 period 6 closing parenthesissubalgebra i s that of Tr $L sub omega ( open H parenthesis ) $ since S angbracketleft $A$ i D s right countably angbracket generated to the power ) of , minus and non closing the parenthesis closure = $Aˆ 0 {\prime Tr (ShDi−n) = 0 for all positive finit e rank operators S( see Section 2 b elow ) . In \primefor allω} positive$ of finit $A$ e rank operators in theweak S open ( parenthesis or see Section 2 b elow closing parenthesis period .. In comparison Tr open parenthesis comparison Tr (ST ) > 0 for STstrong closing parenthesis) operator greater t opology 0 for ( the bicommutant , a von Neumann subalgebra of $ L ( H ) [ 1 3 ,a $ trace\S class2 . 4 operator ] ) . 0 < T ∈ L1(H) with trivial kernel and every finit e rank operator S > 0. a traceThis class property operator of 0 vanishingless T in L to on the the power finit of e 1 rank open operatorsparenthesis ,H closing called parenthesis singularity with , trivial i s essential kernel and for every finit e rank operator S greater 0 period \ [ \ beginthe{ fundamentala l i g n e d } Secondly theorem of , integration if 0 theory< S in NCG\ in ( ConnesL(H), ’ trace theorem\\ [ 1 7 ] , and ThisTheorem property of 4 vanishing. 1 in this on the t ext finit ) e . rank operators The property comma of .. calledsingularity singularity , however comma i, s implies essential that for the the Trfundamental{\omega theorem} (S of integration\ langle theory in NCGD open\rangle parenthesisˆ{ − Connesn } quoteright) = trace Tr theorem{\omega open square} ( bracket\ sqrt 1{ 7 closingS }\ squarelangle functional ( 1 . 6 ) i s either trivial or else i s a non - normal functional on L(H). To see Dbracket\rangle comma andˆ{ −Theoremn 4\ periodsqrt 1} S) \geq 0 . \end{ a l i g n e d }\ ] this , let S b e a sequence of positive in this t ext closingn parenthesis period .. The property of singularity comma however comma implies that the functional open parenthesis 1 period 6 closing parenthesis i s either \noindenttrivial or elseHence i s a non( 1 hyphen . 6 ) normal defines functional a positive on L open linear parenthesis functional H closing parenthesis on $A period , .. A$ To see or this $Aˆ comma{\ letprime S sub n b e\prime a } sequence. $ of positive Thirdly ,

\ [ \mid Tr {\omega } (S \ langle D \rangle ˆ{ − n } ) \mid \ leq C \ parallel S \ parallel , \ f o r a l l S \ in L(H) \ ]

\noindent for a constant $ C > 0 . $ Hence the same formula ( 1 . 6 ) i s the unique uniformly continuous extension t o \quad $ A $ and $ A ˆ{\prime \prime }$ \quad of the positive linear functional \quad ( 1 . 1 ) \quad on $ A . $ \quad Denote t h i s \quad linear functional by

\ begin { a l i g n ∗} \ l e f t . 0 < \Phi {\omega }\ in ( A ˆ{\prime \prime } )\ begin { a l i g n e d } & ∗ \\ &. \end{ a l i g n e d }\ right . \end{ a l i g n ∗}

\ centerline { Fourthly $ , \Phi {\omega }$ is a trace on $Aˆ{\prime \prime } [ 1 4 , $ Theorem1.3] ,[2,p.313] ,[15,p.280] ,[16,p.47] . }

The fundamental difference b etween the functionals ( 1 . 3 ) and ( 1 . 6 ) i s that $ Tr {\omega } (S \ langle D \rangle ˆ{ − n } ) = 0 $ for all positive finit e rank operators $ S ( $ see Section 2 b elow ) . \quad In comparison Tr $ ( ST ) > 0 $ f o r

\noindent a trace class operator $ 0 < T \ in L ˆ{ 1 } ( H ) $ with trivial kernel and every finit e rank operator $ S > 0 . $ This property of vanishing on the finit e rank operators , \quad called singularity , i s essential for the fundamental theorem of integration theory in NCG ( Connes ’ trace theorem [ 1 7 ] , and Theorem 4 . 1 in this t ext ) . \quad The property of singularity , however , implies that the functional ( 1 . 6 ) i s either trivial or else i s a non − normal functional on $L ( H ) . $ \quad To see this , let $S { n }$ b e a sequence of positive 4 S . Lord and F . Sukochev

ﬁnit e rank operators converging upward strongly t o the identity 1 of L(H) and suppose ( 1 . 6 ) i s a normal linear functional on L(H), then

−n −n 0 = sup Trω(SnhDi ) = Trω(hDi ). n

−n Hence either Trω(hDi ) = 0 or else ( 1 . 6 ) is non - normal . 00 ∗ The property of singularity does not exclude the trace 0 < Φω ∈ (A ) ( the restriction of ( 1 . 6 ) t o the von Neumann algebra A00 ⊂ L(H)) from b eing a normal linear functional on A00. 00 There are spectral triples where Φω i s not a normal linear functional on A ( evidently when 00 00 A contains a finit e rank operator ) , and where Φω i s a normal linear functional on A ( see Theorem 4 . 3 ) . Currently we lack , in general , a characterisation of the relationship between D and A that 00 ∗ implies Φω ∈ (A ) i s a normal linear functional . All we know of is a sufficient condition , due t o the authors , see Theorem 6 . 2 in Section 6 . 2 . It has been suggested that A00 containing no finit e rank operators i s a necessary and sufficient condition for normality of Φω. This conj ecture i s open . Non - normality of the linear functional on L(H) defined by ( 1 . 6 ) can have advantages 00 ∗ , as indicated by Connes [ 2 , p . 326 ] . However , normality of the trace 0 < Φω ∈ (A ) i s coincident with our ability t o apply the standard t ools of integration theory t o the formula ( 1 . 1 ) . 00 ∗ 00 In general , the value of the trace 0 < Φω ∈ (A ) on the proj ections of A ,

∗ 2 00 Φω(E),E = E = E ∈ A , (1.7) i s only analogous to an element of the ba space of finit e and finit ely additive functionals on a Borel σ− algebra [ 1 8 , IV . 2 ] . The integral ( 1 . 1 ) i s constructed by limits of finit e linear combina - t ions of the values in ( 1 . 7 ) if and only if the trace Φω i s normal . In this review , we discuss the known ways t o measure the ‘ log divergence of the trace ’ , including residue formulas , heat kernels and Dixmier traces . Each of these ways may result in different functionals in ( 1 . 1 ) . It i s desirable t o know when all these ways agree , and that ( 1 . 1 ) defines a unique linear functional on A. There are a variety of difficulties behind this problem . For example , if V denotes a set of linear functionals on M1,∞ ‘ measuring the log divergence of the trace ’ , the set KV := {T ∈ M1,∞ | f (T ) = const ∀ f ∈ V} −n i s not an ideal [ 1 9 , Remark 3 . 6 ] . Therefore , hDi ∈ KV will not imply −n ahDi ∈ KV for all a ∈ A in general . Specifying hDi−1 alone is not sufficient for a unique value from the formula ( 1 . 1 ) . Additionally , if one picks smaller sets , V, or larger set 0 s , V , of linear functionals , it may be that KV 6= KV 0( discussed in Section 3 ) . Allied t −n o this problem i s the fact that , if a ∈ A i s a selfadj oint operator and ahDi ∈ KV it i s −n −n not necessarily true that the decompositions a+hDi , a−hDi ∈ KV , where a+ and a− are the positive and negative parts of a. Currently , in general , there i s no characterisation , given hDi−n, −n what relationship with a ∗− algebra A is necessary so that Φω(a) = Trω(ahDi ) specifies a unique trace ( meaning the value of Φω(a) i s independent of the linear functional Trω ∈ V). This i s an open prob - lem . Checking uniqueness i s a case −n by case basis which requires identifying Trω(ahDi ) with a known functional , e . g . the Lebesgue integral for a compact Riemannian manifold [ 1 7 , 1 6 ] , or the Hausdorff measure of Fuchsian circles and self - similar fractals [ 2 , IV . 3 ] , [ 2 0 ] , or estimating the singular − n values of the product ahDi . In the review we shall provide some sufficient conditions for normality of the trace 0 < 00 ∗ Φω ∈ (A ) . We review the demonstration of normality for the case of a compact Riemannian manifold ( in Section 4 ) and the noncommutative t orus ( in Section 6 ) . 4 .... S period Lord and F period Sukochev \noindent 4 \ h f i l l S . Lord and F . Sukochev hline Finally , we discuss the role of other positive singular traces on M1,∞, b esides Dixmier finittraces e rank ,operators in the converging formula ( upward 1 . 1 strongly) . We t o the show identity that 1 , of even L open in parenthesis Connes ’ H trace closing theorem parenthesis [ 1 and 7 , suppose open parenthesis 1\ [ period\ rTheorem u l 6 e closing{3em}{ 1parenthesis0.4 ] , pt there}\ i] exist s a normal linear functional on L open parenthesis H closing parenthesis comma then Line 1 0 = supremum Tr sub omega open parenthesis S sub n angbracketleft D right angbracket to the power of minus n closing parenthesis =\noindent Tr sub omegafinit open e parenthesis rank operators angbracketleft converging D right angbracket upward to strongly the powerof tminus o the n identityclosing parenthesis 1 of period $ L Line ( 2n H ) $ and suppose ( 1 . 6 ) i s aHence normal either linear Tr sub functionalomega open parenthesis on $L angbracketleft ( H D ) right , angbracket $ then to the power of minus n closing parenthesis = 0 or else open parenthesis 1 period 6 closing parenthesis is non hyphen normal period \ [ \Thebegin property{ a l i g n of e dsingularity} 0 = does\sup not excludeTr the{\ traceomega 0 less} Capital(S Phi{ subn }\ omegalangle in open parenthesisD \rangle A to theˆ{ − powern of} prime) prime = closingTr {\ parenthesisomega } to( the power\ langle of * openD parenthesis\rangle theˆ restriction{ − n of} open). parenthesis\\ 1 period 6 closing parenthesis nt o\ theend von{ a l Neumanni g n e d }\ ] algebra A to the power of prime prime subset L open parenthesis H closing parenthesis closing parenthesis from b eing a normal linear functional on A to the power of prime prime period .. There are spectral triples where Capital Phi sub omega i s not a normal linear functional on A to the power of prime prime open parenthesis evidently\noindent whenHence A to the either power of $ prime Tr prime{\omega contains} ( \ langle D \rangle ˆ{ − n } ) = 0$ orelse(1.6)isnon − normal . a finit e rank operator closing parenthesis comma and where Capital Phi sub omega i s a normal linear functional on A to the power of primeThe property prime open of parenthesis singularity see Theorem does .. not 4 period exclude 3 closing the parenthesis trace period $ 0 < \Phi {\omega }\ in ( A ˆ{\prime \primeCurrently} we) ˆ lack{ ∗ comma } ( in $ general the restrictioncomma a characterisation of ( 1 . of 6 the ) relationship between D and A that timplies o the Capital von Neumann Phi sub omega algebra in open $ parenthesis A ˆ{\prime A to the\ powerprime of}\ primesubset prime closingL parenthesis ( H to ) the ) power $from of * i s b a eingnormal a linear normal linear functional on functional$ A ˆ{\ periodprime .. All\prime we know} of is. a $ sufficient\quad conditionThere comma due t o arethe spectral authors comma triples see Theorem where ..$ 6 period\Phi 2{\ in Sectionomega 6} period$ i 2s period not a .. normal It has been linear suggested functional that A to onthe power $ A ofˆ{\ primeprime prime containing\prime } no( finit $ e evidently when $ A ˆ{\prime \prime }$ c o n t a i n s arank finit operators e rank i s a operator necessary and ) , sufficient and where condition $ \ forPhi normality{\omega of Capital}$ Phi is sub a omega normal period linear .... This functional conj ecture i on s $ A ˆ{\prime \primeopen period} ( Non $ hyphensee Theorem normality\quad of the linear4 . 3 functional ) . on L open parenthesis H closing parenthesis defined by open parenthesis 1 period 6 closing parenthesis can have advantages comma as Currentlyindicated by we Connes lack open , in square general bracket ,a 2 comma characterisation p period 326 closing of the square relationship bracket period between.. However comma $ D $ normality and $ of A the $ trace that 0 lessi m Capital p l i e s Phi $ sub\Phi omega{\ inomega open parenthesis}\ in A( to the A power ˆ{\prime of prime prime\prime closing} parenthesis) ˆ{ ∗ }$ to the i power s a normal of * i s coincident linear functional . \quad All we know of is a sufficient condition , due t o thewith authors our ability , t see o apply Theorem the standard\quad t ools6 . of 2 integration in Section theory 6 t . o 2 the . formula\quad openIt hasparenthesis been 1 suggested period 1 closing that parenthesis $ A ˆ{\ periodprime \primeIn general}$ comma containing the value no of the finit trace e 0 less Capital Phi sub omega in open parenthesis A to the power of prime prime closing parenthesis to the power of * on the proj ections of A to the power of prime prime comma \noindentEquation:rank open parenthesis operators 1 period i s a 7 closing necessary parenthesis and .. sufficient Capital Phi sub condition omega open for parenthesis normality E closing of parenthesis $ \Phi {\ commaomega E to} the power. $ of\ h * f = i l E l =This E to conjthe power ecture of 2 in i A s to the power of prime prime comma i s only analogous to an element of the ba space of finit e and finit ely additive functionals on \noindenta Borel sigmaopen hyphen . Non algebra− normality open square of bracket the 1 linear .. 8 comma functional IV period 2 on closing $ L square ( bracket H period ) $ .. defined The integral by open ( 1 parenthesis . 6 ) can have advantages , as 1indicated period 1 closing by parenthesis Connes [ i 2s constructed , p . 326 by ] limits . \quad of finitHowever e linear combina , normality hyphen of the trace $ 0 < \Phi {\omega } \ int ions( of the A values ˆ{\prime in open parenthesis\prime } 1 period) ˆ{ 7 ∗ closing }$ parenthesis i s coincident if and only if the trace Capital Phi sub omega i s normal period withIn this our .. review ability comma t o .. apply we .. discuss the standard .. the .. known t ools ways of .. t integration o .. measure the theory .. quoteleft t o log the .. divergence formula .. ( of 1 the . 1 trace ) . quoteright comma \ centerlineincluding residue{ In general formulas comma , the heat value kernels of and the Dixmier trace traces $ 0 period< .. Each\Phi of these{\omega ways may}\ resultin in ( A ˆ{\prime \prime } ) ˆdifferent{ ∗ }$ functionals on the in proj open ections parenthesis of 1 period $Aˆ 1{\ closingprime parenthesis\prime period} .., It $i s desirable} t o know when all these ways agree comma and that open parenthesis 1 period 1 closing parenthesis \ begindefines{ a a l i unique g n ∗} linear functional on A period .. There are a variety of difficulties behind this problem period \PhiFor example{\omega comma} if( V denotes E a) set of , linear E ˆ functionals{ ∗ } = on M E sub = 1 comma E ˆ{ infinity2 }\ quoteleftin A measuring ˆ{\prime the log divergence\prime } of , \ tag ∗{$ ( 1the . trace 7 quoteright ) $} comma the set \endK{ suba l i V g n :∗} = open brace T in M sub 1 comma infinity bar f open parenthesis T closing parenthesis = const forall f in V closing brace i s not .... an ideal .... open square bracket 1 9 comma .... Remark 3 period 6 closing square bracket period .... Therefore comma angbracketleft\noindent i D s right only angbracket analogous to the to power anof element minus n ofin K the sub baV .... space will not of .... finit imply ae angbracketleft and finit elyD right additive angbracket functionals to the power on ofa minus Borel n in $K sub\sigma V for − $ algebra [ 1 \quad 8 , IV . 2 ] . \quad The integral ( 1 . 1 ) i s constructed by limits of finit e linear combina − tall ions a in Aof in the general values period in .. Specifying ( 1 . 7 .. ) angbracketleft if and only D right if the angbracket trace to $ the\Phi power{\ of minusomega 1 ..}$ alone i is s .. normal not .. sufficient . .. for a unique value from the \ hspaceformula∗{\ openf i l parenthesis l } In t h i s 1 period\quad 1review closing parenthesis , \quad periodwe \quad .. Additionallyd i s c u s s comma\quad ifthe one picks\quad smallerknown sets ways comma\quad V commat o or\quad largermeasure the \quad ‘ l o g \quad divergence \quad of the trace ’ , set s comma V to the power of prime comma of linear functionals comma \noindentit may be thatincluding K sub V residue negationslash-equal formulas K sub , heat V prime kernels open parenthesis and Dixmier discussed traces in Section . \quad 3 closingEach parenthesis of these period ways .. Allied may t result o in thisdifferent problem i s functionals the fact that comma in ( if 1 a .in 1 A ) . \quad It i s desirable t o know when all these ways agree , and that ( 1 . 1 ) definesi s a selfadj a unique oint operator linear and a functional angbracketleft on D right $ A angbracket . $ \ toquad the powerThere of are minus a n variety in K sub Vof it difficulties i s not necessarily behind true that this the problem . decompositions \noindenta sub plusFor angbracketleft example D , right if angbracket $ V $ denotes to the power a set of minus of linear n comma functionals a sub minus angbracketleft on $M { D1 right angbracket , \ infty to the}$ power ‘ measuring the log divergence of ofthe minus trace n in K ’ sub , Vthe comma set where a sub plus and a sub minus are the positive and negative parts of a period Currently comma .. in .. general comma .. there .. i s .. no .. characterisation comma .. given .. angbracketleft D right angbracket to the power\ centerline of minus{ n$ comma K { ..V what} ..: relationship = \{ .. withT \ in M { 1 , \ infty }\mid $ f $( T ) =$ const $ \af * o hyphen r a l l $ algebra f $ A\ isin .. necessaryV \} so that$ ..} Capital Phi sub omega open parenthesis a closing parenthesis = Tr sub omega open parenthesis a angbracketleft D right angbracket to the power of minus n closing parenthesis .. specifies .. a unique trace .. open parenthesis meaning \noindentthe value ofi Capital s not Phi\ h f sub i l l omegaan i d open e a l parenthesis\ h f i l l [ a 1 closing 9 , \ parenthesish f i l l Remark .. i s independent 3 . 6 ] . of\ theh f i linearl l Therefore functional Tr $ sub , omega\ langle in V Dclosing\rangle parenthesisˆ{ period − n .. This}\ iin s an openK { probV } hyphen$ \ h f i l l w i l l not \ h f i l l imply $ a \ langle D \rangle ˆ{ − n }\lem periodin ..K Checking{ V }$ uniqueness f o r i s a case by case basis which requires identifying Tr sub omega open parenthesis a angbracketleft D right angbracket to the power of minus n closing parenthesis with \noindenta known functionala l l $ commaa \ in e periodA$ g period in general the Lebesgue . \quad integralS forp e c a i fcompact y i n g \ Riemannianquad $ \ langle manifold openD square\rangle bracketˆ{ 1 −7 comma1 } ..$ 1\quad 6 closingalone square i s bracket\quad commanot \ orquad s u f f i c i e n t \quad for a unique value from the formulathe Hausdorff ( 1 measure . 1 ) . of\ Fuchsianquad Additionally circles and self hyphen , if one similar picks fractals smaller open square sets bracket $ , 2 comma V IV , $ period or 3 larger closing square set s bracket $ , Vcomma ˆ{\ ..prime open square} , bracket $ of 2 linear 0 closing functionals square bracket comma , or estimating itthe may singular be that values of $K the product{ V }\ a angbracketleftnot= K D{ CaseV }\ 1 minusprime n Case( 2 $ period discussed in Section 3 ) . \quad Allied t o this problem i s the fact that , if $ aIn the\ in reviewA we $ shall provide some sufficient .. conditions for normality of the trace .. 0 less iCapital s a selfadj Phi sub omega ointin operator open parenthesis and $ A a to the\ langle power of primeD prime\rangle closingˆ{ parenthesis − n }\ to thein powerK of{ *V period}$ it .. We i s review not the necessarily true that the decompositions demonstration$ a { + of}\ normalitylangle for theD case\rangle of a compactˆ{ − Riemanniann } , a { − } \ langle D \rangle ˆ{ − n }\ in K { V } , $manifold where open $ parenthesis a { + } in$ Section and 4 $ closing a { parenthesis − }$ are and the the noncommutative positive and t orus negative open parenthesis parts of in Section $ a .. 6. closing $ parenthesis period CurrentlyFinally comma , \quad we discussin \ thequad roleg of e n other e r a l positive , \quad singularthe re traces\quad on Mi sub s \ 1quad commano infinity\quad commacharacterisation b esides Dixmier traces , \quad commagiven \quad $ \inlangle the formulaD open\rangle parenthesisˆ{ 1 − periodn 1} closing, $ parenthesis\quad what period\ ..quad We showrelationship that comma even\quad in Conneswith quoteright trace theorem open squarea $ bracket∗ − 1$ 7 comma algebra Theorem $A$ 1 closing is square\quad bracketnecessary comma so there that exist\quad $ \Phi {\omega } ( a ) = Tr {\omega } ( a \ langle D \rangle ˆ{ − n } ) $ \quad s p e c i f i e s \quad a unique trace \quad ( meaning the value of $ \Phi {\omega } ( a ) $ \quad i s independent of the linear functional $ Tr {\omega } \ in V ) . $ \quad This i s an open prob − lem . \quad Checking uniqueness i s a case by case basis which requires identifying $ Tr {\omega } ( a \ langle D \rangle ˆ{ − n } ) $ with a known functional , e . g . the Lebesgue integral for a compact Riemannian manifold [ 1 7 , \quad 1 6 ] , or the Hausdorff measure of Fuchsian circles and self − similar fractals [ 2 , IV . 3 ] , \quad [ 2 0 ] , or estimating the singular values of the product $\ l e f t . a \ langle D \rangle\ begin { a l i g n e d } & − n \\ &. \end{ a l i g n e d }\ right . $

In the review we shall provide some sufficient \quad conditions for normality of the trace \quad $ 0 < $ $ \Phi {\omega }\ in ( A ˆ{\prime \prime } ) ˆ{ ∗ } . $ \quad We review the demonstration of normality for the case of a compact Riemannian manifold ( in Section 4 ) and the noncommutative t orus ( in Section \quad 6 ) .

Finally , we discuss the role of other positive singular traces on $ M { 1 , \ infty } , $ b esides Dixmier traces , in the formula ( 1 . 1 ) . \quad We show that , even in Connes ’ trace theorem [ 1 7 , Theorem 1 ] , there exist Measure Theory in Noncommutative Spaces .... 5 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 5 5 other positive singular traces rho on the compact operators comma which are not Dixmier traces comma such \ [ that\ r u rhol e {3em open}{ parenthesis0.4 pt }\ P] closing parenthesis = Res open parenthesis P closing parenthesis open parenthesis up to a positive constant closing parenthesis for a classical pseudo hyphen differential P of order minus n other positive singular traces ρ on the compact operators , which are not Dixmier traces , such on a n hyphen dimensional compact Riemannian manifold comma where Res is the Wodzicki residue period that ρ(P ) = Res (P )( up to a positive constant ) for a classical pseudo - differential P of order \noindentNotation other positive singular traces $ \rho $ on the compact operators , which are not Dixmier traces , such that−n $on\rho a n− dimensional( P ) compact =$ Res Riemannian $ ( P manifold ) ( , $where upto Res a is positivethe Wodzicki constant residue ) . for a classical pseudo − differential ThroughoutNotation .. H .. denotes .. a .. separable .. complex .. Hilbert .. space comma L open parenthesis H closing parenthesis open parenthesis resp$P$ period of Row order 1 infinity $ − Rown 2 equiv $ . to the power of infinity open parenthesis H closing parenthesis comma Throughout H denotes a separable complex Hilbert space , L(H)( resp onRow a 1 $ p nRow− 2 equiv$ dimensional . to the power of compact p open parenthesis Riemannian H closing manifold parenthesis , where comma Res p in is open the square Wodzicki bracket 1residue comma infinity . closing ∞ p parenthesis.L closingL∞ parenthesis(H),L L ..p( denotesH), p ∈ the[1, linear∞)) boundeddenotes open the parenthesis linear bounded resp period ( resp compact . compact comma , Schatten Schatten endash von Neumann ≡ ≡ \noindentopen parenthesisNotation see .. open parenthesis 1 period 2 closing parenthesis closing parenthesis closing parenthesis .. operators .. on .. H period .. Throughout– von Neumann .. the t ext .. ( M see sub 1 ( comma 1 . 2 infinity ) ) ) equiv operators M sub 1 comma on infinityH. openThroughout parenthesis H the closing t ext parenthesis .. denotes .. the ideal\noindent ofM1,∞ Throughout≡ M1,∞(H\)quaddenotes$ H $ the\quad idealdenotes of \quad a \quad s e p a r a b l e \quad complex \quad H i l b e r t \quad space ∞ p $,compactcompact L operators ( operators H defined ) defined in($resp$.L open parenthesis in ( 1 . 5 1 ) period or\ begin ( 2 5 .closing{ 2array ) b parenthesis elow}{ c}\ .infty or Denote open\\\ parenthesis by equiv` 2( periodresp\end{.c, 2array closing c0, ` })Lˆ parenthesis{\ infty b elow} (H period ..) Denote ,the $ by bounded l to the power ( resp of infinity . convergent open parenthesis , convergent resp period to zero c comma, p− summable c sub 0 comma ) sequences l to the power . Similarly of p closing parenthesis the bounded ∞ p $Lopen\,begin for parenthesis a{ array regular resp}{ Borelc} periodp \\\ mea convergent -equiv sure comma space\end{ convergentarray(X, µ)},LLˆ{ top( zeroX,} µ)( comma(resp H p hyphen.L )(X, , summable µ) p,Cb\(Xin closing),C[0 parenthesis(X)) 1 , sequences\ infty period) .. Similarly) $ denotes\quad commadenotes the for aessentially regular the Borel linear bounded mea bounded hyphen ( resp (.p resp− integrable . compact , continuous , Schatten and−− boundedvon Neumann , continuous (sure seeand space\quad vanishing open( parenthesis1 at. 2 infinity ) ) X ) comma)\ (quad equivalence muo p closing e r a t classeso rparenthesis s \quad of ) functionson comma\quad L on to$ theX. H powerWhen . $of infinityX\quadis a Riemannian openThroughout parenthesis\quad X commathe mu t closingext \quad parenthesis$ M {manifold1 open , parenthesis and\ inftydx i resps}\ the period volumeequiv L to form theM power{ on1X, of p , open\ parenthesisinfty } X( comma H mu ) closing $ \quad parenthesisdenotes comma\quad C subthe b open i d parenthesis e a l o f X closing parenthesis comma C sub 0 open parenthesis X closing parenthesis closing parenthesis .. denotes the essentially bounded \noindentopen parenthesiscompact resp operators period p hyphen defined integrable in ( comma 1 . 5 continuous ) or ( 2 and . 2 bounded ) b elow comma . \ continuousquad Denote and vanishing by $ \ ate l infinityl ˆ{\ closinginfty } wewriteLp(X) ≡ Lp(X, dx). parenthesis($ resp open $. parenthesis c ,equivalence c { 0 } , \ e l l ˆ{ p } ) $ the bounded (classes resp of . closing convergent parenthesis , convergent functions on X to period zero .. When $ , X is p a Riemannian− $ summable manifold ) and sequences dx i s the volume . \quad formSimilarly on X comma , for a regular Borel mea − surespace2 Dixmier $( X , traces\mu ) , L ˆ{\ infty } (X, \mu ) ($ resp $. Lˆ{ p } weFor write a L separable to the power complex of p open Hilbert parenthesis space X closingH, denote parenthesis by µn( equivT ), L n to∈ the power, the of singular p open parenthesisvalues X comma dx closing parenthesis(X, period\mu ),C { b } (X),C { 0 } ( XN ) ) $ \quad denotes the essentially bounded ( respof $ . p − $ integrable , continuous and bounded , continuous and vanishing at infinity ) ( equivalence 2 .. Dixmier .. traces ∗ 1/2 classesa compact of ) functions operator T ( onthe $X singular . values $ \quad are theWhen eigenvalues $ X $ ofis the a Riemannian operator | T manifold| = (T T ) and $ dx $ i s the volume form on Forarranged a .... separable with .... multiplicity complex Hilbert in decreasing .... space .... order H comma [ 1 1 , ....§1 ]denote ) . by Define mu n theopen logarithmic parenthesis T average closing parenthesis comma n in N comma$ X .... , $ the .... singular values .... of a compact operator T open parenthesis the singular values are the eigenvalues of the operator bar T bar = open parenthesis T to the power \ begin { a l i g n ∗} n=1 of * T closing parenthesis to the power of 1 slash∞ 2 1 X we w rite L ˆ{ p } α((X){µn(T )} \)equiv := { L ˆ{ pµn}(T )(}∞ X,T ∈ , L∞ dx. ) . (2.1) arranged with multiplicity in decreasingn order= 1 open squarelog(1 + bracketk) 1 1 commak=1 S 1 closing square bracket closing parenthesis period .. Define the\end logarithmic{ a l i g n ∗} average k Equation:Then open parenthesis 2 period 1 closing parenthesis .. alpha open parenthesis open brace mu n open parenthesis T closing parenthesis Row\noindent 1 infinity2 Row\quad 2 n =Dixmier 1 closing\ brace.quad :t = r a open c e s brace 1 divided by log open parenthesis 1 plus k closing parenthesis sum from n = 1 to k mu n open parenthesis T closing parenthesis closing brace sub k = 1 to the power of infinity comma T in L to the power of infinity period \noindentThen For a \ h f i l l s e p a r a b l e \ h f i l l complex Hilbert \ h f i l∞ l space \ h f i l l $ H , $ \ h f i l l denote by $ \mu n ( TM )1,∞ ,:= M n1,∞(H\ in) = {TN|k T k ,1, $∞:=\ suph f iα l( l{µnthe(T )}\ h f i l l )ksingular < ∞} values \(2h.2) f i l l o f Equation: open parenthesis 2 period 2 closing parenthesis .. M subk 1 comma infinityn = 1 : = M sub 1 comma infinity open parenthesis H closing parenthesis = open brace T bar bar T bar sub 1 comma infinity : = supremum k alpha open parenthesis open brace mu n open parenthesis T closing\noindentdefines parenthesisa a compact Banach Row 1 infinity ideal operator of Row compact 2 $n = T 1 operators closing ( $ brace. the whose singular k less sequence infinity values closing of partial arebrace the sums eigenvalues of singular values of the operator $ \mid T defines\midi s ofa Banach order= (ideal logarithm T of ˆcompact{ ∗ . } operators WeT refer ) ˆwhose{ to1 the sequence / recent 2 of} paper partial$ ofsums Pietsch of singular [ 2 values 1 ] , discussing the arrangedi s oforigin order with of logarithm this multiplicity object period ..in We mathematics in refer decreasing to the recent . In orderpaper [ 2 2 of ] [ Pietsch, 1J .1 Dixmier , open\S square1 constructed] ) bracket . \quad 2 .. a 1Define non closing - normal square the logarithmic bracket comma average discussing the originsemifinite of this trace ( a Dixmier trace ) on L(H) using the weight \ beginobject{ a in l i mathematics g n ∗} period In open square bracket 2 2 closing square bracket comma J period Dixmier constructed a non hyphen normal \alpha ( \{\mu n ( T ) \}\ begin { array }{ c}\ infty \\ n = 1 \end{ array }) : = \{ semifinite trace open parenthesis a Dixmier n=1 \ f r a c { 1 }{\ log ( 1 + k ) }\1sum Xˆ{ n = 1 } { k }\mu n ( T ) \} ˆ{\ infty } { k trace closing parenthesis on L openTr parenthesis(T ) := ω({ H closing parenthesisµn(T )}∞ using), the T > weight0, (2.3) = 1 } ,T \ in L ˆ{\ωinfty } log(1. \ tag + k)∗{$ ( 2k=1 . 1 ) $} Equation: open parenthesis 2 period 3 closing parenthesis .. Trk sub omega open parenthesis T closing parenthesis : = omega open parenthesis open\end{ bracea l i g 1 n divided∗} by log open parenthesis 1 plus k closing parenthesis sum from n = 1 to k mu n open parenthesis T closing parenthesis ∞ closingwhere brace subω i k s = a 1 state to the on power` ofassociated infinity closing t o a parenthesis translation comma and T dilation greater 0 invariant comma state on R. The \noindentwheresequence omegaThen i ( s a 2 state . 1 on ) , l whileto the power bounded of infinity for T ....∈ associated M1,∞, i ts o not a translation generally and convergent dilation invariant . There state on R period .... The ∞ sequenceare operators open parenthesis0 < 2 T period∈ M1 1,∞ closingfor which parenthesis the comma sequence whileα bounded({µn(T )} forn=1 T) in∈ Mc( subcalled 1 comma Tauberian infinity comma i s not generally convergent\ beginoperators{ a lperiod i g n ∗} ..) , There and for are operators the Tauberian operators we want a Dixmier trace to be equal to the limit M { 1 , \ infty∞ } : = M { 1 , \ infty } ( H ) = \{ T \mid \ parallel T \ parallel { 1 0 lesslim Tk→∞ in Mα( sub{µn( 1T comma)}n=1)k. infinityHowever for which , the the linearsequence span alpha of open the parenthesisset of Tauberian open brace operators mu n open does parenthesis not T closing parenthesis closing, \forminfty brace an sub} ideal n: = 1 of to = M the1\, power∞sup. For of{ the infinityk semifinite}\ closingalpha parenthesisdomain( of\{\ in a cDixmier openmu parenthesis tracen to( called b Te an Tauberian ideal) \}\ ofbegin operators compact{ array closing}{ c parenthesis}\ infty comma\\ andn for =operators 1 \end ,{ thearray use}) of k < \ infty \}\ tag ∗{$ ( 2 . 2 ) $} ∞ \endthe{aa Tauberian l state i g n ∗} ω operatorson ` such we want that a Dixmierω(ak) = tracelimk a tok when be equalak to∈ thec( known limit limint as a k generalised right arrow infinity limit ) alpha cannot open parenthesis open brace mu n openb e parenthesis avoided . T closing Note parenthesis that | ω(ak closing) |≤k a bracek k∞, subhence n = 1 to the power of infinity closing parenthesis k period \noindentHowever commadefines .. the a linear Banach span ideal of the set of of compact Tauberian operators operators does whose not form sequence an ideal of of M partial sub 1 comma sums infinity of singular period values iFor s of the ordersemifinite logarithm domain of a . Dixmier\quad traceWe refer to b e an to ideal the of recent compact paper operators of comma Pietsch the use [ 2of \quad 1 ] , discussing the origin of this | Tr (T ) |≤k T k ,T ∈ M . (2.4) objecta state omega in mathematics on l to the power . In of infinity [ 2 2 such ]ω , that J . omega Dixmier1,∞ open parenthesisconstructed1,∞ a sub a k non closing− parenthesisnormal semifinite = limint k a sub trace k when ( a a Dixmiersub k in ctrace open parenthesis ) on $L known (as a generalised H ) $ limit using closing the parenthesis weight cannot b e A Dixmier trace Trω i s positive by construction and continuous by ( 2 . 4 ) , i . e avoided period ..∗ Note that bar omega open parenthesis a sub k closing parenthesis∗ bar less or equal bar a sub k bar sub infinity comma .0 < Trω ∈ M . That it is a trace follows from the identity µn(U TU) = µn(T ) for any unitary hence\ begin { a l i g n ∗} 1,∞ U ∈ L(H). That TrEquation:{\omega open} parenthesis( T 2 period ) : 4 closing = parenthesis\omega ..( bar Tr\{\ sub omegaf r a c { open1 }{\ parenthesislog T( closing 1 parenthesis + k bar) }\ lesssum or equalˆ{ barn =T bar 1 sub} 1{ commak }\ infinitymu comman ( T in T M sub ) 1 comma\} ˆ{\ infinityinfty period} { k = 1 } ),T > 0 , \ tag ∗{$ ( 2 .A 3Dixmier ) $ trace} Tr sub omega i s positive by construction and continuous by open parenthesis 2 period 4 closing parenthesis comma i period e\end period{ al 0 i gless n ∗} Tr sub omega in M sub 1 comma infinity to the power of period to the power of * That it is a trace follows from the identity mu n open parenthesis U to the power of * TU closing parenthesis = mu n open parenthesis T closing\noindent parenthesiswhere for any $ \ unitaryomega U$ in i L opensastateon parenthesis H $ closing\ e l l parenthesisˆ{\ infty period}$ ..\ Thath f i l l associated t o a translation and dilation invariant state on $ R . $ \ h f i l l The

\noindent sequence ( 2 . 1 ) , while bounded for $ T \ in M { 1 , \ infty } , $ i s not generally convergent . \quad There are operators $ 0 < T \ in M { 1 , \ infty }$ for which the sequence $ \alpha ( \{\mu n ( T) \} ˆ{\ infty } { n = 1 } ) \ in c ( $ called Tauberian operators ) , and for the Tauberian operators we want a Dixmier trace to be equal to the limit $ \lim { k \rightarrow \ infty } \alpha ( \{\mu n ( T ) \} ˆ{\ infty } { n = 1 } ) k . $ However , \quad the linear span of the set of Tauberian operators does not form an ideal of $ M { 1 , \ infty } . $ For the semifinite domain of a Dixmier trace to b e an ideal of compact operators , the use of

\noindent a s t a t e $ \omega $ on $ \ e l l ˆ{\ infty }$ such that $ \omega ( a { k } ) = \lim { k } a { k }$ when $ a { k }\ in c ( $ known as a generalised limit ) cannot b e avoided . \quad Note that $ \mid \omega ( a { k } ) \mid \ leq \ parallel a { k }\ parallel {\ infty } , $ hence

\ begin { a l i g n ∗} \mid Tr {\omega } (T) \mid \ leq \ parallel T \ parallel { 1 , \ infty } ,T \ in M { 1 , \ infty } . \ tag ∗{$ ( 2 . 4 ) $} \end{ a l i g n ∗}

A Dixmier trace $ Tr {\omega }$ i s positive by construction and continuous by ( 2 . 4 ) , i . e $ . 0 < Tr {\omega }\ in M ˆ{ ∗ } { 1 , \ infty ˆ{ . }}$ That it is a trace follows from the identity $ \mu n ( U ˆ{ ∗ } TU ) = \mu n ( T ) $ for any unitary $U \ in L ( H ) . $ \quad That 6 S . Lord and F . Sukochev it is singular ( vanishes on finit e rank operators , in fact on any trace class operator ) follows from ∞ 1 α({µn(S)}n=1) ∈ c0 for any finit e rank operator S( resp .S ∈ L ). The basic facts stated in the introduction now follow ( for the fourth see the cit ed references ) . For the third notice that k ST k1,∞≤ k S k k T k 1, ∞( hence | Trω(ST ) |≤k S kk T k1,∞) for all S ∈ L(H),T ∈ M1,∞. The non - trivial feature of Trω is linearity . Dixmier ’ s sp ecification that ω is constructed from a translation and dilation invariant state on R provides sufficient conditions for linearity . Later formulations of the Dixmier trace considered wider or narrower specifications suited t o their context . Unfortunately , the sp ecifications are a mess . We will now introduce a wide range of set s of linear functionals , all of which play some part in known results concerning the Dixmier trace . Let S(`∞) denote the states on `∞, i . e . the positive linear functionals σ such that σ((1, 1, ...)) = 1. Let S([1, ∞))( resp .S([0, ∞))) denote the states of L∞([1, ∞))( resp .L∞([0, ∞))). A state on `∞( resp .L∞([1, ∞)),L∞([0, ∞))) i s singular if it vanishes on finite sequences ( resp . functions with a . e . compact support ) . It is easy to show a state σ on `∞( resp .L∞([1, ∞)),L∞([0, ∞))) i s singular if and only if

sup ∞ lim inf an ≤ σ({an}n=1) ≤ lim ∞an, an > 0 ∀n ∈ n→∞ n→ N

( resp . ess − limt→ ∞ inf f(t) ≤ σ(f) ≤ ess − limt→∞ sup f(t), f > 0). That is , a state i s singular if and only if it is an extension t o `∞( resp .L∞([1, ∞)),L∞([0, ∞))) of the ordinary limit of convergence sequences ( resp . limit at inﬁnity of convergent at inﬁnity functions ) . The Hahn – Banach theorem guarantees that these linear extensions , called ∞ gener - alised limits , exist . Denote by S∞(` )( resp .S∞([1, ∞)),S∞([0, ∞))) the set of singular states ( or generalised limits ) . Denote by SC∞([1, ∞)) ( resp .SC∞([0, ∞))) the same construct with functions from L∞([1, ∞))( resp .L∞([0, ∞))) replaced by continuous functions from Cb([1, ∞))

(resp.Cb([0, ∞))).

Let dxe, x ≥ 0, denote the ceiling function . For j ∈ N deﬁne the maps `∞ → `∞ by

∞ ∞ Tj({ak}k=1) := {ak+j}k=1, (2.5) ∞ ∞ D ({a } ) := {a −1 } (2.6) j k k = 1 dj ke k = 1, k=1 1 X C({a }∞ ) := { a }∞ . (2.7) k k=1 n k n=1 n

∞ Deﬁne subsets of S∞(` ) by

∞ BL := {ω ∈ S(` ) | ω ◦ Tj = ω∀j ∈ N}, (2.8) ∞ ∞ CBL := {ω ∈ S(` ) | ω = σ ◦ C, σ ∈ S∞(` )}, (2.9) B(C) := {ω ∈ S(`∞) | ω ◦ C = ω}, (2.10) ∞ D := {ω ∈ S (`∞) | (ω ◦ D − ω)(α({µn(T )} )) = 0∀T ∈ M }, (2.11) 2 ∞ 2 n = 1 1,∞ ∞ DL := {ω ∈ S(` ) | ω ◦ Dj = ω∀j ∈ N}, (2.12) where α i s from ( 2 . 1 ) . The set BL i s the classical set of Banach limits [ 2 3 , 2 4 ] . The elements of CBL are called Ces a` ro – Banach limits and the elements of B(C) are called Ces a` ro invariant Banach limits . We note that ( as the names suggest ) 6 .... S period Lord and F period Sukochev \noindenthline 6 \ h f i l l S . Lord and F . Sukochev it is singular open parenthesis vanishes on finit e rank operators comma in fact on any trace class operator closing parenthesis follows from \ [ \ r u l e {3em}{0.4 pt }\ ] alpha open parenthesis open brace mu n open parenthesisB(C) ( CBL S closing( BL parenthesis closing brace sub n = 1(2 to.13) the power of infinity closing parenthesis in c sub 0 for any finit e rank operator S open parenthesis resp period S in L to the power of 1 closing parenthesis period .. The basic factsand stated in \noindentthe introductionit is now singular follow open ( vanishesparenthesis for on the finit fourth e see rank the cit operators ed references , inclosing fact parenthesis on any period trace .. Forclass thethird operator notice )that follows from bar ST bar sub 1 comma infinity less or equal bar S bar bar T bar 1 comma infinity open parenthesis hence bar Tr sub omega open parenthesis\noindent ST$ closing\alpha parenthesis( bar\{\ less ormu equaln bar S (DL bar( barSD2 T. ) bar sub\} 1ˆ comma{\ infty infinity} { closingn parenthesis = 1 }(2 for.14)) all S\ in in L openc parenthesis{ 0 }$ Hfor closing any parenthesis finit e comma rank operator T in M sub 1 $S comma ( infinity $ resp period $ . S \ in L ˆ{ 1 } ) . $ \quad The basic facts stated in theThe introduction non hyphen trivial now feature follow of Tr ( sub for omega the is fourth linearity seeperiod the .. Dixmier cit ed quoteright references s sp ecification ) . \quad thatFor omega the is constructed third notice from that $a translation\ parallel and dilationST \ invariantparallel state{ on1 R provides , \ infty sufficient}\ conditionsleq for\ parallel linearity periodS ....\ Laterparallel \ parallel T \ parallel 1formulations , \ infty of the Dixmier( $ hence trace considered $ \mid widerTr or{\ narroweromega specifications} ( ST suited ) t o their\mid \ leq \ parallel S \ parallel \ parallelcontext periodT \ parallel { 1 , \ infty } ) $ f o r a l l $ S \ in L(H),T \ in M { 1 , Unfortunately\ infty } comma. $ the sp ecifications are a mess period .. We will now introduce a wide range of set s of linear functionals comma all of which play some part in known results concerning the Dixmier trace period \ hspaceLet S open∗{\ f parenthesis i l l }The non l to the− powertrivial of infinity feature closing of parenthesis $ Tr {\ denoteomega the states}$ on is l to linearity the power of . infinity\quad commaDixmier i period ’ s e period sp ecification the that positive$ \omega linear$ functionals is constructed sigma such from that sigma open parenthesis open parenthesis 1 comma 1 comma period period period closing parenthesis closing parenthesis \noindent= 1 perioda Let translation S open parenthesis and open dilation square bracket invariant 1 comma state infinity on closing $ R parenthesis $ provides closing sufficient parenthesis open conditions parenthesis for resp linearityperiod . \ h f i l l Later S open parenthesis open square bracket 0 comma infinity closing parenthesis closing parenthesis closing parenthesis denote the states of L to the power\noindent of infinityformulations open parenthesis of open the square Dixmier bracket trace 1 comma considered infinity closing wider parenthesis or narrower closing parenthesis specifications open parenthesis suited resp t o period their L to the power of infinity open parenthesis open square bracket 0 comma infinity closing parenthesis closing parenthesis closing parenthesis period A\noindent state context . on l to the power of infinity open parenthesis resp period L to the power of infinity open parenthesis open square bracket 1 comma infinity\ hspace closing∗{\ f parenthesis i l l } Unfortunately closing parenthesis , the comma sp ecifications L to the power of are infinity a mess open . parenthesis\quad We open will square now bracket introduce 0 comma a infinity wide closingrange of set s of parenthesis closing parenthesis closing parenthesis i s singular if it vanishes on finite sequences open parenthesis resp period functions \noindentwith a periodlinear e period functionals compact support , all closing of which parenthesis play period some .. part It is easy in knownto show results a state sigma concerning on l to the the power Dixmier of infinity trace open . parenthesis resp period L to the power of infinity open parenthesis open square bracket 1 comma infinity closing parenthesis closing parenthesis commaLet $ L toS the ( power\ ofe l infinity l ˆ{\ openinfty parenthesis} ) $ open denote square bracket the states 0 comma on infinity $ \ closinge l l ˆ{\ parenthesisinfty closing} , parenthesis$ i . e closing . the parenthesis positive linear functionals $ \isigma s singular$ if such and only that if $ \sigma ((1,1,...))$ $=limint n 1 right .$Let$S arrow infinity inf a sub ( n [less or 1 equal , sigma\ infty open parenthesis) ) open ($resp$. brace a sub n closing S brace ( sub [ n = 0 1 to , the power\ infty of infinity) ) closing )$ parenthesis denote less the or states equal limint of n $Lˆ right{\ arrowinfty sub infinity} ( to the [ power 1 of , supremum\ infty a sub) n comma ) ($ a sub n resp greater $. 0 forall Lˆ n {\ infty } in( N [ 0 , \ infty ) ) ) .$Astate onopen $ parenthesis\ e l l ˆ{\ respinfty period} ..( ess $ hyphen resp limint $ . t right L ˆ arrow{\ infty sub infinity} ( inf f [ open 1 parenthesis , \ infty t closing parenthesis) ) less , or L equal ˆ{\ sigmainfty } open( parenthesis [ 0 , f closing\ infty parenthesis) less ) or equal) $ ess i hyphens singular limint t if right it arrow vanishes infinity on sup finite f open parenthesis sequences t closing ( resp parenthesis . functions comma f greaterwith a 0 closing. e . parenthesis compact supportperiod ) . \quad It is easy to show a state $ \sigma $ on $ \ e l l ˆ{\ infty } ( $That resp is comma $ . a state L ˆ i{\ s singularinfty if} and( only [if it is 1 an extension , \ infty t o l to the) power ) of , infinity L ˆ open{\ infty parenthesis} ( resp period [ 0 L to , the power\ infty of) infinity ) open ) $ parenthesis open square bracket 1 comma infinity closing parenthesis closing parenthesis comma L to the power of infinity open parenthesisi s singular open square if and bracket only 0 comma if infinity closing parenthesis closing parenthesis closing parenthesis of the ordinary limit of convergence sequences open parenthesis resp period limit at infinity of convergent at infinity \ [ functions\lim { closingn \ parenthesisrightarrow period\ ..infty The Hahn}\ endashinf Banacha { n theorem}\ leq guarantees\sigma that these( linear\{ extensionsa { n comma}\} ..ˆ{\ calledinfty gener} { n =hyphen 1 } ) \ leq \lim { n \rightarrow }ˆ{\sup } {\ infty } a { n } , a { n } > 0 \ f o r a l l n alised\ in limitsN comma\ ] exist period .. Denote by S sub infinity open parenthesis l to the power of infinity closing parenthesis open parenthesis resp period S sub infinity open parenthesis open square bracket 1 comma infinity closing parenthesis closing parenthesis comma S sub infinity open parenthesis open square bracket 0 comma infinity closing parenthesis closing parenthesis closing parenthesis the set of singular states \ centerlineopen parenthesis{( resp or generalised . \quad limitse s s closing $ − parenthesis \lim { periodt ..\ Denoterightarrow by SC sub} {\ infinityinfty open} parenthesis$ i n f $ open f square ( bracket t ) 1 comma\ leq infinity\sigma closing( parenthesis f ) closing\ leq parenthesis$ e s s $ open− parenthesis \lim { respt period\rightarrow SC sub infinity\ infty open parenthesis}$ sup open $ f square ( bracket t ) 0 comma , infinityf > closing0 parenthesis ) . $ closing} parenthesis closing parenthesis the same construct with functions from L to the power of infinity open parenthesis open square bracket 1 comma infinity closing parenthesis closing parenthesis open parenthesis\noindent respThat period is L , to a the state power i of sinfinity singular open parenthesis if and only open square if it bracket is an 0 comma extension infinity t closing o $ parenthesis\ e l l ˆ{\ closinginfty parenthesis} ( $ closingresp parenthesis $ . L ˆreplaced{\ infty by continuous} ( [ functions 1 from , C\ infty sub b open) parenthesis ) , open L ˆsquare{\ infty bracket} 1 comma( [ infinity 0 closing , \ parenthesisinfty closing) ) parenthesis ) $ ofopen the parenthesis ordinary resp limit period of C sub convergence b open parenthesis sequences open square ( resp bracket . limit 0 comma at infinity infinity closing of parenthesis convergent closing at parenthesis infinity closing parenthesisfunctions period ) . \quad The Hahn −− Banach theorem guarantees that these linear extensions , \quad called gener − alisedLet .. ceilingleft limits x , ceilingright exist . \ commaquad Denote x greater by equal $ 0 S comma{\ infty denote the} ceiling( \ functione l l ˆ{\ periodinfty .. For} j) in N ( define $ resp the maps $ .l to the S {\ infty } power( [ of infinity 1 ,right\ arrowinfty l to the)),S power of infinity by{\ infty } ( [ 0 , \ infty ) ) ) $ the set of singular states (Equation: or generalised open parenthesis limits 2 period ) . \quad 5 closingDenote parenthesis by .. $ T SC sub{\ j openinfty parenthesis} ( open [ brace 1 a , sub k\ infty closing brace) sub ) k = ( 1 $ to the resp power$ . of SC infinity{\ closinginfty parenthesis} ( [ : = 0open brace , \ ainfty sub k plus) j closing ) brace ) $ sub the k =same 1 to construct the power of withcomma to the power of infinity D subfunctions j open parenthesis from open $ Lbrace ˆ{\ infty a sub k} Row( 1 infinity [ 1 Row 2 , k =\ 1infty closing brace.) :) = open ($ brace resp a sub $. ceilingleft Lˆ j{\ to theinfty power} of([ minus 10 k ceilingright , \ infty Row 1 infinity) ) Row ) 2 $ k = replaced 1 to the power by ofcontinuous comma closing functions brace. 2 period from 6 closing $ C parenthesis{ b } ( Equation: [ 1 open , parenthesis\ infty 2 period) ) 7 $ closing parenthesis .. C open parenthesis open brace a sub k closing brace sub k = 1 to the power of infinity closing parenthesis : = open brace 1 divided by n sum from k = 1 to n a sub k closing brace sub n = 1 to the power of infinity period \ beginDefine{ a subsets l i g n ∗} of S sub infinity open parenthesis l to the power of infinity closing parenthesis by (Equation: resp open . parenthesis C { b } 2 period( [ 8 closing 0 parenthesis , \ infty .. BL))). : = open brace omega in S open parenthesis l to the power of infinity closing\end{ a parenthesis l i g n ∗} bar omega circ T sub j = omega forall j in N closing brace comma Equation: open parenthesis 2 period 9 closing parenthesis .. CBL : = open brace omega in S open parenthesis l to the power of infinity closing parenthesis bar omega = sigma circ C comma sigma in\ centerline S sub infinity{ Let open\ parenthesisquad $ \ ll to c e the i l powerx of\ infinityr c e i l closing, parenthesis x \geq closing0 brace , $ comma denote Equation: the ceiling open parenthesis function 2 period . \quad 10 For closing$ j parenthesis\ in N$ .. B define open parenthesis the maps C closing $ \ e parenthesis l l ˆ{\ infty : = open}\ bracerightarrow omega in S open\ e l l parenthesisˆ{\ infty l to} the$ power by } of infinity closing parenthesis bar omega circ C = omega closing brace comma Equation: open parenthesis 2 period 1 1 closing parenthesis .. D sub 2 : = open brace\ begin omega{ a l i gin n S∗} sub infinity open parenthesis l to the power of infinity closing parenthesis bar open parenthesis omega circ D sub 2 minus omegaT { closingj } parenthesis( \{ opena { parenthesisk }\} alphaˆ{\ openinfty parenthesis} { k open = brace 1 } mu n) open : parenthesis = \{ T closinga { parenthesisk + Row j }\} 1 infinityˆ{\ Rowinfty } { k =2 n = 1 1 ˆclosing{ , }}\ brace.tag closing∗{$ ( parenthesis 2 . = 0 5 forall ) T $}\\ in MD sub{ 1 commaj } ( infinity\{ closinga brace{ k }\}\ commabegin Equation:{ array open}{ parenthesisc}\ infty 2 period\\ k 1 =2 closing 1 \ parenthesisend{ array ..}) DL : : = open = brace\{ omegaa {\ in Sl openc e i l parenthesisj ˆ{ − l to the1 } powerk of\ infinityr c e i l closing}\}\ parenthesisbegin { array bar}{ omegac}\ circinfty D sub\\ j = omegak = forall 1 j ˆ in{ N, closing}\end brace{ array comma}( 2 . 6 ) \\ C( \{ a { k }\} ˆ{\ infty } { k = 1 } ) :where = alpha\{\ i sf rfrom a c { open1 }{ parenthesisn }\sum 2 periodˆ{ k 1 closing = parenthesis 1 } { n period} a ..{ Thek }\} set BL iˆ s{\ theinfty classical} set{ ofn Banach = limits 1 } open. \ tag square∗{$ ( bracket2 . 2 .. 7 3 comma ) $} 2 4 closing square bracket period .. The elements \endof{ CBLa l i g are n ∗} called Ces grave-a ro endash Banach limits and the elements of B open parenthesis C closing parenthesis are called Ces grave-a ro invariant \noindentBanach limitsDefine period subsets We note that of open $ S parenthesis{\ infty as} the names( \ suggeste l l ˆ{\ closinginfty parenthesis} ) $ by Equation: open parenthesis 2 period 13 closing parenthesis .. B open parenthesis C closing parenthesis subsetneq CBL subsetneq BL \ beginand { a l i g n ∗} BLEquation: : = open\{\ parenthesisomega 2 period\ in 14 closingS( parenthesis\ e l l ..ˆ DL{\ subsetneqinfty } D sub) 2\ periodmid \omega \ circ T { j } = \omega \ f o r a l l j \ in N \} , \ tag ∗{$ ( 2 . 8 ) $}\\ CBL : = \{\omega \ in S( \ e l l ˆ{\ infty } ) \mid \omega = \sigma \ circ C, \sigma \ in S {\ infty } ( \ e l l ˆ{\ infty } ) \} , \ tag ∗{$ ( 2 . 9 ) $}\\ B ( C ) : = \{\omega \ in S( \ e l l ˆ{\ infty } ) \mid \omega \ circ C = \omega \} , \ tag ∗{$ ( 2 . 10 ) $}\\ D { 2 } : = \{ \omega \ in S {\ infty } ( \ e l l ˆ{\ infty } ) \mid ( \omega \ circ D { 2 } − \omega )( \alpha ( \{\mu n ( T ) \}\ begin { array }{ c}\ infty \\ n = 1 \end{ array })) = 0 \ f o r a l l T \ in M { 1 , \ infty }\} , \ tag ∗{$ ( 2 . 1 1 ) $}\\ DL : = \{\omega \ in S( \ e l l ˆ{\ infty } ) \mid \omega \ circ D { j } = \omega \ f o r a l l j \ in N \} , \ tag ∗{$ ( 2 . 1 2 ) $} \end{ a l i g n ∗}

\noindent where $ \alpha $ isfrom(2.1). \quad The set $ BL $ i s the classical set of Banach limits [ 2 \quad 3 , 2 4 ] . \quad The elements of $CBL$ are called Ces $ \grave{a} $ ro −− Banach limits and the elements of $B ( C ) $ are called Ces $ \grave{a} $ ro invariant Banach limits . We note that ( as the names suggest )

\ begin { a l i g n ∗} B(C) \ subsetneq CBL \ subsetneq BL \ tag ∗{$ ( 2 . 13 ) $} \end{ a l i g n ∗}

\noindent and

\ begin { a l i g n ∗} DL \ subsetneq D { 2 } . \ tag ∗{$ ( 2 . 14 ) $} \end{ a l i g n ∗} Measure Theory in Noncommutative Spaces .... 7 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 7 7 For a greater 0 define the maps L to the power of infinity open parenthesis open square bracket 0 comma infinity closing parenthesis closing parenthesis\ [ \ r u l e { right3em}{ arrow0.4 ptL to}\ the] power of infinity open parenthesis open square bracket 0 comma infinity closing parenthesis closing parenthesis by ∞ ∞ Line 1 T sub a open parenthesisFor a > f0 closingdefine parenthesis the maps openL ([0 parenthesis, ∞)) → L t([0 closing, ∞)) parenthesisby : = f open parenthesis t plus a closing parenthesis\ centerline comma{For open $ parenthesis a > 0 2 period $ define 1 5 closing the parenthesis maps $ LineL ˆ{\ 2 Dinfty sub a open} ( parenthesis [ 0 f closing , parenthesis\ infty open)) parenthesis\rightarrow t Ta(f)(t) := f(t + a), (2.15) Lclosing ˆ{\ parenthesisinfty } : =( f open [ parenthesis 0 , a\ toinfty the power) of minus ) $ 1 t by closing} parenthesis comma open parenthesis 2 period 16 closing parenthesis −1 Line 3 P sub a open parenthesis f closing parenthesisD opena(f)( parenthesist) := f(a t ), closing(2.16) parenthesis : = f open parenthesis t to the power of a closing \ [ \ begin { a l i g n e d } T { a } (f)(t):=f(t+a),(2.1a parenthesis comma open parenthesis 2 period 1 7 closingP parenthesisa(f)(t) := f Line(t ), 4 C(2. open17) parenthesis f closing parenthesis open parenthesis t closing 5 ) \\ parenthesis : = 1 divided by t integral sub 0 to the power of t1 fZ opent parenthesis s closing parenthesis ds period open parenthesis 2 period 18 closingD { parenthesisa } (f)(t):=f(aˆ Line 5 Define L to the power ofC minus(f)(t) 1 := : L to thef(s) powerds. {(2 of −.18) infinity1 } opent parenthesis ) , open ( square 2 bracket . 16 1 comma ) \\ infinity t closingP { parenthesisa } (f)(t):=f(tˆ closing parenthesis right arrow L to the power0 of infinity{ a open} parenthesis) , ( open 2 square . bracket 1 0 7 comma ) \\ infinity closing −1 ∞ ∞ parenthesisC ( closing f parenthesis) ( t by Line ) 6 L :Define to the =L power\ f: rL a ofc([1{ minus,1∞}{)) 1→ opent L}\([0 parenthesis,int∞))byˆ{ gt closing} { 0 parenthesis} f open ( parenthesis s ) ds t closing . parenthesis ( 2 . 18 ) \\ : = g open parenthesis e to the power of t closing parenthesis L−1(g)(t) := g(et) Defineand comma L if Gˆ{ : − L to the1 } power: of L infinity ˆ{\ infty open parenthesis} ( open [ square1 , bracket\ infty 0 comma)) infinity closing\rightarrow parenthesis closingL ˆ{\ parenthesisinfty } ∞ ∞ right( arrow [and 0 L ,if to the,G : powerL\ infty([0, of∞ infinity)) →)L open([0 ), ∞ parenthesis)) by, set\\ open square bracket 0 comma infinity closing parenthesis closing parenthesis comma set LEquation: ˆ{ − open1 } parenthesis(g)(t):=g(eˆ 2 period 19 closing parenthesis .. L open parenthesis{ t } G closing) \end parenthesis{ a l i g n e d :}\ L] to the power of infinity open parenthesis open square bracket 1 comma infinity closing parenthesis closing parenthesis right arrow L to the power of infinity open parenthesis L(G): L∞([1, ∞)) → L∞([1, ∞)),G 7→ L ◦ G ◦ L−1. (2.19) open square bracket 1 comma infinity closing parenthesis closing parenthesis comma G arrowright-mapsto L circ G circ L to the power of minus \noindent and,if $G : Lˆ{\ infty } ( [ 0 , \ infty )) \rightarrow L ˆ{\ infty } 1 periodSimilarly , if σ ∈ S([0, ∞)) set (Similarly [ 0 comma , if\ sigmainfty in S open)parenthesis ) , $ open s e t square bracket 0 comma infinity closing parenthesis closing parenthesis set Equation: open parenthesis 2 period 20 closing parenthesis .. L open parenthesis sigma closing parenthesis in S open parenthesis open square \ begin { a l i g n ∗} bracket 1 comma infinity closing parenthesis closingL(σ) ∈ parenthesisS([1, ∞)), comma σ 7→ σ ◦ sigmaL−1. arrowright-mapsto sigma circ L(2 to.20) the power of minus 1 period LThese ( definitions G ) imply : L L open ˆ{\ parenthesisinfty } sigma( closing [ 1 parenthesis , \ infty circ L open)) parenthesis\rightarrow G closing parenthesisL ˆ{\ =infty L open} parenthesis([ 1 , \ infty )),G \mapsto L \ circ G \ circ L ˆ{ − 1 } . \ tag ∗{$ ( 2 . 19 sigma circThese G closing definitions parenthesis imply periodL(σ ..) ◦ ItL( iG s) known = L(σ L◦ openG). parenthesisIt i s known T subL a(T closinga) = D parenthesis−a and L =(D Da) sub = P e− to1 , the power of minus a and ) $} e a L opensee parenthesis [ 2 5 D , § sub1 . a 1 closing ] . parenthesis Define subsets = P sub of aS to∞([0 the, ∞ power)) and of minusS∞([1 1, ∞ comma)) by \endsee{ opena l i g n square∗} bracket 2 .. 5 comma S 1 period 1 closing square bracket period .. Define subsets of S sub infinity open parenthesis open square bracket 0 comma infinity closing parenthesis closing parenthesis and S sub infinity open parenthesis open square bracket 1 comma infinity \noindent Similarly , if $ \sigma \ in S ( [ 0 , \ infty ) ) $ s e t closing parenthesis closing parenthesis by BL[0, ∞) := {ω ∈ S([0, ∞)) | ω ◦ Ta = ω∀a > 0}, (2.21) Equation: open parenthesis 2CBL period[0, 2∞ 1) closing:= {ω ∈ parenthesisS([0, ∞)) | ..ω BL= σ open◦ C, square σ ∈ S bracket([0, ∞ 0))} comma, infinity closing(2.22) parenthesis : = open brace \ begin { a l i g n ∗} ∞ omega in S open parenthesis open square bracketB 0( commaC)[0, ∞ infinity) := {ω ∈ closingS([0, ∞ parenthesis)) | ω ◦ C = closingω}, parenthesis bar(2. omega23) circ T sub a = omega forallL( a greater\sigma 0 closing brace) \ commain S Equation: ( open [ parenthesis 1 , \ 2infty period 22 closing)), parenthesis\sigma .. CBL open\mapsto square bracket\sigma 0 comma\ circ infinity DL[0, ∞) := {ω ∈ S([0, ∞)) | ω ◦ D = ω∀a > 0}, (2.24) Lclosing ˆ{ − parenthesis1 } :. =\ tag open∗{ brace$ ( omega 2 in . S open 20 parenthesis ) $} open square bracketa 0 comma infinity closing parenthesis closing parenthesis bar\end omega{ a l i g = n ∗} sigma circ C comma sigma in S sub infinityDL open[1, ∞ parenthesis) := {L(ω) open| ω ∈ squareBL[0, ∞ bracket)}, 0 comma infinity(2.25) closing parenthesis closing parenthesis closing brace comma Equation: open parenthesisCDL[1, ∞ 2) period := {L(ω 23) | closingω ∈ CBL parenthesis([0, ∞))} .., B open parenthesis(2.26) C closing parenthesis open \noindent These definitions imply $ L ( \sigma ) \ circ L ( G ) = L ( \sigma \ circ square bracket 0 comma infinity closing parenthesisD :( =C open)[1, ∞ brace) := { omegaL(ω) | ω in∈ SB open(C)([0 parenthesis, ∞))}, open square bracket(2.27) 0 comma infinity closing Gparenthesis ) . closing $ \quad parenthesisItisknown bar omega circ $L C = omega ( T closing{ a brace} ) comma = Equation: D { e open ˆ{ − parenthesisa }}$ 2 period and 24 $ closing L ( parenthesis D { a ..} PDL[1, ∞) := {L(ω) | ω ∈ BL[0, ∞) ∩ DL[0, ∞)}, (2.28) DL) open = square P { bracketa ˆ{ 0− comma1 }} infinity, $ closing parenthesis : = open brace omega in S open parenthesis open square bracket 0 comma infinity closingsee [ parenthesis 2 \quad closing5 , \S parenthesisMDL1 .[1 1, ∞ ] bar) . := omega\{quadL(ω) circ|Defineω ∈ DBL sub[0 subsets, a∞ =) ∩ omegaDL[0 of, forall∞) $∩ a SB greater(C{\)[0, ∞infty 0)} closing. } brace( comma [(2 0.29) Equation: , \ infty open parenthesis) ) $ 2and period $ 25S closing{\ infty parenthesis} ( .. DL [ open 1 square , \ bracketinfty 1 comma) ) infinity $ by closing parenthesis : = open brace L open parenthesis omega ∞ closingA parenthesis subscript bar of omegac for in any BL of open the square set s bracket ( 2 . 2 0 1 comma ) – ( infinity 2 . 2 9 closing ) denotes parenthesis the replacement closing brace commaof L − Equation: open parenthesis \ begin { a l i g n ∗} 2 periodfunctions 26 closing by parenthesisCb− functions .. CDL , open i . squaree .BLc bracket[0, ∞) denotes 1 comma the infinity translation closing parenthesis invariant : states = open on brace the L open parenthesis omega BL [∗ 0 , \ infty ) : = \{\omega \ in S ( [ 0 , \ infty )) \mid \omega closingC parenthesis− algebra barC omegab([0, ∞)) inand CBL it open is a parenthesis subset of openSC∞ square([0, ∞)) bracket. 0 comma infinity closing parenthesis closing parenthesis closing brace\ circ commaWeT Equation:{ nowa } define= open the\ parenthesisomega Dixmier\ 2 tracesf period o r a l l corresponding 27 closinga > parenthesis0 t o\} the .. Dsingular, open\ tag parenthesis∗{ states$ ( ( 2 C 2 . closing 8 ). – parenthesis ( 2 2 . 1 open ) $}\\ squareCBL bracket [ 1 comma0 , 1 infinity )\ ,infty ( 2 closing . 2 1 ) parenthesis – ( : 2 . = 29 : = )\{\ and open subscript braceomega L openc \versions parenthesisin S . omega ( [ closing 0 parenthesis , \ infty bar omega)) in B open\mid parenthesis\omega C closing= ∗ parenthesis\sigma open\ circLet parenthesisgC,b open e\ square thesigma bracket monotonically\ in 0 commaS {\ infinityinfty decreasing closing} parenthesis( left [- continuous closing0 , parenthesis\ infty rearrangement closing)) brace comma\} Equation:, \ tag ∗{ open$ ( parenthesis2 .of 22 2| periodg |, )where $28}\\ closingB parenthesisg (∈ C .. ) PDL [ open 0 square , bracket\ infty 1 comma) infinity : closing= \{\ parenthesisomega : = open\ in braceS L open ( parenthesis [ 0 omega, \ infty closing parenthesis)) bar\mid omega in\omega BL open square\ circ bracketC 0 = comma\omega infinity closing\} parenthesis, \ tag ∗{$ cap ( DL 2 open . square 23 bracket ) $}\\ 0 commaDL L∞([0, ∞)) [2 6]. Define infinity[ 0 closing , parenthesis\ infty closing) : brace = comma\{\ Equation:omega open\ in parenthesisS ( 2 period [ 29 0 closing , parenthesis\ infty ..)) MDL open\mid square\ bracketomega 1 \ circ D { a } = \omega \ f o r a l l a Z t> 0 \} , \ tag ∗{$ ( 2 . 24 ) $}\\ DL [ 1 comma infinity closing parenthesis : = open∗ brace L open1 parenthesis∗ omega closing parenthesis bar omega in BL open square bracket 0 comma , \ infty ) : = \{ αt(L(g ) := \omega g ()s)ds,\mid t ≥ 1\ (2omega.30) \ in BL [ 0 , \ infty ) \} infinity closing parenthesis cap DL open square bracketlog(1 + 0t) comma infinity closing parenthesis cap B open parenthesis C closing parenthesis open , \ tag ∗{$ ( 2 . 25 ) $}\\ CDL [1 1 , \ infty ) : = \{ L( \omega ) \mid square bracket 0 comma infinity closing parenthesis closing brace period ∞ \omegaA subscript( the\ continuousin of c forCBL any versionof ( the set [ of s open ( 0 2 . parenthesis 1 , ) )\ .infty 2 Define period)) 2 the 1 closing floor\} parenthesis map , p\fromtag endash∗{`$ open (t o bounded parenthesis 2 . Borel 26 2 period ) 2 $}\\ 9 closingD(C parenthesis denotes) [functions the 1 replacement , on\ infty[0 of, ∞ L)( toor the)[1 power, ∞ :)) by of = infinity\{ hyphenL( functions\omega by C sub) b hyphen\mid \omega \ in B(C)( [functions 0 , comma\ infty i period)) e period BL\} sub, c\ opentag ∗{ square$ ( bracket 2 0 . comma 27 infinity ) $}\\ closingPDL parenthesis [ 1 denotes , the\ infty translation) invariant : = states\{ onL( the C to\omega the power of) * hyphen\mid algebra\omega C sub b\ in open parenthesisBL [ open 0 square , \ bracketinfty 0 comma) \ infinitycap closingDL [parenthesis 0 , closing\ infty ∞ parenthesis) \} , \ tag ∗{$ ( 2 . 28 ) $}\\ MDL [ 1 , \ infty ) : = \{ L( \omega ) \mid \omega \ in BL [∞ 0 ,X \ infty ) \cap DL [ 0 , \ infty ) \cap B and it is a subset of SC sub infinityp({ openak}k=1 parenthesis)(t) := openakχ[k,k square+1)(t bracket), t ≥ 0 comma (or1) infinity closing parenthesis (2.31) closing parenthesis period (We C now ) define [ the Dixmier 0 , traces\ infty corresponding) \} t o the singular. \ tag ∗{ states$ ( open 2 parenthesis . 29 2 period ) $ 8} closing parenthesis endash open parenthesis k = 1 2\end period{ a l 1 i g .. n 1∗} closing parenthesis comma open parenthesis 2 period 2 1 closing parenthesis endash open parenthesis 2 period 29 closing parenthesis and subscript c versions period \noindentLet .. g toAsubscript the power of * ..of b e $ .. c the $ .. formonotonically any of ..the decreasing set s left( 2 hyphen . 2 1 continuous ) −− ( 2 .. rearrangement . 2 9 ) denotes .. of .. the bar g replacement bar comma .. of where$ L ˆ ..{\ g ininfty } − $ functions by $C { b } − $ functionsLine 1 L to the , i power . e of $ infinity . BL open{ parenthesisc } [ open 0 square , bracket\ infty 0 comma) $ infinity denotes closing the parenthesis translation closing parenthesis invariant open states square on the bracket$ C ˆ{ 2 ∗ 6 closing } − $ square algebra bracket period $ C Define{ b } Line( 2 alpha [ sub 0 t open , \ parenthesisinfty g) to the ) $power of * closing parenthesis : = 1 divided by logand open it parenthesis is a subset 1 plus of t closing $ SC parenthesis{\ infty integral} sub( 1 [ to the 0 power , of\ tinfty g to the power) of ) * open . $ parenthesis s closing parenthesis ds comma t greater equal 1 open parenthesis 2 period 30 closing parenthesis Weopen now parenthesis define the the continuous Dixmier versiontraces of corresponding open parenthesis 2 t period o the 1 closing singular parenthesis states closing ( 2 parenthesis . 8 ) −− period( 2 .. . Define 1 \quad the floor1 ) map , ( 2 . 2 1 ) −− p( from 2 .l to 29 the ) power and ofsubscript infinity t o bounded $ c $ Borel versions functions . on open square bracket 0 comma infinity closing parenthesis open parenthesis or open square bracket 1 comma infinity closing parenthesis closing\ hspace parenthesis∗{\ f i l l } byLet \quad $ g ˆ{ ∗ }$ \quad b e \quad the \quad monotonically \quad decreasing left − continuous \quad rearrangement \quad o f \quad $ \infinitymid Equation:g \mid open, parenthesis $ \quad 2 periodwhere 31\quad closing parenthesis$ g \ in ..$ p open parenthesis open brace a sub k closing brace sub k = 1 to the power of infinity closing parenthesis open parenthesis t closing parenthesis : = sum a sub k chi sub open square bracket k comma k plus 1 closing\ [ \ begin parenthesis{ a l i g n e open d } L parenthesis ˆ{\ infty t closing} ( parenthesis [ 0 comma , t\ greaterinfty equal) 0 open ) parenthesis [ 2 or 6 1 closing ] parenthesis. Define k =\\ 1 \alpha { t } ( g ˆ{ ∗ } ) : = \ f r a c { 1 }{\ log ( 1 + t ) }\ int ˆ{ t } { 1 } g ˆ{ ∗ } ( s ) ds , t \geq 1 ( 2 . 30 ) \end{ a l i g n e d }\ ]

\noindent ( the continuous version of ( 2 . 1 ) ) . \quad Define the floor map $ p $ from $ \ e l l ˆ{\ infty }$ t o bounded Borel functions on $ [ 0 , \ infty ) ( $ or $ [ 1 , \ infty ) ) $ by

\ begin { a l i g n ∗} \ infty \\ p ( \{ a { k }\} ˆ{\ infty } { k = 1 } ) ( t ) : = \sum a { k } \ chi { [ k , k + 1 ) } ( t ) , t \geq 0 ( or 1 ) \ tag ∗{$ ( 2 . 31 ) $}\\ k = 1 \end{ a l i g n ∗} 8 .... S period Lord and F period Sukochev \noindenthline88 \ h f i l l S . Lord and F . Sukochev S . Lord and F . Sukochev and the restriction map r from bounded Borel functions on open square bracket 0 comma infinity closing parenthesis open parenthesis or open\ [ \ squarer u l e {3em bracket}{0.4 1 comma pt }\ infinity] closing parenthesis closing parenthesis t o l to the power of infinity Equation:and the open restriction parenthesis map 2 periodr from 32 closing bounded parenthesis Borel functions.. r open parenthesis on [0, ∞)( f closingor [1, parenthesis∞)) t o `∞ sub n : = f open parenthesis n closing parenthesis comma n in N period \noindentWe have theand following the restriction set s of unitarily map invariant $ r weights $ from on M bounded sub 1 comma Borel infinity functions : on $ [ 0 , \ infty ) ( $ or $ [ 1 , \ infty ) ) $ t o $ \ e l l ˆ{\ infty }$ V sub B = open brace tr sub omega open parenthesisr(f)n := T closingf(n), parenthesis n ∈ N. : = omega circ alpha sub t circ p(2 open.32) parenthesis open brace mu n openWe parenthesis have the T closing following parenthesis set s of closing unitarily brace invariant closing parenthesis weights comma on M T1,∞ greater: 0 bar omega in B closing brace comma B = Case 1 DL\ begin open{ squarea l i g n bracket∗} 1 comma infinity closing parenthesis comma DL sub c open square bracket 1 comma infinity closing parenthesis comma Caser 2 ( CDL f D open ) parenthesis{ n } : C closing = f parenthesis ( n tothe ) power , ofn open\ in squareN. bracket\ tag sub∗{ open$ ( square 2 bracket . 321 comma ) $ to} the power of\end 1 comma{ a l i g n infinity∗} sub infinity sub closing parenthesis comma to the powerVB of= closing{trω(T ) parenthesis := ω ◦ αt ◦ commap({µn(T CDL)}), T D > open0 | ω parenthesis∈ B}, C closing parenthesis sub c open square bracket 1 comma to the power of c open square bracketDL[1 1, ∞ comma),DL infinityc[1, ∞) sub, infinity sub closing parenthesis \noindent We have the following set s of unitarily invariant weights on $ M { 1 , \ infty } : $ comma to the power of closing parenthesis comma open parenthesis 2 period 33 closing parenthesis[ Case 3c PDL[1, open square bracket 1 comma B = braceex − braceex − braceleftmid − braceex − braceex − braceleftbt CDL 1, ∞), CDL ∞), (2.33) infinity closing parenthesis comma MDL open square bracket 1 comma infinity closingD parenthesis(C)[1, ∞), Equation:D(C)c[1 open, ∞) parenthesis, 2 period 34 closing\ begin parenthesis{ a l i g n ∗} .. V sub B = open brace Tr sub omega open parenthesis T closingPDL[1 parenthesis, ∞),MDL :[1 =, ∞ omega) circ r circ alpha sub t circ p openV { parenthesisB } = open\{ bracet rmu{\ n openomega parenthesis} ( T Tclosing ) parenthesis : = closing\omega brace closing\ circ parenthesis\alpha comma{ t T}\ greatercirc 0 bar omegap ( in B \{\mu n ( T ) \} ),T V>B = {0Trω(\Tmid) := ω ◦\romega◦ αt ◦ p({µn\ in(T )}),B T > 0\}| ω ∈ B, },B\\\ l e= fD t .2, B = braceex−braceex−braceleftmid −braceex−braceex−braceleftbt \ begin { a l i g n e d } & closing brace comma B = D sub 2 comma W sub Y = open brace f sub omega open parenthesis T closing parenthesis(2.34) : = omega circ alpha DLsub e t [ circ p1 open , parenthesis\ infty open) brace , mu DL n open{ c parenthesis} [ 1 T closing , parenthesis\ infty closing), brace\\ closing parenthesis comma T greater 0 bar& omega CDL in Y{ closingD(C) brace comma} Yˆ{ =[ Case}ˆ{ 1 BL1 open , } square{ [ bracket 1W 0 ,Y comma}\= {fωinfty(T infinity) := ω ◦{\ closingαetinfty◦ p parenthesis({µn}(Tˆ{)})),, Tcomma > 0 |}ω BL{∈ subY),}, c open} squareCDL { D (C) }ˆ{ c [ 1 , } { c [ 1 , }\ infty {\ infty }ˆ{ ), } { ), } ( 2 . 33 bracket 0 comma infinity closing parenthesis comma Case 2 CBL open square bracket 0 commaBL[0 infinity, ∞),BL closingc[0, ∞ parenthesis), comma CBL sub c ) \\  open square bracket 0 comma infinity closing parenthesis comma open parenthesisY 2= periodCBL 35 closing[0, ∞),CBL parenthesisc[0, ∞ Case), 3(2 B.35) open parenthesis C& closing PDL parenthesis [ 1open square, \ infty bracket 0 comma) , infinity MDL closing [ parenthesis 1 , \ commainfty B open) \ parenthesisend{ a l i g n Ce d closing}\ right parenthesis. \\ V sub{ B c} open=  B(C)[0, ∞),B(C) [0, ∞) square\{ Tr bracket{\ 0omega comma} infinity( closing T ) parenthesis : = Equation:\omega Y =\ BLcirc commar CBL\ commacirc B\ openalpha parenthesis{ t c}\ C closingcirc parenthesisp ( period\{ \mu n ( T ) \} ),T > 0 \mid \omega \ in B \} , B = D { 2 } , \ tag ∗{$ ( open parenthesis 2 period 36 closing parenthesis .. W sub Y = open braceWY = F{ subFω( omegaT ) := ω open◦ r ◦ parenthesisαet ◦ p({µn T(T closing)}), T > parenthesis0 | ω ∈ Y }, : = omega circ r2 circ .alpha 34 sub e )t circ $}\\ p openW { parenthesisY } = open\{ bracef mu{\ n openomega parenthesis} ( T T closing ) parenthesisY := BL, = CBL, closing\omega B(C brace).(2.36) closing\ circ parenthesis\alpha comma{ e T} t \ circ p ( \{\mu n ( T ) \} ),T > 0 \mid \omega \ in Y \} , \\ Y greaterBy 0 bar linear omega extension in Y closing each brace of comma the unitarily invariant weights defines a positive trace with do = By\ l elinear f t \{\ extensionbegin { eacha l i g nof e thed } unitarily& BL invariant [ 0 weights , .. defines\ infty a positive) trace , BL with{ doc hyphen} [ 0 , \ infty ), \\ &- CBL main [M1,∞ 0. , \ infty ) , CBL { c } [ 0 , \ infty ) , ( 2 . 35 ) \\ main MThe sub reader 1 comma should infinity rightly period b e b ewildered by this variety of Dixmier traces . Before we &The Breader ( should C rightly ) b [ e b ewildered0 , by\ infty this variety),B(C) of Dixmier traces period .. Before{ c we} explain[ 0 , \ infty ) \end{ a l i g n e d }\ right . \\ W { Y } = \{explainF {\ theomega origin} of so( many T options) : , = let us\omega clarify the\ circ picturer .\ circ From Appendix\alpha { e A} t \ circ p thewe origin have of so many options comma let us clarify the picture period .. From Appendix .. A .. we have ( V\{\ sub DLmu openn square ( bracket T 1 ) comma\} infinity),T closing parenthesis> 0 =\ Vmid sub DL\omega sub c open\ in squareY bracket\} 1 comma, \ tag infinity∗{$ Y closing = BLparenthesis , V CBL sub D sub , 2 B supset ( shortparallel C ) supset . V( sub 2 PDL . sub open 36 square )$} bracket 1 comma infinity closing parenthesis W sub BL = W \end{ a l i g n ∗} V = V sub BL open square bracket 1 comma infinity closing parenthesisDL[1,∞ =) W subDLc[1 BL,∞) sub c open square bracket 1 comma infinity closing parenthesis cup V sub CDL open square bracket 1 comma infinityVD2 closing⊃ q parenthesis⊃ VPDL =[1,∞ V) sub CDL sub c open square bracket 1 comma infinity closing parenthesis\noindent Equation:By linear open extensionparenthesis 2 eachperiod of37 closing the unitarily parenthesis .. invariant shortparallel weights W sub CBL\quad = W subdefines CBL open a positive square bracket trace 1 comma with do − WBL = WBL[1,∞) = WBL [1,∞) infinitymain closing $ M parenthesis{ 1 , =\ Winfty sub CBL} sub. $ c open square bracket 1 commac infinity closing parenthesis cup V sub D open parenthesis C ∪ closing parenthesis open square bracket 1 comma infinity closing parenthesis = V sub D open parenthesis C closing parenthesis sub c open square The reader should rightly b e b ewildered by this variety of Dixmier traces . \quad Before we explain bracket 1 comma infinity closing parenthesis shortparallelVCDL supset[1,∞) V= subVCDL MDLc[1,∞ sub) open square bracket 1 comma infinity closing parenthesis W the origin of so many options , let us clarify the picture . \quad From Appendix \quad A \quad we have sub B open parenthesis C closing parenthesis = W sub B open parenthesis C closingq parenthesis open square bracket(2.37) 1 comma infinity closing parenthesis = W sub B open parenthesis C closing parenthesis sub c open square bracket 1 comma infinity closing parenthesis WCBL = WCBL[1,∞) = WCBL [1,∞) \ beginWhy{ dida l i such g n ∗} a variety result ? .. Connes quoteright trace theorem .. openc square bracket 1 7 closing square bracket .. used the generalised ∪ limitsV { origiDL hyphen [ 1 , \ infty ) } = V { DL { c } [ 1 , \ infty ) }\\ V { D { 2 }}\supset \ shortparallel \supset V { PDL { [ 1 , \ infty ) }}\\ W { BL } = W { BL [ 1 , nally sp ecified by Dixmier .. open square bracket 2VD 2( closingC)[1,∞) square= VD(C bracket)c[1,∞) period .. Connes lat er introduced Dixmier traces defined by the\ infty set s ) } = W { BL { c } [ 1 , \ infty ) }\\\cup \\ V { CDL [ 1 , \ infty ) } q ⊃ VMDL[1,∞) =CDL V sub{ CDL c open{ squarec } bracket[ 1 1 comma , infinity\ infty closing) parenthesis}\\\ shortparallel open square bracket\ tag ∗{ 2$ comma ( pp 2 period . 303 37 endash ) $ 308}\\ closingW { squareCBL } W = W = W =bracket W comma{ CBL aft er [ explaining 1 , open\ infty squareB bracket(C) } 2B=( commaC)[1 W,∞) { p periodCBLB(C)c[1{ 305,∞c) closing} [ square 1 bracket , \ infty comma following) }\\\ Dixmiercup \\ openV square{ D bracket( CWhy 2 2 ) closing did [ such square 1 a bracket variety , \ infty comma result that?) any}Connes omega= V ’ in trace D{ subD(C) theorem 2 [ 1{ 7 ]c } used[ the 1 generalised , \ infty ) }\\\ shortparallel \supsetwaslimits sufficientV origi t{ oMDL - define nally a{ spDixmier[ ecified 1 trace by , period Dixmier\ infty .. In the [ 2) same 2}}\\ ] . lo cationW Connes{ ..B(C) open lat square er introduced bracket} 2= Dixmiercomma W p{ period tracesB 308 ( closing C ) square [ bracket 1 .., Connes\definedinfty introduced by) the} = set s WCDL{ B(C)[1, ∞)[2, pp . 303 –{ 308c } ] ,[ aft er 1 explaining , \ infty [ 2 , p .) 305} ] , following \end{ a l i g n ∗} c theDixmier idea of measurability [ 2 2 ] , that associated any ω t∈ oD the2 was set sufficient of traces defined t o define by CDL a Dixmier sub c open trace square . bracket In the 1 samecomma lo infinity closing parenthesis open parenthesiscation see [ 2 the , p next . 308 ] Connes introduced the idea of measurability associated t o the set \noindent Why did such a variety result $ ? $ \quad Connes ’ trace theorem \quad [ 1 7 ] \quad used the generalised limits origi − sectionof traces closing defined parenthesis by periodCDLc[1, ∞)( see the next section ) . nallyTranslation spTranslation ecified invariance invarianceby in Dixmier in quoteright\ Dixmierquad [ s original 2’ s 2 original ] sp . ecification\quad sp ecificationConnes is redundant lat is redundant er comma introduced hence , hence the set Dixmier the of funcset of hyphen traces defined by the set s $ CDL { c } [ 1 , \ infty ) [ 2 ,$ pp.303 −− 308 ] , aft er explaining [ 2 , p . 305 ] , following Dixmier [ 2 2 ] , that any t ionalsfunc V - sub t ionals DL openVDL square[1,∞) bracketare generally 1 comma termed infinity theclosing ( original parenthesis ) Dixmier .. are generally traces ,termed while the the open more parenthesis original closing parenthesis$ \omegarestrictive Dixmier\ in tracesD comma{ 2 }$ while the more restrictive was sufficient t o define a Dixmier trace . \quad In the same lo cation \quad [ 2 , p . 308 ] \quad Connes introduced setset V subVCDL CDLc[1 sub,∞) cwe open t ermed squareConnes bracket 1 – comma Dixmier infinity traces closing , in parenthesis [ 2 7 ] , t .... o wedistinguish t ermed Connes them endash . The Dixmier t traces comma in .... openthe squareext idea bracket of measurability 2 7 closing square associated bracket comma t o t theo distinguish set of them traces period defined .... The by t ext $ .... CDL open{[ square 2c 7} ] bracket[ 1 2 7 closing , \ infty square bracket) (also $initiated see thethe next identifications in ( 2 . 3 7 ) , where our aim was t o identify Dixmier and s ealso c t iConnes o initiated n ) . – the Dixmier identifications traces in with open subsets parenthesis of singular 2 period symmetric3 7 closing parenthesis functionals comma [ 2 where 8 , our 2 9aim , was 3 t o identify Dixmier and Connes0 endash ] , i . e .V = W Translation invarianceDL[1,∞) inBL Dixmier ’ s original sp ecification is redundant , hence the set of func − Dixmierand tracesV with subsets= W of. singularThe form symmetric of the functionals functionals openW square bracketi s simpler 2 .. 8 commathan it .. looks 2 9 comma in ( 2 .. . 3 0 closing square bracket t i o n a l sCDL $ Vc[1,∞{) DLCBL [ 1 , \ infty ) }$ \quadBL,CBLare generally termed the ( original ) Dixmier traces , while the more restrictive comma3 i 6period ) . e period V sub DL open square bracket 1 comma infinity closing parenthesis = W sub BL and V sub CDL sub c open square bracket 1 comma infinity closing parenthesis = W sub CBL period .... The form of the functionals W sub\noindent BL commas e CBL t $ i s V simpler{ CDL than{ itc looks} in[ open 1 parenthesis , \ infty 2 period 3) 6} closing$ \ h parenthesis f i l l we t period ermed Connes −− Dixmier traces , in \ h f i l l [ 2 7 ] , t o distinguish them . \ h f i l l The t ext \ h f i l l [ 2 7 ] \noindent also initiated the identifications in ( 2 . 3 7 ) , where our aim was t o identify Dixmier and Connes −− Dixmier traces with subsets of singular symmetric functionals [ 2 \quad 8 , \quad 2 9 , \quad 3 0 ] , i . e $ . V { DL [ 1 , \ infty ) } = W { BL }$

\noindent and $ V { CDL { c } [ 1 , \ infty ) } = W { CBL } . $ \ h f i l l The form of the functionals $ W { BL , CBL }$ i s simpler than it looks in ( 2 . 3 6 ) . Measure Theory in Noncommutative Spaces .... 9 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 9 9 Indeed comma as a result of open square bracket 2 7 closing square bracket and open square bracket 2 .. 9 closing square bracket comma one\ [ can\ r u write l e {3em any}{ Dixmier0.4 pt }\ trace] open parenthesis up t o a constant closing parenthesis as F subIndeed omega , as open a result parenthesis of [ 2 T 7 closing ] and [ parenthesis 2 9 ] , one = omega can write parenlefttp-parenleftbt any Dixmier trace braceleftbt-braceleftmid-bracelefttp ( up t o a constant 1 divided by N sum from) as k = 1 to 2 to the power of N mu k open parenthesis T closing parenthesis bracerighttp-bracerightmid-bracerightbt infinity N = 1 to the\noindent power of parenrightbt-parenrighttpIndeed , as a result comma of [ T 2 greater 7 ] and0 [ 2 \quad 9 ] , one can write any Dixmier trace ( up t o a constant ) as for some Banach limit omega comma and any Connesk=1 endash Dixmier trace using the same formula except with \ [F {\omega } ( T ) = \omega1 X ( \{\ f r a c { 1 }{ N }\sum ˆ{ k = 1 } { 2 ˆ{ N }}\mu a Ces grave-a ro endash Banach limitF ( omegaT ) = ω in({ CBL periodµk(T )}∞N = 1) , T > 0 k ( T ) \}\ infty { Nω } = 1N ˆ{ ) } ,T > 0 \ ] The sets V sub D open parenthesis C closing parenthesis2N open square bracket 1 comma infinity closing parenthesis and V sub MDL open squarefor bracket some 1 comma Banach infinity limit closingω, and parenthesis any Connes were – introduced Dixmier trace with a using different the intention same formula period .. except The paper with open square bracket 2 .. 5 closinga square Ces bracketa` ro – Banach limit ω ∈ CBL. \noindentby A periodfor Carey some comma Banach J period limit Phillips $ comma\omega and the, $ second and author any Connes comma demonstrated−− Dixmier the trace residue using and heat the kernel same formu formula hyphen except with a Ces $ \graveThe{ setsa} $VD( roC)[1−−,∞) andBanachVMDL limit[1,∞) were $ \omega introduced\ in withCBL a different . $ intention . The lationspaper of the [ 2 Dixmier 5 ] trace comma and did so with the typ e I von Neumann algebra open parenthesis L open parenthesis H closing parenthesis comma Tr closing parenthesis replaced \ hspaceby∗{\ A .f Carey i l l }The , J s . e Phillips t s $ V , and{ D the second ( C author ) [ , demonstrated 1 , \ infty the residue) } and$ and heat kernel$ V { MDL [ 1 , withformu any semifinite - lations von Neumannof the Dixmier algebra open trace parenthesis , and did N comma so with tau the closing typ parenthesis e I von with Neumann faithful algebranormal semifinite trace tau comma ..\ infty see ) }$ were introduced with a different intention . \quad The paper [ 2 \quad 5 ] (L(H), Tr ) replaced with any semifinite von Neumann algebra (N , τ) with faithful normal also open square bracket 1 9 comma .. 3 1 comma .. 1 6 closing square bracket period .. The generalised s hyphen numbers rho s open semifinite trace τ, see also [ 1 9 , 3 1 , 1 6 ] . The generalised s− numbers ρs(T ) parenthesis\noindent Tby closing A . parenthesis Carey , of J a . tau Phillips hyphen compact , and operator the second T in N comma author open , demonstratedsquare bracket 8 closing the residue square bracket and comma heat kernel provide formu − of a τ− compact operator T in N , [8], provide a generalisation of singular values , such that lationsa generalisation of the of Dixmier singular values trace comma , and such did that so rho with s open the parenthesis typ e I T von closing Neumann parenthesis algebra = p open $ parenthesis ( L ( open H brace ) mu n , $ ρs(T ) = p({µn(T )}∞ ) for the pair (L(H), Tr ) . To use the generalised s− numbers , one openTr ) parenthesis replaced T closing parenthesism=1 closing brace sub m = 1 to the power of infinity closing parenthesis for the pair open parenthesis L open replaces the sequence ( 2 . 1 ) by the function ( 2 . 3 0 ) . For the semifinite pair (N , τ) parenthesiswith any H semifinite closing parenthesis von comma Neumann Tr closing algebra parenthesis $ ( period N , \tau ) $ with faithful normal semifinite trace only the functionals V ( and all continuous variants in ( 2 . 3 3 ) $ \Totau use the, generalised $ \quad ssee hyphenDL numbers[1,∞) comma one replaces the sequence open parenthesis 2 period 1 closing parenthesis by the function opena l s oparenthesisand [ 1 ( 9 2 . , 2 3\ period 5quad ) ) 3can3 0 closing1 be , defined\quad parenthesis1 . 6 period] . \quad .. For The generalised $ s − $ numbers $ \rho s ( T ) $the o semifinite fThe a $ set pair\tauMDL open[1− parenthesis, ∞$), which compactoperator N i comma s non - tau empty closing $T$ [ 2 parenthesis 5 , in p . only74 $N ] ,the was functionals, a constraint [ 8 V sub ] used DL ,$open in [ 2square provide 5 bracket 1 comma infinity closinga generalisation] parenthesis t o identify open the parenthesisof Dixmier singular and trace allvalues continuouswith residues , such variants thatof a in zeta open $ function\ parenthesisrho s and 2 period( heat T kernel 3 3 closing) asymptotics = parenthesis p ( ( \{\mu n ( T)andsee open\} Section parenthesisˆ{\ 5infty b 2 elow period} { ) .m 3 Later 5 closing = results 1 parenthesis} on)$ zeta closing forthepair functions parenthesis [ 1 can $(5 ] be enabled defined L a period( residue H formulation) ,$ Tr). ToThe usefor set all the MDL Dixmier generalised open square traces bracket in $ the s 1 set comma− V$PDL numbersinfinity[1,∞). It closing i , s still one parenthesis open replaces whether comma the a which sequenceresidue i s non formulation ( hyphen 2 . 1 empty )exists by open the square function bracket ( 2 2 .. . 5 3 0 ) . \quad For commathefor semifinite p period all functionals 74 closing pair square in $ the ( bracket set N comma , \ wastau a constraint) $ only used thein open functionals square bracket $ 2 .. V 5 closing{ DL square [ bracket 1 , t o identify\ infty ) } ( $theV and DixmierDL[1, all∞). trace continuousWhile with residues residue variants of aformulations zeta function in ( 2 wereand . 3 heat able 3 ) kernel to drop asymptotics the Ces opena` ro parenthesis invariance see condition Section 5 ( which b elow closing parenthesis period Later∗ results on zeta functions open square bracket 1 .. 5 closing square bracket enabled a residue formulation\noindenti s unnecessary forand all Dixmier ( 2 . for 3 traces the5 ) in )weak can− beKaramata defined theorem . [ 1 5 , p . 2 71 ] ) , a lat er heat kernel theformulation set V sub PDL due open t square o A bracket . Sedaev 1 comma dropped infinity the closing power parenthesis invariance period requirement It i s still open ( as whether [ 1 a 5 residue , formulation exists for allTheset functionalsProposition $MDL in the 4set . [ 3 ] i1 s unnecessary , \ infty for the) heat ,$ kernel which ) . i snon In [ 3 2− ]empty , heat kernel [ 2 \quad formulations5 , p . 74 ] , was a constraint used in [ 2 \quad 5 ] t o identify thewere Dixmier derived trace for with Connes residues – Dixmier of traces a zeta in function the set s andV heatand kernelV asymptotics. ( see Section 5 V sub DL open square bracket 1 comma infinity closing parenthesis periodCDLc ....[1,∞ While) residueD(C)c formulations[1,∞) were able to drop the Ces grave-a rob invariance elowThe ) condition . set LaterD open results(C)[1 parenthesis, ∞) onwas zeta which used functions in [ 1 [ 6 1 , \quad§ 15 . ] 3 enabled ] . Finally a residue , the formulation set B(C) for all Dixmier traces in thei s unnecessarywas s e t introduced $ V for{ PDL the in weak the [ topaper the 1 power [ 3 , 4 ] of\ withinfty * hyphen an application Karamata) } . $ theorem t o It measurability i open s still square open( bracket see the whether 1 next 5 comma section a residue p period 2 formulation 71 closing square exists for all functionals in the set bracket). closing parenthesis comma a lat er heat kernel formulation due \noindentt o A periodIt will$ Sedaev V b{ e convenientdroppedDL [ the power1 to simplify , invariance\ infty the requirement schematic) } open. ( 2 $ . parenthesis 3\ h 7 f ) i lof l DixmierWhile as open residue square traces bracket . formulations Set 1 .. 5 comma were Proposition able 4 to period drop the Ces 3$ closing\grave square{a} $ bracket ro i invariance s unnecessary condition ( which for the heat kernel closing parenthesis period .. In open square bracket 3 2 closing1 square bracket comma heat kernel formulations were \noindent i s unnecessary for the $ weak ˆ{ ∗ } − $ KaramataV theorem:= V , [ 1 5 , p(2. 38) 2 71 ] ) , a lat er heat kernel formulation due derived for Connes endash Dixmier traces 2 D2 tin o the A set. Sedaev s V sub CDL dropped sub c theopenpower square bracket invariance 1 comma requirement infinity closing ( parenthesis as [ 1 \ andquad V sub5 D, open Proposition parenthesis 4 C. closing 3 ] parenthesis i s unnecessary V1 := VDL[1,∞) = VDL [1,∞) = WBL [1,∞) = WBL[1,∞) = WBL, (2.39) subfor c open the square heat bracket kernel 1 comma ) . \quad infinityIn closing [ 3 parenthesis 2 ]c , heat period kernelc formulations were derived for Connes −− Dixmier traces V := V = V = W = W = W , (2.40) inThe the set set .. Dopen s $V parenthesis{ 2 CDL CCDL closing{ [1c,∞}) parenthesis[CDL 1c[1,∞ open) , squareCBL\ inftyc[1 bracket,∞) ) 1CBL} comma$[1, and∞) infinity $CBL V closing{ D(C) parenthesis .. was used{ c in} .. open[ 1 square , \ infty ) } . $ V := V = V = W = W = W . (2.41) bracket 1 .. 6 comma .. S ..3 1 periodD(C)[1 3,∞ closing) D square(C)c[1,∞ bracket) B period(C)c[1,∞ ..) FinallyB(C comma)[1,∞) .. theB(C) set B open parenthesis C closing parenthesis .. was introduced in the The schematic ( 2 . 3 7 ) is then simplied to Dixmier traces (V ), Connes – Dixmier traces Thepaper s e t open\quad square$ bracketD ( 3 4 C closing ) square [ bracket 1 , with\ infty an application) $ t o\quad measurability1 was used open in parenthesis\quad [ see 1 the\quad next6 section , \quad closing\S \quad 1 . 3 ] . \quad F i n a l l y , \quad the s e t (V ) and Ces a` ro invariant Dixmier traces (V ): parenthesis$ B (2 period C ) $ \quad was introduced in the3 paperIt will [ b e3 convenient 4 ] with to an simplify application the schematic t o open measurability parenthesis 2 period ( see 3 7 the closing next parenthesis section of Dixmier ) . traces period .. Set

Equation: open parenthesis 2 period 38V closing1 parenthesis⊃ V1 ..⊃ V V12 divided⊃ Vby3 2 : = V sub D sub 2 comma Equation: open parenthesis 2 period\ centerline 39 closing{ It parenthesis will b .. e V convenient sub 1 : = V sub to DL simplify open square the bracket schematic 1 comma ( infinity 2 . 3 closing 7 ) parenthesis of Dixmier = V traces sub DL sub . \quad c openSet square} 2 ∪ . (2.42) bracket 1 comma infinity closing parenthesis = W sub BL sub c open square bracket 1 comma infinity closing parenthesis = W sub BL open square\ begin bracket{ a l i g n 1∗} comma infinity closing parenthesis = W sub BL commaVPDL Equation:[1,∞) open parenthesis 2 period 40 closing parenthesis .. V sub 2V : =\ Vf r sub a c { CDL1 }{ open2 square} : bracket = 1 V comma{ D infinity{ 2 }} closing, parenthesis\ tag ∗{$ ( = V 2 sub CDL . sub 38 c open ) $ square}\\ V bracket{ 1 } 1 comma: = infinity V closing{ DL [ 1It i , s known\ inftyWB(C) (} W=CBL ( V W{BLDL, which{ c i} s not[ trivial 1 , see\ [infty 2 7 ] and) [} 3 4= ] . W Hence{ BL the{ c } [ 1 , parenthesisinclusions = W sub CBL sub c open square bracket 1 comma infinity closing parenthesis = W sub CBL open square bracket 1 comma infinity closing\ infty parenthesis) } = = W W sub{ CBLBL comma [ Equation: 1 , open\ infty parenthesis) } 2 period= W 41 closing{ BL parenthesis} , \ tag ..∗{ V$ sub ( 3 : 2 = V sub . D 39 open parenthesis ) $}\\ V { 2 } are strict , i . e . V V V . We do not know whether V 1 ⊃ V is strict , i . C: closing = parenthesis V { CDL open [ square 1 bracket) , 2\ 1infty comma) 3 infinity) } closing= V parenthesis{ CDL ={ Vc sub} D2[ open 1 parenthesis1 , \ infty C closing) parenthesis} = W sub{ c openCBL { c } [ 1e . , \ infty ) } = W { CBL [ 1 , \ infty ) } = W {anythingCBL } i, s \ tag ∗{$ ( 2 . squaregained bracket 1from comma the infinity larger closing set of parenthesis generalised = W limits sub B . open In some parenthesis NCG C papers closing parenthesis Dixmier traces sub c open are square bracket 1 comma infinity40 ) closing $}\\ parenthesisV { 3 } = W: sub = B open V { parenthesisD ( C C closing ) parenthesis [ 1 open , square\ infty bracket) 1} comma= infinity V { D(C) closing parenthesis ={ Wc } [ 1defined , \ withinfty vague) reference} = W t o{ theB(C) typ e of generalised{ c } limit[ b 1 eing , used\ infty , relying) on} the= W { B(C sub B openassumption parenthesis of C measurability closing parenthesis ( next period section ) to make the distinction irrelevant . To our )The [ schematic 1 , open\ infty parenthesis) } 2 period= W 3 7{ closingB(C) parenthesis is} then. \ simpliedtag ∗{$ to ( Dixmier 2 traces . 41 open parenthesis) $} V sub 1 closing \end{knowledgea l i g n ∗} , the classes of functionals in ( 2 . 4 2 ) represent the known relevant distinctions parenthesisin measuring comma Connes the ‘ endash log divergence Dixmier traces of the open trace parenthesis ’ using V the sub 2idea closing of Dixmier parenthesis . and Ces a-grave ro invariant Dixmier traces open parenthesis V sub 3 closing parenthesis : \noindentV sub hlineThe 1 supset schematic V sub 1 ( supset 2 . V 3 sub 7 ) 2 supset is then V sub simplied 3 Equation: to open Dixmier parenthesis traces 2 period $ ( 42 closing V { parenthesis1 } ) .. , 2 $ cup Connes period V−− Dixmier traces sub$ ( PDL V open{ 2 square} ) bracket $ and 1 comma infinity closing parenthesis CesIt i s $ known\grave W sub{a} B$ open ro parenthesis invariant C closing Dixmier parenthesis traces subsetneq $ ( W V sub{ CBL3 } subsetneq) : W $ sub BL comma which i s not trivial comma see open square bracket 2 7 closing square bracket and open square bracket 3 4 closing square bracket period Hence the inclusions \ beginare strict{ a l i commag n ∗} .... i period e period V sub 1 supsetneq V sub 2 supsetneq V sub 3 period .... We do not know whether V 1 divided by 2 supsetV {\ V subr u l 1e { ....3em is}{ strict0.4 commapt }} ....1 i period\supset e periodV .... anything{ 1 }\ i ssupset V { 2 }\supset V { 3 }\\ 2 \cup . \ tag ∗{$ ( 2gained . from 42 the ) larger $}\\ setV of{ generalisedPDL [ limits 1 period , In\ infty some NCG) papers} Dixmier traces are defined \endwith{ a vaguel i g n ∗} reference t o the typ e of generalised limit b eing used comma relying on the assumption of measurability open parenthesis next section closing parenthesis to make the distinction irrelevant period .. To our knowledge comma the classes\noindent of It i s known $W { B(C) }\ subsetneq W { CBL }\ subsetneq W { BL } , $ which i s not trivial , see [ 2 7 ] and [ 3 4 ] . Hence the inclusions functionals in open parenthesis 2 period 4 2 closing parenthesis represent the known relevant distinctions in measuring the quoteleft log divergence\noindent are s t r i c t , \ h f i l l i . e $ . V { 1 }\ supsetneq V { 2 }\ supsetneq V { 3 } . $ \ hof f i lthe l We trace do quoteright not know using whether the idea of $ Dixmier V \ f r period a c { 1 }{ 2 }\supset V { 1 }$ \ h f i l l i s s t r i c t , \ h f i l l i . e . \ h f i l l anything i s \noindent gained from the larger set of generalised limits . In some NCG papers Dixmier traces are defined with vague reference t o the typ e of generalised limit b eing used , relying on the assumption of measurability ( next section ) to make the distinction irrelevant . \quad To our knowledge , the classes of functionals in ( 2 . 4 2 ) represent the known relevant distinctions in measuring the ‘ log divergence of the trace ’ using the idea of Dixmier . 1 0 S . Lord and F . Sukochev

2 . 1 Singular symmetric functionals It would b e remiss t o ﬁnish a section on Dixmier traces without mentioning singular symmetric functionals . Consider the following well - known formula for the canonical trace , a special case of the Lidskii

formula3 : N X Tr(T ) = lim µk(T ), T > 0, (2.43) N→∞ k=1 which defines a weight that is finit e on the ideal L1 of trace class operators . The formula ( 2 . 43 ) makes the positivity and trace property evident ( recall µk(U ∗TU) = µk(T ) for all unitaries U ∈ L(H) and T ∈ L∞), and makes it easy t o see S 7→ Tr (ST ),S ∈ L(H),T ∈ L1, is a positive continuous linear functional on L(H) by the same arguments as for the formula ( 2 . 3 ) . Linearity is not immediately evident from ( 2 . 4 3 ) as it was not for ( 2 . 3 ) ( but is P ∞ evident from the definition of the trace Tr (T ) = mhhm | T hmi, {hm}m=1 an orthonormal basis of H, which will be of relevance in Section 6 ) . Both formula ( 2 . 4 3 ) and ( 2 . 3 ) have the general format

f({µn(T )}), (2.44) where f i s a positive linear functional on the sequence space `1 in the case of Tr , and the sequence space

∞ ∞ ∗ m1,∞ := {x = {xn}n=1 ∈ ` | sup α(x )k < ∞}, k ∗ ∞ where α is the sequence from ( 2 . 1 ) and x is the sequence {| xn |}n=1 rearranged in nonincreasing order , in the case of Trω. ∗ ∗ A Calkin sequence space j i s a subset of c0 such that x ∈ j and y ≤ x implies y ∈ j( called the Calkin property ) [ 1 1 , p . 1 8 ] . It i s known that a subset J of compact ∗ operators i s a two - sided ideal if and only if the sequence space jJ = {x ∈ c0 | x = ∞ {µn(T )}n=1, T ∈ J } is a Calkin space [ 3 7 ] . A symmetric sequence space is a Calkin space e with norm k · ke such that ∗ ∗ y ≤ x im - plies k y k e ≤ k x ke ∀x ∈ e. A symmetric ideal of compact operators E i s a two sided ideal with a norm k · k E such that k STV k E ≤ k S kk T kE k V k, ∀S, V ∈ L(H),T ∈ E. It is known ( actually , only recently when [ 3 8 ] extended the ﬁnit e dimensional result of J . von Neumann in [ 1 2 ] ) that E i s a symmetric ideal if and only if jE is a symmetric sequence space and that ∞ 1 k T k E =k {µn(T )}n=1 k j. For example ` ( resp .m1,∞) is the symmetric sequence space associ - 1 ated t o the symmetric ideal L ( resp .M1,∞). These corresp ondences allow us to form the equation ( 2 . 4 4 ) for any symmetric ideal ∗ E and using 0 < f ∈ jE. The functional deﬁned on E+ by ( 2 . 4 4 ) i s positive , unitarily invariant , continuous ( in the symmetric norm k · kE ), but not necessarily linear . To obtain linearity , and so derive a trace on E, one introduces the notions of symmetric and fully symmetric functionals . If e i s a symmetric sequence space , then a functional 0 < f ∈ e∗ is called symmetric if y∗ ≤ x∗ implies f(y∗) ≤ f(x∗). If x, y ∈ e, we write y ≺≺ x( and say y i s submajorised by x) when

NN X ∗ X ∗ yn ≤ xn ∀N ∈ N. n = 1 n = 1 1 0 .... S period Lord and F period Sukochev \noindent 1 0 \ h f i l l S . Lord and F . Sukochev hline 3There are Lidksii type formulations for the Dixmier trace , which for sake of brevity we will not discuss in this text 2 period, see [ 1 3 .. 5 ]Singular and [ 3 3 symmetric ] . For spectral functionals asymptotic formulae see [ 3 6 , 1 6 ] . \ [ It\ r would u l e {3em b e remiss}{0.4 tpt o}\ finish] a section on Dixmier traces without mentioning singular symmetric functionals period Consider the following well hyphen known formula for the canonical trace comma a special case of the Lidskii \noindentLine 1 formula2 . to 1 the\quad powerSingular of 3 : Line symmetric 2 N Line 3 Tr functionals open parenthesis T closing parenthesis = limint N right arrow infinity sum k = 1 mu k open parenthesis T closing parenthesis comma T greater 0 comma open parenthesis 2 period 43 closing parenthesis \noindentwhich definesIt a would weight bthat e is remiss finit e on t the o ideal finish L to a the section power of 1 on oftrace Dixmier class operators traces withoutperiod .. The mentioning formula open singular parenthesis symmetric 2 period 43 closing parenthesis \noindentmakes thefunctionals positivity and trace . property evident .. open parenthesis recall mu k open parenthesis U to the power of * TU closing parenthesis = mu k open parenthesis T closing parenthesis .. for all unitaries \ hspaceU in L∗{\ openf i parenthesisl l } Consider H closing the followingparenthesis and well T in− Lknown to the power formula of infinity for the closing canonical parenthesis trace comma , and a makes special it easy case t o ofsee Sthe Lidskii mapsto-arrowright Tr open parenthesis ST closing parenthesis comma S in L open parenthesis H closing parenthesis comma T in L to the power of\ [ 1\ begin comma{ isa l a i g positive n e d } formula ˆ{ 3 } : \\ Ncontinuous\\ linear functional on L open parenthesis H closing parenthesis by the same arguments as for the formula open parenthesis 2 period 3 closingTr parenthesis ( T ) period = \lim { N \rightarrow \ infty }\sum { k = 1 }\mu k ( T ) , T Linearity> 0 is not , immediately ( 2 evident . 43 from ) open\end parenthesis{ a l i g n e d 2}\ period] 4 3 closing parenthesis as it was not for open parenthesis 2 period 3 closing parenthesis open parenthesis but is evident from the definition of the trace Tr open parenthesis T closing parenthesis = sum sub m angbracketleft h sub m bar Th sub m right angbracket comma\noindent open bracewhich h sub defines m closing a weightbrace sub that m = 1 is to the finit power e of on infinity the idealan orthonormal $ L ˆ{ basis1 } of$ H comma of trace which class operators . \quad The formula ( 2 . 43 ) makeswill be the of relevance positivity in Section and 6 closingtrace parenthesis property period evident \quad ( r e c a l l $ \mu k ( U ˆ{ ∗ } TU ) = \muBothk formula ( open T parenthesis ) $ \quad 2 periodfor 4 3all closing unitaries parenthesis and open parenthesis 2 period 3 closing parenthesis have the general format Equation: open parenthesis 2 period 44 closing parenthesis .. f open parenthesis open brace mu n open parenthesis T closing parenthesis closing\noindent brace closing$ U parenthesis\ in L comma ( H ) $ and $ T \ in L ˆ{\ infty } ) , $ and makes it easy t o see $ Swhere\mapsto f i s a positive$ Tr linear $ functional ( ST on ) the sequence , S space\ in l toL(H),T the power of 1 in the case of Tr comma\ in andL ˆ the{ 1 sequence} , $ is a positive continuousspace linear functional on $ L ( H ) $ by the same arguments as for the formula ( 2 . 3 ) . m sub 1 comma infinity : = braceleftbig x = open brace x sub n closing brace sub n = 1 to the power of infinity in l to the power of infinity barLinearity supremum is k alpha not openimmediately parenthesis evident x to the power from of ( * 2 closing . 4 parenthesis 3 ) as it k less was infinity not for bracerightbig ( 2 . 3 comma ) ( but is evident from thewhere definition alpha is the of sequence the tracefrom open Tr parenthesis $ ( T 2 period ) 1 = closing\sum parenthesis{ m }\ andlangle x to the powerh of{ *m is}\ the sequencemid Th open{ bracem }\ bar xrangle sub, n\{ bar closingh { bracem }\} sub n =ˆ{\ 1 toinfty the power} { ofm infinity = rearranged 1 }$ an in nonincreasing orthonormal basis of $H , $ which order comma in the case of Tr sub omega period \noindentA Calkin sequencewill be space of j relevance i s a subset of in c sub Section 0 such that 6 ) x . in j and y to the power of * less or equal x to the power of * implies y in j open parenthesis called \ centerlinethe Calkin property{Bothformula closing parenthesis ( 2 . 4 open 3) square and bracket ( 2 . 1 3 1 comma ) have p periodthe general 1 8 closing format square bracket} period It i s known that a subset J of compact operators i s a two hyphen sided \ beginideal{ ifa and l i g n only∗} if the sequence space j sub J = open brace x in c sub 0 bar x to the power of * = open brace mu n open parenthesis T closingf ( parenthesis\{\ closingmu bracen sub( n T = 1 to ) the\} power), of comma\ tag to∗{ the$ power ( of2 infinity . T 44 in J closing ) $} brace .. is a Calkin \endspace{ a l openi g n ∗} square bracket 3 7 closing square bracket period A symmetric sequence space is .. a Calkin space e with norm .. bar times bar sub e .. such that y to the power of * less or equal x to the power\noindent of * .. imwhere hyphen $ f $ i s a positive linear functional on the sequence space $ \ e l l ˆ{ 1 }$ in the case of Tr , and the sequence spaceplies .. bar y bar e less or equal bar x bar sub e forall x in e period .. A symmetric ideal of compact .. operators E i s a two sided ideal with a norm .. bar times bar E .. such that .. bar STV bar E less or equal bar S bar bar T bar sub E bar V bar comma forall S comma V in\ [ L m open{ parenthesis1 , \ Hinfty closing} parenthesis: = comma\{ Tx in E = period\{ .. It isx known{ n }\} ˆ{\ infty } { n = 1 }\ in \ e l l ˆ{\ infty } \midopen parenthesis\sup { k actually}\alpha comma .. only( recently x ˆ{ ∗ when } ..) open k square< bracket\ infty 3 8 closing\} square, \ ] bracket .. extended the finit e dimensional result .. of J period von Neumann in open square bracket 1 2 closing square bracket closing parenthesis that E i s a symmetric ideal if and only if j E is a symmetric sequence space\noindent and thatwhere $ \alpha $ is the sequence from ( 2 . 1 ) and $ x ˆ{ ∗ }$ is the sequence $ \{\mid x bar{ n T}\ bar Emid = bar open\} ˆ brace{\ infty mu n open} { parenthesisn = T 1 closing}$ rearranged parenthesis closing in nonincreasing brace sub n = 1 to the power of infinity bar j sub epsilon periodorder .. For , in example the case l to the of power $ Tr of 1 open{\omega parenthesis} . resp $ period m sub 1 comma infinity closing parenthesis is the symmetric sequence space associ hyphen A Calkinated t o the sequence symmetric space ideal L $ to jthe $ power i s of a 1 subset open parenthesis of $ respc { period0 }$ M sub such 1 comma that infinity $ x closing\ in parenthesisj $ and period $ y ˆ{ ∗ } \ leqThesex corresp ˆ{ ∗ ondences }$ i m allow p l i e s us to $ form y the\ in equationj open ( $ parenthesis c a l l e d 2 period 4 4 closing parenthesis for any symmetric ideal E and theusing Calkin 0 less f propertyin j sub E to ) the [ power 1 1 ,of p period . 1 to 8 the ] power. It of i * s .. known The functional that a defined subset .. on $ E J sub $ plus of .. compact by .. open operatorsparenthesis 2 iperiod s a two − sided 4ideal 4 closing if parenthesis and only .. i if s .. the positive sequence comma .. space unitarily $ invariant j { J comma} = \{ x \ in c { 0 }\mid x ˆ{ ∗ } = \{\continuousmu openn parenthesis ( T ) in the\} symmetricˆ{\ infty norm} ..{ barn times = bar 1 sub ˆ{ E, closing}} T parenthesis\ in commaJ \} but$ not\quad necessarilyi s a linear Calkin period .. Tospace obtain [ linearity 3 7 ] comma . and so derive a trace on E comma one introduces the notions of symmetric and fully symmetric functionals period A symmetricIf e i s a symmetric sequence sequence space space is comma\quad thena aCalkin functional space 0 less f $ in e e to$ the with power norm of * is\quad called symmetric$ \ parallel if y to the\cdot power of *\ parallel less or { e }$ equal\quad x tosuch the power that of $ * y ˆ{ ∗ } \ leq x ˆ{ ∗ }$ \quad im − pimplies l i e s \quad f open parenthesis$ \ parallel y to they power\ parallel of * closing parenthesise \ leq less or\ parallel equal f open parenthesisx \ parallel x to the power{ e }\ of *f closing o r a l l parenthesisx \ in periode . .. $ If x\ commaquad A y insymmetric e comma we ideal write y of prec compact prec x open\quad parenthesisoperators and say $ y E i s $ submajorised i s a two by x sided closing ideal parenthesis when withN N asum norm y sub\quad n to the$ power\ parallel of * less or equal\cdot sum x\ subparallel n to the powerE $ of\ *quad forall Nsuch in N that period\ nquad = 1 n =$ 1\ parallel hline STV \ parallel E 3 sub\ leq There\ areparallel Lidksii typeS formulations\ parallel for the Dixmier\ parallel trace commaT \ whichparallel for sake{ ofE brevity}\ weparallel will not discussV in\ thisparallel , \ f otext r a l commal S,V see open square\ in bracketL(H),T 3 5 closing square bracket and\ in openE square . bracket $ \quad 3 3 closingI t i s square known bracket period For spectral asymptotic( a c t u a lformulae l y , \quad see openonly square recently bracket 3 when 6 comma\quad 1 6 closing[ 3 8 square ] \quad bracketextended period the finit e dimensional result \quad o f J . von Neumann in [ 1 2 ] ) that $ E $ i s a symmetric ideal if and only if $ j E $ is a symmetric sequence space and that

\ hspace ∗{\ f i l l } $ \ parallel T \ parallel E = \ parallel \{\mu n ( T ) \} ˆ{\ infty } { n = 1 }\ parallel j {\ epsilon } . $ \quad For example $ \ e l l ˆ{ 1 } ( $ resp $ . m { 1 , \ infty } ) $ is the symmetric sequence space associ −

\noindent ated t o the symmetric ideal $ L ˆ{ 1 } ( $ resp $ . M { 1 , \ infty } ) . $

These corresp ondences allow us to form the equation ( 2 . 4 4 ) for any symmetric ideal $ E $ and using $ 0 < f \ in j ˆ{ ∗ } { E ˆ{ . }}$ \quad The functional defined \quad on $ E { + }$ \quad by \quad ( 2 . 4 4 ) \quad i s \quad p o s i t i v e , \quad unitarily invariant , continuous ( in the symmetric norm \quad $ \ parallel \cdot \ parallel { E } ) , $ but not necessarily linear . \quad To obtain linearity , and so derive a trace on $ E , $ one introduces the notions of symmetric and fully symmetric functionals .

\noindent If $ e $ i s a symmetric sequence space , then a functional $ 0 < f \ in e ˆ{ ∗ }$ is called symmetric if $ y ˆ{ ∗ } \ leq x ˆ{ ∗ }$ implies $f ( yˆ{ ∗ } ) \ leq f ( x ˆ{ ∗ } ) . $ \quad I f $ x , y \ in e , $ we w rite $ y \prec \prec x ( $ andsay $y$ i s submajorised by $x ) $ when

\ begin { a l i g n ∗} NN \\\sum y ˆ{ ∗ } { n }\ leq \sum x ˆ{ ∗ } { n }\ f o r a l l N \ in N. \\ n = 1 n = 1 \\\ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

$ 3 { There }$ are Lidksii type formulations for the Dixmier trace , which for sake of brevity we will not discuss in this text , see [ 3 5 ] and [ 3 3 ] . For spectral asymptotic formulae see [ 3 6 , 1 6 ] . Measure Theory in Noncommutative Spaces .... 1 1 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 1 1 1 1 A symmetric sequence space e is called fully symmetric if x in e and y prec prec x implies y in e \ [ open\ r u lparenthesis e {3em}{0.4 the pt Calkin}\ ] property with rearrangment order replaced by Hardy endash Littlewood endash P acute-o lya submajori hyphen sation .. open square bracket 3 9 closing square bracket closing parenthesis period .. A functional 0 less f in e to the power of * .. on a fully symmetricA symmetric sequence space sequence e is .. called space fullye is called fully symmetric if x ∈ e and y ≺≺ x implies \noindenty ∈ Ae symmetric( the Calkin sequence property space with rearrangment $ e $ is called order replaced fully symmetric by Hardy – if Littlewood $ x \ –in P e $ and $ y \prec symmetric if y prec prec x implies f open parenthesis y to the power of * closing parenthesis∗ less or equal f open parenthesis x to the power of\prec * closingo´ lyax$ parenthesis submajori implies period - sation $y .. Both\ l[into 3 9 the ] )e power . $ Aof1 functional and m sub 10 comma< f infinity∈ aree fullyon asymmetric fully symmetric sequence ( thesequence Calkin space propertye is with called rearrangment fully order replaced by Hardy −− Littlewood −− P $ \acute{o} $ lya submajori − spaces period .. The ideal E is called fully∗ symmetric∗ if j E is a fully1 symmetric sequence space period .. Finally comma s aa t symmetric i osymmetric n \quad functional[ if 3y 9≺≺ 0 ] lessx )implies .f in\quad e tof the(yA) power functional≤ f of(x *) is. calledBoth $ singular0 ` and< if fmf vanishes1,∞ \arein onfully alle symmetricˆfinit{ ∗ e sequences}$ \ sequencequad periodon a fully symmetric sequence space $ eIn $ openspaces i s square\quad . bracket Thecalled ideal2 .. 8 fully commaE is called 2 9 comma fully .. symmetric 3 0 closing square if jE bracketis a fully and symmetric open square bracketsequence 4 0 space comma . p period 77 closing square bracketFinally to the power , of 4 comma it was shown open parenthesis singular closing parenthesis symmetric linear functionals 0 less f in j sub E to \noindenta symmetricsymmetric functional if $0 y < f \∈prece∗ is called\prec singularx$ if impliesf vanishes $f on all finit( yˆ e sequences{ ∗ } ) . \ leq f ( x ˆ{ ∗ } the power of * 4 )correspond . $ In\quad [ t 2 o openBoth 8 , parenthesis 2 9 $ ,\ e l 3 lsingular 0ˆ{ ] and1 closing}$ [ 4 and 0 parenthesis , p $. m77] positive,{it1 was linear , shown\ unitarilyinfty ( singular} invariant$ are ) symmetric functionals fully symmetric on linear fully symmetric sequence ideals E 0 < f ∈ j∗ spacesusingfunctionals the . formula\quad openThe parenthesis idealE correspond $ 2 E period $ t is o4 4 ( called closing singular parenthesis fully ) positive symmetric period linear unitarily if $ j invariant E $ is functionals a fully symmetric sequence space . \quad F i n a l l y , Nowon comma fully thesymmetric relevant question ideals E i susing whether the all formula Dixmier ( traces 2 . 4 open 4 ) . parenthesis V sub 1 closing parenthesis on M sub 1 comma infinity are\noindent constructedNowa from ,symmetric the relevant functional question i s $ whether 0 < allf Dixmier\ in tracese ˆ{ ∗(V }1)$on isM called1,∞ are constructedsingular if $ f $ vanishes on all finit e sequences . singularfrom .. singular symmetric functionals symmetric .. 0 functionals less f in m * sub0 1 comma< f infinity∈ m period∗1,∞. .. TheThe .. answer answer is .. is yes period yes . .. In .. fact comma .. we nowIn [ knowIn 2 \quad fact8 , , we2 9 now , \ knowquad 30]and[40,p $. 77 ]ˆ{ 4 } , $ it was shown ( singular ) symmetric linear functionals $ 0f openf<({µn parenthesis(Tf )}) defines\ in openj bracea ˆDixmier{ mu ∗ } n{ open traceE } parenthesis$ if and only T closing if 0 < parenthesis f ∈ m∗1,∞ closingis a singular brace closingfully parenthesissymmetric defines a Dixmier trace if and onlycorrespond iffunctional 0 less f in t m o( * a sub ( result singular 1 comma of the infinity ) second positive is a author singular linear , Nfully . Kalton symmetric unitarily and A invariant . Sedaev , tfunctionals o appear ) . on ( fully Fully symmetric ideals $ Efunctional $) symmetric open parenthesis singular a functionalsresult of the second on author ( fully comma ) N symmetric period Kalton and ideals A period thus Sedaev generalise comma t o appear closing parenthesis periodusingthe .. open the constructionparenthesis formula ( Fully 2 . closing of 4 Dixmier 4 )parenthesis . ( other symmetric generalisations exist , see [ 3 1 , 4 1 ] ) singular. functionalsAnother relevant .. on .. open question parenthesis i s fully whether closing all parenthesis positive .. linear symmetric unitarily .. ideals invariant .. thus .. generalise singular .. the .. construction .. of NowDixmier ,functionals the relevant on questionM1,∞ constructed i s whether from all singular Dixmier symmetric traces functionals $ ( V { are1 Dixmier} ) $ traces on $ M { 1 , \ infty }$ areopen constructed. parenthesis The answer other from generalisations i s no . There .. exist are comma singular .. see symmetric .. open square functionals bracket 3 1 comma0 < f ∈ ..m 4∗ 11 closing,∞ which square are bracket closing parenthesis periods i n gnot u.. lAnother a r fully\quad symmetric relevantsymmetric .. question , yet functionalsf i( s{µn whether(T )}) defines all\ positivequad a singular$ 0 < positivef \ linearin m{ ∗ } { 1 , \ infty } . $ \quad The \quad answer i s \quad yes . \quad In \quad f a c t , \quad we now know unitarily invariant functional on M1,∞ [40, p . 79 ] , i . e . we now know linear unitarily invariant .. singular functionals .. on M sub∗ 1 comma infinity .. constructed from singular symmetric \noindentfunctionals are$ f Dixmierthere ( exist traces\{\ singular periodmu .. traces Then answer (0 < T ρ i s∈ no M ) period1,∞ \}that .. There are) $ not are defines Dixmier singular asymmetric traces Dixmier . functionals trace ( 2 . 45 if ) and only if $ 0 < 0 lessf In f particular in\ in m *m sub{ 1 ∗ , comma [ } 4{ 01 , infinity p . ,79 which ]\ informsinfty are not} us$ fully that is symmetric therea singular exists comma a fully nonyet f open- symmetrictrivial parenthesis symmetric open brace functional mu n open parenthesis T closing parenthesis0 < f closingp ∈ brace closing parenthesis defines a singular positive linear \noindent functional (1 a∞ result of the second author , N1 .∞ Kalton and A . Sedaev , t o appear ) . \quad ( Fully ) symmetric unitarilym∗1,∞ invariantsuch that functionalfp({ n } onn=1 M) sub = 0 1for comma the harmonic infinity open sequence square bracket{ n }n 4=1 0∈ commam1,∞( pand period hence 79 closing it i s square not bracket comma i period singular functionals5 \quad on \quad ( f u l l y ) \quad symmetric \quad i d e a l s \quad thus \quad g e n e r a l i s e \quad the \quad construction \quad o f Dixmier e periodfully we nowsymmetric know ). Setting f = f1 + fp, where f1 is any symmetric functional on m1,∞ for (there otherwhich exist generalisations singular traces 0 less\quad rho ine M x isub s t 1 , comma\quad infinitysee \ toquad the power[ 3 1 of *, that\quad are4 not 1 Dixmier ] ) . traces\quad periodAnother .. open relevant parenthesis\quad 2 question i s whether all positive linear unitarily1 ∞ invariant \quad singular functionals \quad on $ M { 1 , \ infty }$ \quad constructed from singular symmetric periodf 451( closing{ }n=1 parenthesis) = 1, yields a symmetric functional which cannot be fully symmetric . The positive functionalsn are Dixmier traces . \quad The answer i s no . \quad There are singular symmetric functionals In particularsingular commatrace ρ opendefined square by bracketρ(T ) 4 = 0 (f comma1 + fp)( p{ periodµn(T )} 79) is closing not a square Dixmier bracket trace informs , yet usρ that(T ) = there 1 for exists a non hyphen trivial symmetric$ 0any< functional positivef 0compact\ in less fm sub{ operator ∗ p in } { 1T with , \ singularinfty } values$ which given are by not the harmonic fully symmetric sequence , ( yetsuch $ f ( \{\mu nm ( *as sub Tthe 1 comma inverse ) \} infinity of the) such $ squarethat defines rootf sub of ap open the singular Laplacian parenthesis positive onopen the brace linear 1 - 1 t divided orus R by/Z) n. closing brace sub n = 1 to the power of infinity closing3 parenthesis Measurability = 0 for the harmonic sequence open brace 1 divided by n closing brace sub n = 1 to the power of infinity in m sub 1 comma infinity\noindentMeasurability open parenthesisunitarily , andas invariant defined hence it by i s notfunctional Connes [ 2 , on p . 308 $ M ] ,{ i s1 the ,notion\ infty that } [ 4 0 ,$ p.79],i.e.wenowknow fully symmetric to the power of 5 closing parenthesis period .. Setting f = f 1 plus f sub p comma where f 1 is any symmetric functional on m\ hspace sub 1 comma∗{\ f i infinityl l } there for which exist singular traces $ 0 < \rho \ in M ˆ{ ∗ } { 1 , \ infty }$ that are not Dixmier traces . \quad ( 2 . 45 ) 0 f 1 open parenthesis open brace 1 dividedTr byω( nT ) closing = Trω0 brace(T ) ∀ subω, ω n∈ =B, 1 to the power of infinity closing parenthesis(3.1) = 1 comma yields a symmetric\noindent functionalIn particular which cannot , be[ 4 fully 0symmetric , p . 79 period ] informs .... The us positive that there exists a non − trivial symmetric functional $ 0singularwhere< tracef B {i rho sp some defined}\ subsetin by$ rho of open generalised parenthesis limits T closing which parenthesis defines a Dixmier = open parenthesis trace ( see f 1 the plus previous f sub p closing parenthesis open parenthesissection open ) brace. Specifically mu n open parenthesis , Connes T used closing the parenthesis set B = CDL closingc[1, brace∞) [2closing, p . 305parenthesis ] . isA not compact a Dixmier trace comma yet rho \noindent $m{ ∗ } { 1 , \ infty }$ such that $ f { p } ( \{\ f r a c { 1 }{ n }\} ˆ{\ infty } { n open parenthesisoperator TT closing∈ M1,∞ parenthesissatisfying = 1( for3 . 1 ) is called B− measurable . =any 1 positive} With) compact reference= 0 operator $ t for o the T the with noncommutative harmonic singular values sequence given integral by the $ (\{\ 1harmonic . 1f ) r a ,sequence c one{ 1 desires}{ openn }\} thatparenthesis theˆ{\ products suchinfty as } { n = 1 }\ in −n m the{ a1 inversehDi , ofare the\ infty square(B−)} rootmeasurable( of $ the and Laplacian , hence so on it thatthe i 1 s hyphen not the t orus formula R slash Z ( closing 1 . 1 ) parenthesis i s independent period 00 fully3 ..of Measurability the $ symmetric generalised ˆ limit{ 5 } used) . . $That\quad the projSetting ections $f in A =( using f 1the extension + f { (p 1} . 6 ),$ where $f 1$ −n isMeasurability any) symmetric have commaa unique functional as defined value corresponding by on Connes $ m open{ tsquare1 o the , bracket ‘ infinitesimal\ infty 2 comma}$ p ’ f period o rhD whichi 308parallels closing square the ideabracket that comma i s the notion that Equation:the Lebesgue open parenthesis measurable 3 period set 1 s closing are those parenthesis with a .. unique Tr sub omega‘ size ’ open based parenthesis on the metric T closing . parenthesis = Tr sub omega prime open\noindent parenthesis$ T f closing 1 parenthesis ( \{\ forallfDefine r a omegac { 1 the}{ commaTauberiann }\} omega toˆ{\operators theinfty power of} prime{ n in = B comma 1 } ) = 1 , $ yields a symmetric functional which cannot be fully symmetric . \ h f i l l The positive where B i s some subset of generalised limits which defines a Dixmier trace open parenthesis see the previous \noindent singular trace $ \rho $ defined by $n=1\rho ( T ) = ( f 1 + f { p } )( section closing parenthesis period+ .. Specifically comma Connes1 X used the set∞ B = CDL sub c open square bracket 1 comma infinity closing T := {0 < T ∈ M1,∞|{ µn(T )} ∈ c}. (3.2) parenthesis\{\mu openn square ( bracket T ) 2 comma\} p) period $ is 305 notlog(1 closing a + Dixmiersquarek) bracket tracek=1 period , yet .. A compact $ \rho operator( T ) = 1 $ f o r k anyT in positive M sub 1 comma compact infinity operator satisfying open $ T parenthesis $ with singular3 period 1 closing values parenthesis given by is called the B harmonic hyphen measurable sequence period ( such as theWith inverse reference of t o the the noncommutative square root of integral the open Laplacian parenthesis on 1 the period 1 − 1torus closing parenthesis $R / comma Z one ) desires .$ that the products a angbracketleft4These Dtexts right called angbracket symmetric to the ( resp power . rearrangement of minus n invariant ) the functionals which are here called fully symmetric \noindentare( .. resp open .3 symmetric parenthesis\quad )Measurability . B hyphen closing parenthesis measurable comma .. so .. that .. the .. formula .. open parenthesis 1 period 1 closing parenthesis5All .. isequences s .. independent in the unit .. of ball the of generalisedm1,∞ are limit submajorised .. used period by the harmonic sequence . A non - trivial conti \noindent Measurability , as defined by Connes [ 2 , p . 308 ] , i s the notion that That- the nuous proj functional ectionsin on Am to1,∞ thethat power vanishes of prime on the prime harmonic open parenthesis sequence is notusing fully the symmetric extension , open else non parenthesis - triviality 1 periodis 6 closing parenthesis closingcontradicted parenthesis . .. have a unique value corresponding t o \ beginthe quoteleft{ a l i g n ∗} infinitesimal quoteright .. angbracketleft D right angbracket to the power of minus n parallels the idea that the Lebesgue measurableTr {\omega set s are} those( with T ) = Tr {\omega }\prime (T) \ f o r a l l \omega , \omega ˆ{\prime } \ ina uniqueB, quoteleft\ tag ∗{ size$ quoteright ( 3 based . 1on the ) metric $} period \endDefine{ a l i theg n ∗} Tauberian operators Line 1 T to the power of plus : = open brace 0 less T in M sub 1 comma infinity vextendsingle-vextendsingle-vextendsingle open brace 1 divided\noindent by logwhere open parenthesis $ B $ 1 i plus s somek closing subset parenthesis of generalised sum from n = 1 limits to k mu n which open parenthesis defines T a closing Dixmier parenthesis trace closing ( see brace the sub previous ks = e c1 t to i o then ) power . \quad of infinitySpecifically in c closing brace , Connes period open used parenthesis the set 3 period $ B 2 = closing CDL parenthesis{ c } Line[ 2 hline 1 , \ infty )[ 24 sub , $ These p . texts 305 called ] . symmetric\quad A open compact parenthesis operator resp period rearrangement invariant closing parenthesis the functionals which are here called$ T fully\ in M { 1 , \ infty }$ satisfying ( 3 . 1 ) is called $B − $ measurable . symmetric open parenthesis resp period symmetric closing parenthesis period With5 sub reference All sequences t o in the the unit noncommutative ball of m sub 1 comma integral infinity ( are1 . submajorised 1 ) , one by desires the harmonic that sequence the products period .. A $ non a hyphen\ langle trivial Dconti\ hyphenrangle ˆ{ − n }$ arenuous\quad functional$ ( on m B sub− 1 comma) $ infinity measurable that vanishes , \ onquad the harmonicso \quad sequencethat is\quad not fullythe symmetric\quad commaformula else\ nonquad hyphen( 1 triviality . 1 ) \quad i s \quad independent \quad of the generalised limit \quad used . isThat the proj ections in $ A ˆ{\prime \prime } ( $ using the extension ( 1 . 6 ) ) \quad have a unique value corresponding t o thecontradicted ‘ infinitesimal period ’ \quad $ \ langle D \rangle ˆ{ − n }$ parallels the idea that the Lebesgue measurable set s are those with a unique ‘ size ’ based on the metric .

\ centerline { Define the Tauberian operators }

\ [ \ begin { a l i g n e d } T ˆ{ + } : = \{ 0 < T \ in M { 1 , \ infty }\arrowvert \{\ f r a c { 1 }{\ log ( 1 + k ) }\sum ˆ{ n = 1 } { k }\mu n ( T ) \} ˆ{\ infty } { k = 1 }\ in c \} . ( 3 . 2 ) \\ \ r u l e {3em}{0.4 pt }\end{ a l i g n e d }\ ]

$ 4 { These }$ texts called symmetric ( resp . rearrangement invariant ) the functionals which are here called fully symmetric ( resp . symmetric ) .

$ 5 { All }$ sequences in the unit ball of $m { 1 , \ infty }$ are submajorised by the harmonic sequence . \quad A non − trivial conti − nuous functional on $ m { 1 , \ infty }$ that vanishes on the harmonic sequence is not fully symmetric , else non − triviality is contradicted . 1 2 .... S period Lord and F period Sukochev \noindenthline1 21 2 \ h f i l l S . Lord and F . Sukochev S . Lord and F . Sukochev By construction comma for a subset of generalised limits B subset S sub infinity open parenthesis l to the power of infinity closing parenthesis comma\ [ \ r u l e {3em}{0.4 pt }\ ] Tr sub omega open parenthesis T closing parenthesis = limint k right arrow infinity 1 divided by log open parenthesis 1 plus k closing B ⊂ S (`∞), parenthesisBy construction sum from n = ,1 for to k a mu subset n open of parenthesis generalised T closing limits parenthesis∞ forall omega in B comma T in T to the power of plus period \noindent By construction , for a subset of generalised limits $ B \subset S {\ infty } ( \ e l l ˆ{\ infty } Hence the Tauberian operators are B hyphen measurablen=1 for any B subset S sub infinity open parenthesis l to the power of infinity closing ) , $ 1 X + parenthesis period .. We know fromTrω open(T ) = parenthesis lim 2 periodµn 42( closingT ) ∀ω parenthesis∈ B,T ∈ T . k→∞ log(1 + k) in the previous section there are three open parenthesisk possibly five closing parenthesis main varieties of Dixmier trace period .. Then comma\ [ Tr {\omega } ( T ) = \lim { k \rightarrow \ infty }\ f r a c { 1 }{\ log ( 1 + k B− B ⊂ S (`∞). ) }\a prioriHencesum commaˆ the{ n thereTauberian = are 1three} operators{ openk }\ parenthesis aremu n possiblymeasurable ( five T closing for ) any parenthesis\ f o r a l l ∞ versions\omega ofWe measurability know\ in fromB,T period ( 2 . .. Define\ in T ˆ{ + } . \K] sub42 )i : in = the open previous brace T sectionin M sub there 1 comma are three infinity ( bar possibly f open five parenthesis ) main T varieties closing parenthesis of Dixmier = trace const . forall f in V sub i closing brace commaThen ,i =a 1 priori divided, by there 2 sub are comma three 1 comma ( possibly 2 comma five )3 versionscomma PDL of measurability open square bracket . 1 Define comma infinity closing parenthesis open K := {T ∈ M | f (T ) = const ∀ f ∈ V }, i = 1 1, 2, 3,PDL[1, ∞) (3.3) parenthesis 3 period 3 closing parenthesisi 1,∞ i 2 , \noindent Hence+ the Tauberian operators are $ B − $ measurable for any $ B \subset S {\ infty } andand let K let subK i todenote the power0 < of T plus∈ Ki denote. We concentrate 0 less T in K on sub positive i period operators.... We concentrate for the on moment positive . operators Then , for the moment period .... Then( \ commafrome l l ˆ ({\ 2 .infty 42 ) ,} we) know . that $ \quad Weknow from ( 2 . 42 ) infrom the open previous parenthesis section 2 period there 42 closing are parenthesis three ( comma possibly we know five that ) main varieties of Dixmier trace . \quad Then , a priori , there are three+ ( possibly five ) versions+ + of measurability+ . \quad Define T to the power of plus subset 1T from hline⊂ 1 toK+ K to the⊂ power K1 of⊂ plus K subset2 ⊂ K sub K3 1 to the power of plus subset K sub 2 to the power of plus subset K sub 3 to the power of plus 2 ∩ (3.4) \ hspace2 cap open∗{\ f parenthesis i l l } $ K 3{ periodi } 4 closing: = parenthesis\{ T \ in M { 1 , \ infty }\mid $ f $ ( T ) = $ const $ \ f o r a l l $ f $ \ in V { i }\} , i = \ f r a c { 1 }{ 2 } { , } 1 , 2 , 3 , K sub PDL open square bracket 1 comma infinity closingK+ parenthesis to the power of plus PDLwhere [ the 1 inclusions , may\ infty or may) not be ( strict 3 period .PDL .. 3 In[1 open,∞ )) $ square bracket 2 7 closing square bracket comma .. our main result was that Kwhere sub 2 to the the inclusions power of plus may = T or to may the power not be of plus strict comma . In [ 2 7 ] , our main result was that \noindent+ and+ l e t $ K ˆ{ + } { i }$ denote $ 0 < T \ in K { i } . $ \ h f i l l We concentrate on positive operators for the moment . \ h f i l l Then , identifyingK2 = ConnesT , identifying quoteright Connes definition ’ of definition measurability of measurability in open square in bracket [ 2 ] for 2 closing positive square operators bracket withfor positive operators with the Tauberianthe Tauberian operators . Specifically we introduced the notion of the Ces a` ro limit \noindentoperatorsproperty periodfrom [ 2 ..( 7Specifically 2 , p . . 42 97 ) ] we and , we introduced any know subset that the notion of Banach of the Ceslimits grave-aB satisfying ro limit property this property open square has bracketB− 2 7 comma p period 97 closingmeasurable square bracket positive and any operators ( using the equivalence VDL[1,∞) = WBL) equivalent t o Tauberian \ [subset Toperators ˆ{ of+ Banach}\ .subset limits The B set satisfying1 ˆCBL{\ r thisof u l Ces e property{3ema` }{ro0.4 has – pt B hyphen}} { K measurable ˆ{ + }}\ positivesubset operatorsK open ˆ{ + parenthesis} { 1 }\ usingsubset K ˆ{ + } { 2 } \subsettheBanach equivalenceK ˆlimits{ V+ sub} satisfy DL{ 3 open}\ the] square Ces bracketa` ro limit 1 comma property infinity . closing From parenthesis the paper = ofW subE . BL Semenov closing parenthesis and the equivalent t o Tauberian operatorssecond period .. The set CBL of Ces grave-a ro endash + + Banachauthor limits [ 3 satisfy 4 ] , the we Ces now a-grave know ro that limit propertyK3 6= T period. From From thethis paper , the of schematic E period Semenov ( 3 . and4 ) the can second be \ hspaceauthorreduced∗{\ openf i square lto l } $ bracket 2 \ 3cap 4 closing( square 3 bracket . 4 comma ) $ we now know that K sub 3 to the power of plus negationslash-equal T to the power of plus period .. From this comma the schematic open parenthesis 3 period 4 closing parenthesis can be reduced to \ [T K to ˆ{ the+ power} { ofPDL plus = [ 1 from 1 hline , to\ Kinfty to the power) }\ of] plus = K sub 1 to the power of plus = K sub 2 to the power of plus subsetneq K + + + + sub 3 to the power of plus Equation:T open= parenthesis 1K+ 3 period= K 51 closing= parenthesisK2 ( K ..3 2 cap comma K sub PDL open square bracket 1 comma infinity closing parenthesis to the power of plus 2 ∩ , (3.5) \noindentso that thewhere set of positive the inclusions measurable operators may or according may not to be Connes strict endash . \ Dixmierquad In traces [ 2 open 7 ] parenthesis , \quad andour any main result was that K+ $ Klarger ˆ{ + set} of{ Dixmier2 } = traces T closing ˆ{ + parenthesis} , $ i s equivalent to the TauberianPDL operators[1,∞) comma but the set of positive identifyingmeasurable operators Connes for ’ Ces definition grave-a ro invariant of measurability Dixmier traces in is strictly [ 2 ] larger for than positive the Tauberian operators with the Tauberian o p e rso a t o that r s . the\quad set ofSpecifically positive measurable we introduced operators the according notion to of Connes the Ces – Dixmier $ \grave traces{a} ($ and ro limit property [ 2 7 , p . 97 ] and any operatorsany larger period ..set The of Ces Dixmier a-grave traces ro limit ) property i s equivalent is sufficient to the for equality Tauberian with operators the Tauberian , but operators the set comma of it subseti s an open of questionBanach as limits t o whether $ B the $ property satisfying is a necessary this condition property open has parenthesis $ B the− $ set measurableB open parenthesis positive C closing operators parenthesis ( using positive measurable operators for Ces a` ro invariant Dixmier traces is strictly larger than ofthe Ces grave-a equivalence ro $ V { DL [ 1 , \ infty ) } = W { BL } ) $ equivalent t o Tauberian operators . \quad The s e t $CBL$the Tauberian of Ces $ operators\grave{a} .$ The ro −− Ces a` ro limit property is sufficient for equality with the invariantTauberian = sub t operators o the Banach , it sub i Tauberian s an open limits question does not as operators t o whether satisfy i thes also property the an open is to a the necessary power of property and ? period K sub 3 to the power of plus equal-negationslash T to the power of plus closing parenthesis period .... Whether K sub PDL open square bracket condition ( the set B(C) of Ces a` ro 1\noindent comma infinityBanach closing limits parenthesis satisfy to the power the Ces of plus $ ....\grave i s {a} $ ro limit property . From the paper of E . Semenov and the second invariant= Banach limitsdoesnotoperatorssatisfy thepropertyand?. K+ 6= T +). Whether It i s interestingtothe t o note thatTauberian some notions of measurability canisalso excludeanopen the Tauberian3 opera hyphen \noindentK+ author [ 3 4 ] , we now know that $Kˆ{ + } { 3 }\not= T ˆ{ + } . $ \quad From this , the schematic ( 3 . 4 ) can be reduced to t orsPDL period[1,∞ From) the last section on symmetric functionals comma since the set of Dixmier traces corresp ondsi s t o the setIt ofi fully s interesting symmetric functionals t o note that 0 less some f in m notions * sub 1 commaof measurability infinity comma can the exclude Tauberian the operators Tauberian T to the power of plus .. are the\ beginopera{ a l i g - n ∗} Tset ˆ{t of+ ors 0} less .= From T in 1 M the ˆ sub{\ last 1r u comma section l e {3em infinity}{ on0.4 symmetric such pt }} that{ thefunctionalsK ˆ open{ + parenthesis}} , since= K 2the period ˆ{ set+ of 4} .... Dixmier{ 1 4 closing} = traces parenthesis K corresp ˆ{ + i} s{ independent2 }\ subsetneq of the fully K ˆ{ + } { 3 }\\ 2 \cap , \ tag ∗{$ ( 3 . 5 ) $}\\ K ˆ{ + } { PDL [ 1+ , \ infty ) } symmetriconds functional t o the f periodset of fully symmetric functionals 0 < f ∈ m∗1,∞, the Tauberian operators T \endGiven{area l i .... g the n open∗} parenthesis 2 period 4 5 closing parenthesis comma .... we could try t o widen this notion period .... For example comma .... let .... Vset sub of sym0 < to T the∈ M power1,∞ such of plus that .... the to b ( e 2 the .4 set .... of all 4 ) i s independent of the fully symmetric \noindentpositivefunctional singularso thatf. traces the rho set on M of sub positive 1 comma infinity measurable such that operators rho open parenthesis according T closing to Connesparenthesis−− =Dixmier f open parenthesis traces open ( and brace any larger set of Dixmier traces ) i s equivalent to the Tauberian operators+ , but the set of positive mu n openGiven parenthesis ( 2 . 4 T 5 closing) , we parenthesis could try closingt o widen brace this closing notion parenthesis . For forallexample T greater , let 0V forsym someto b symmetric e the set measurablefunctionalof all 0 less operators f in m * sub for 1 comma Ces $ infinity\grave period{a} ..$ Then ro set invariant Dixmier traces is strictly larger than the Tauberian o p e r a t o r s . \quad The Ces $ \grave{a} $ ro limit property is sufficient for equality with the Tauberian operators , it K subpositive sym to singular the power traces of plus :ρ =on openM brace1,∞ such 0 less that T in Mρ(T sub) 1 = commaf({µn( infinityT )})∀T bar > 0 rhofor open some parenthesis symmetric T closing parenthesis = const i s an open question as t o whether the property is a necessary condition ( the set $ B ( C ) $ forall rhofunctional in V sub sym0 < to f ∈ them∗ power1,∞. ofThen plus closing set brace period o fWe Ces are informed $ \grave by{ opena} $ square ro bracket 4 0 comma p period 79 closing square bracket that T to the power of plus negationslash-propersubset + + K sub sym to the power of period toK the power:= {0 < of T plus∈ M ..1 So,∞ comma| ρ(T ) = widening const∀ρ the∈ V notion}. of measurable in this \noindent $ i n v a r i a n t { = } {symt o the } Banach { Tauberiansym } limits does not { o p e r a t o r s } s a t i s f y { i manner excludes some Tauberian operators period We+ are indirectly+ informed by this result that there T 6⊂ K . sexist aWe l s singular o are} informedthe traces{ onan Mby sub [ 4 open 1 0 comma , p}ˆ .{ 79 infinityproperty ] that that do} notandsym arise{ ? from}So ,. a widening generalised K ˆ{ + the limiting} { notion3 procedure}\ ofne measurable appliedT ˆ{ + } ) . $ \ h f i l l Whether $ Kt o ˆ{in the+ this sequence} { mannerPDL open excludes[ brace 1 1 divided some , \ Tauberian byinfty log open) operators parenthesis}$ \ h f . i1 lWe plusl i are n s closing indirectly parenthesis informed sum by sub this k = result 1 to the power of n mu k open parenthesisthat T there closing parenthesis closing brace sub n = 1 to the power of period to the power of infinity \ hspace ∗{\ f i l l } It i s interesting t o note that some notions of measurability can exclude the Tauberian opera −

\noindent t ors . From the last section on symmetric functionals , since the set of Dixmier traces corresp onds t o the set of fully symmetric functionals $ 0 < f \ in m{ ∗ } { 1 , \ infty } , $ the Tauberian operators $ T ˆ{ + }$ \quad are the

\noindent s e t o f $ 0 < T \ in M { 1 , \ infty }$ such that the ( 2 . 4 \ h f i l l 4 ) i s independent of the fully symmetric functional $ f . $

\noindent Given \ h f i l l ( 2 . 4 5 ) , \ h f i l l we could try t o widen this notion . \ h f i l l For example , \ h f i l l l e t \ h f i l l $ V ˆ{ + } { sym }$ \ h f i l l to b e the s e t \ h f i l l o f a l l

\noindent positive singular traces $ \rho $ on $ M { 1 , \ infty }$ such that $ \rho (T ) = f ( \{\mu n ( T ) \} ) \ f o r a l l T > 0 $ for some symmetric functional $ 0 < f \ in m{ ∗ } { 1 , \ infty } . $ \quad Then s e t

\ [ K ˆ{ + } { sym } : = \{ 0 < T \ in M { 1 , \ infty }\mid \rho ( T ) = const \ f o r a l l \rho \ in V ˆ{ + } { sym }\} . \ ]

\noindent Weare informed by [ 4 0 , p . 79 ] that $Tˆ{ + }\not\subset K ˆ{ + } { sym ˆ{ . }}$ \quad So , widening the notion of measurable in this manner excludes some Tauberian operators . We are indirectly informed by this result that there

\noindent exist singular traces on $ M { 1 , \ infty }$ that do not arise from a generalised limiting procedure applied

\noindent t o the sequence $ \{\ f r a c { 1 }{\ log ( 1 + n ) }\sum ˆ{ n } { k = 1 }\mu k ( T ) \} ˆ{\ infty } { n = 1 ˆ{ . }}$ exist singular traces on M1,∞ that do not arise from a generalised limiting procedure applied 1 Pn ∞ t o the sequence { log(1+n) k=1 µk(T )}n=1. Measure Theory in Noncommutative Spaces .... 1 3 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 1 3 1 3 In summary comma the b est way t o avoid difficulties with the notion of measurability when one \ [ i\ sr using u l e {3em Dixmier}{0.4 traces pt }\ to] define the noncommutative integral open parenthesis 1 period 1 closing parenthesis i s to have the Tauberian Line 1In condition summary : Line , 2 the a 1 b divided est way by 2 t angbracketleft o avoid difficulties D right angbracket with the to notion the power of measurability of minus n a 1 divided when by 2 in T to the power of plus comma forall 0 less a in A period open parenthesis 3 period 6 closing parenthesis In summaryone i s , using the b Dixmier est way traces t o to avoid define difficulties the noncommutative with the integral notion ( 1 of . measurability1 ) i s to have the when one HoweverTauberian comma we know of no general crit eria involving the * hyphen algebra A and the selfadj oint operator D ithat s using demonstrates Dixmier open traces parenthesis to define 3 period the6 closing noncommutative parenthesis besides integral direct examination ( 1 . 1 )of isingular s to values have period the Tauberian .. Finding such a criteria condition : \ [ \ begin { a l i g n e d } c o n d i t i o n : \\ i s an open problem period .... The linear1 span T1 of T to the power of plus i s not an ideal of compact operators comma indeed we aknow\ fK r a sub c { i1 comma}{ 2 i}\ = 1 commalangle 2 commaa DhDi− 3\n commaranglea ∈ T + is,ˆ not{∀ −0 an< ideal an∈} A for. a any(3.6)\ classf r a c of{ Dixmier1 }{ 2 trace}\ periodin T ˆ{ + } , \ f o r a l l 2 2 0 There< area sp\ ecificin situationsA . where ( open 3 parenthesis . 6 ) 3 period\end{ a 6 l closingi g n e d }\ parenthesis] i s implied without having t o check singular values period However , we know of no general crit eria involving the ∗− algebra A and the selfadj oint op- Theerator functionalsD that defined demonstrates by open parenthesis ( 3 . 6 ) besides 1 period direct 1 closing examination parenthesis of are singular all traces values on the . unital Finding * hyphen algebra A period If \noindentsuch aHowever criteria , we know of no general crit eria involving the $ ∗ − $ algebra $ A $ and the selfadj oint operator angbracketleft D right angbracket to the power of minus n i s a Tauberian+ $ Doperator $i s an and open Tr sub problem omega open. parenthesis The linear angbracketleft span T Dof rightT angbracketi s not an to ideal the power of compact of minus operators n closing parenthesis , greater 0 comma thenthat theindeed demonstrates traces we from open ( parenthesis 3 . 6 ) 1 besides period 1 closing direct parenthesis examination can be normalised of singular by the values same constant . \quad Finding such a criteria independentknow K ofi, i omega= 1, 2, comma3, is not an ideal for any class of Dixmier trace . \noindentEquation:Therei open s are an parenthesis sp open ecific problem 3 situations period .7 closing\ whereh f i l l parenthesis (The 3 . linear6 ) ..i s Tr implied span sub omega $T$without open of parenthesishaving $Tˆ t o{ a check angbracketleft+ }$ singular i s not D right an angbracket ideal of to compact the operators , indeed we values . The functionals defined by ( 1 . 1 ) are all traces on the unital ∗− algebra A. If hDi−n power of minus n closing parenthesis divided by Tr−n sub omega open parenthesis angbracketleft D right angbracket to the power of minus n closing\noindenti parenthesis s a Tauberianknow sub $ period K operator{ i } and,Tr iω(hD =i ) 1> 0, then , 2 the traces , 3 from ,$ ( 1 . is 1 notanideal) can be normalised foranyclass ofDixmiertrace . A uniqueby the trace same state constant on A automatically independent implies of theω, trace states in open parenthesis 3 period 7 closing parenthesis have the same value Therefor all are omega sp in ecific D sub 2 situations period .. Hence where comma (3 for . all 6 a ) greater i s implied 0 comma a without to the power having of 1 slash t o 2 check angbracketleft singular D right values angbracket . The functionals defined by ( 1 . 1 ) are all traces−n on the unital $ ∗ − $ algebra $A .$ If $ \ langle to the power of minus n a to the power of 1 slash 2 in TTr toω(a thehDi power) of plus and condition open parenthesis 3 period 6 closing parenthesis i s D \rangle ˆ{ − n }$ i s a Tauberian −n (3.7) satisfied period .. The Trω(hDi ) . operatorprinciple of and a unique $ Tr open{\ parenthesisomega } singular( closing\ langle parenthesisD trace\rangle i s alsoˆ{ the − constructn } for) measurability> 0 of , pseudo $ then hyphen the differential traces from ( 1 . 1 ) can be normalised by the same constant independentoperatorsA unique of order of trace minus $ state\omega n period on A So,automatically far$ as we know commaimplies these the situations trace states of unique in ( 3 trace . 7 comma ) have b the esides same very strong ge hyphen ometricvalue conditions involving Hochschild cohomology and axioms for noncommutative manifolds 1/2 −n 1/2 + \ beginopenfor{ squarea all l i g nω bracket∗}∈ D2. 2 commaHence p , period for all 309a closing > 0, a squarehDi bracketa ∈ comma T and open condition square bracket ( 3 . 36 comma ) i s satisfied p period .1 60 closing square bracket provide\ f r a c The{ theTr only principle{\ handleomega ofon a} open unique( parenthesis a ( singular\ langle 3 period ) trace 6 closingD i s\ alsorangle parenthesis the constructˆ{ period − n for} measurability) }{ Tr {\ ofomega pseudo} ( \ langle D \rangleThe- unsatisfactory differentialˆ{ − n feature operators} ) } of{ this of. situation order}\ tag ∗{− comman.$So ( far from 3 as a . wemeasure know 7 hyphen ) , these $} theoretic situations point of view unique comma trace i s ,that b \enda unique{esidesa l i g n trace∗} very often strong enables ge more - ometric general conditions traces than Dixmier involving traces Hochschild on compact cohomology operators to be and axioms for usednoncommutative in open parenthesis manifolds 1 period 1 closing [ 2 , p parenthesis . 309 ] , [ to 3 obtain , p . 1 the 60 same ] provide result open the onlyparenthesis handle see on the ( sections 3 . 6 ) b elow closing parenthesis period\noindent. WhereA i s unique the definitive trace rationale state on $ A $ automatically implies the trace states in ( 3 . 7 ) have the same value that DixmierThe unsatisfactory traces should solely feature appear of in this connection situation with , the from noncommutative a measure- integral theoretic ? point of view , \noindent3 periodi s that 1f .. o rDecomposition a a unique l l $ \ traceomega of measurable often\ in enables operatorsD more{ 2 } general. $ traces\quad thanHence Dixmier , for traces all $ on a compact> 0 , a ˆ{ 1 / 2 }\Weoperators concentratedlangle toD in be the\ used previousrangle in ( sectionˆ 1{ . − 1 ) on ton identifying obtain} a ˆ the{ those1 same positive / result 2 ..}\ compact ( seein the ..T sectionsoperators ˆ{ + } b for$ elow and ) . condition Where ( 3 . 6 ) i s satisfied . \quad The principlewhichi s Tr the sub of definitive omega a unique open rationale parenthesis ( singular that T closing Dixmier ) trace parenthesis traces i s = also should const the for solely various construct appear Tr subfor in omega connection measurability in the classes with V the sub of 1 pseudo comma V− subdifferential 2 comma Voperators sub 3noncommutative period of We orderknow that integral $ general− n? T in . M $ sub So 1 comma far as infinity we know , these situations of unique trace , b esides very strong ge − ometricpolarise3 . 1comma conditions Decomposition i period involving e period of Hochschild measurable cohomology operators and axioms for noncommutative manifolds [T 2 =We , T p sub concentrated . 1 309 minus ] T , sub [ in 3 2 the plus , p previous iT . sub 1 60 3 sectionminus ] provide iT on sub identifying 4 the comma only 0 less those handle T subpositive j on in M ( sub 3compact . 1comma 6 ) . infinity operators comma j = 1 comma 2 comma 3 commafor 4 comma which Trω(T ) = const for various Trω in the classes V1, V2, V3. We know that general Theand unsatisfactoryT that∈ M1,∞ polarise feature , i . e . of this situation , from a measure − theoretic point of view , i s that aEquation: unique traceopen parenthesis often enables 3 period 8 more closing general parenthesis traces .. Tr sub than omega Dixmier open parenthesis traces T on closing compact parenthesis operators = omega to open be parenthesis openused brace in 1 ( divided 1 . 1 by ) log to open obtainT parenthesis= T1 the− T2 1+ same plusiT3 N− result closingiT4, 0 parenthesis< ( T seej ∈ M the1 sum,∞, sections from j = k 1, =2, 13 b, to4, elow N mu k ) open . Where parenthesis i s T the sub 1definitive closing parenthesis rationale minusthat muand Dixmier k openthat parenthesis traces should T sub 2 closing solely parenthesis appear plus in i connection mu k open parenthesis with the T sub noncommutative 3 closing parenthesis integral minus i mu $ k open ? $ parenthesis T sub 4 closing parenthesis closing brace sub k = 1 to the power of infinity closing parenthesis period \noindent 3 . 1 \quad Decomposition of measurable operators For convenience comma denote k=1 1 X mu-tilde k open parenthesisTr (T ) T = closingω({ parenthesis :µk =(T mu) − kµk open(T ) parenthesis + iµk(T ) − Tiµk sub(T 1) closing}∞ ). parenthesis minus(3.8) mu k open parenthesis T \noindent We concentratedω inlog(1 the + previousN) section1 on2 identifying3 those4 k=1 positive \quad compact \quad operators for sub 2 closing parenthesis plus i mu k open parenthesisN T sub 3 closing parenthesis minus i mu k open parenthesis T sub 4 closing parenthesis which $ Tr {\omega } ( T ) =$ const for various $Tr {\omega }$ in the classes $V { 1 } commaFor T in convenience L to the power , denote of infinity open parenthesis H closing parenthesis period ,VIt i s evident{ 2 } if T,V in T : ={ span3 } sub. C $ open Weknow parenthesis that T to general the power of $ plus T closing\ in parenthesisM { 1 open , parenthesis\ infty i}$ period e period T sub 1polarise comma T sub , 2 i comma . e . T sub 3 comma T sub 4 in T to the power of plus closing parenthesis comma or µk˜ (T ) := µk(T ) − µk(T ) + iµk(T ) − iµk(T ),T ∈ L∞(H). T in T-tilde : = open brace T in M sub1 1 comma2 infinity3 vextendsingle-vextendsingle-vextendsingle4 open brace 1 divided by log open \ [ TIt = i s evident T { 1 if }T ∈ − T :=T span{ 2(T}+)(+i . e iT.T ,T{ ,T3 },T −∈ T +iT), or { 4 } , 0 < T { j }\ in M { 1 , parenthesis 1 plus N closing parenthesis sumC from k = 1 to1 N2 mu-tilde3 4 k open parenthesis T closing parenthesis closing brace sub k = 1 to the power\ infty of infinity} ,j=1,2,3,4, in c closing brace supset T comma \ ] k=1 then 1 X T ∈ T˜ := {T ∈ M | { µk˜ (T )}∞ ∈ c} ⊃ T , Tr sub omega open parenthesis T closing parenthesis1,∞ log(1 = limint + N) N right arrowk=1 infinity 1 divided by log open parenthesis 1 plus N closing N parenthesis\noindent sumand from that k = 1 to N mu-tilde k open parenthesis T closing parenthesis period then \ begin { a l i g n ∗} k=1 Tr {\omega } ( T ) = \omega ( \{\1 fX r a c { 1 }{\ log ( 1 + N ) }\sum ˆ{ k = Trω(T ) = lim µk˜ (T ). 1 } { N }\mu k ( T { 1 } ) N−→∞ log(1 \mu + Nk) ( T { 2 } ) + i \mu k ( T { 3 } ) − i \mu k ( T { 4 } ) \} ˆ{\ infty } {Nk = 1 } ). \ tag ∗{$ ( 3 . 8 ) $} \end{ a l i g n ∗}

\noindent For convenience , denote

\ [ \ tilde {\mu} k ( T ) : = \mu k ( T { 1 } ) − \mu k ( T { 2 } ) + i \mu k ( T { 3 } ) − i \mu k ( T { 4 } ),T \ in L ˆ{\ infty } (H ). \ ]

\noindent It i s evident if $T \ in T : = span { C } ( T ˆ{ + } ) ( $ i . e $ . T { 1 } ,T { 2 } ,T { 3 } ,T { 4 }\ in T ˆ{ + } ) , $ or

\ [T \ in \ tilde {T} : = \{ T \ in M { 1 , \ infty }\arrowvert \{\ f r a c { 1 }{\ log ( 1 + N ) }\sum ˆ{ k = 1 } { N }\ tilde {\mu} k ( T ) \} ˆ{\ infty } { k = 1 } \ in c \}\supset T, \ ]

\noindent then

\ [ Tr {\omega } ( T ) = \lim { N \rightarrow \ infty }\ f r a c { 1 }{\ log ( 1 + N ) }\sum ˆ{ k = 1 } { N }\ tilde {\mu} k ( T ) . \ ] 14 .... S period Lord and F period Sukochev \noindenthline1414 \ h f i l l S . Lord and F . Sukochev S . Lord and F . Sukochev Hence Tr sub omega open parenthesis T closing parenthesis = .... const for T in T-tilde sub period .... Does the reverse implication hold for the\ [ various\ r u l e { classes3em}{0.4 pt }\ ] Tr sub omega in V sub 1 comma V sub 2 comma˜ V sub 3 ? RecentHence workTr byω( theT ) second = authorconst comma for T N∈ periodT. KaltonDoes comma the reverse and A period implication Sedaev comma hold for t o the appear various comma indicates that the \noindentpropositionclassesHence $ Tr {\omega } ( T ) = $ \ h f i l l const f o r $ T \ in \ tilde {T} { . }$ \ h f i l l Does the reverse implication hold for the various classes f open parenthesis T closing parenthesis = const forall f in V sub 1 double stroke right arrow open brace 1 divided by log open parenthesis 1\ begin plus N{ closinga l i g n ∗} parenthesis sum from k = 1 to N mu-tilde k open parenthesis T closing parenthesis closing brace sub k = 1 to the power of Tr ∈ V , V , V ? infinityTr {\ in comega open parenthesis}\ in 3V period{ 1 9 closing} ,V parenthesis{ω 2 }1 2,V3 { 3 } ? \end{ a l i g n ∗} i s false periodRecent .. In work the next by the section second on symmetric author , subideals N . Kalton we briefly , and describ A . Sedaev e the strictness , t o appear of the , indicates inclusions in the following schematic : \ hspacethat∗{\ thef i l l } Recent work by the second author , N . Kalton , and A . Sedaev , t o appear , indicates that the Equation:proposition open parenthesis 3 period 10 closing parenthesis .. T-tilde subsetneq K sub 1 subsetneq K sub 2 subsetneq K sub 3 period The place of K 1 divided by 2 open parenthesis and1 KP subk=1 PDL open∞ square bracket 1 comma infinity closing parenthesis closing parenthesis f (T ) = const ∀ f ∈ V1 ⇒ { µk˜ (T )} ∈ c (3.9) i s false . In the next i\ snoindent open periodproposition log(1+N) N k=1 Thesection strictness on of symmetric the inclusion subideals T-tilde subsetneq we briefly K sub describ 1 and the e the result strictness open parenthesis of the 3 period 5 closing parenthesis for positive elements impliesf $(inclusions T ) in the =$ following const schematic $ \ f o r a l : l $ f $ \ in V { 1 }\Rightarrow \{\ f r a c { 1 }{\ log ( 1span + sub N C open ) }\ parenthesissum ˆ{ Kk sub i= to the 1 power} { N of}\ plus closingtilde {\ parenthesismu} k subsetneq ( T K sub ) i comma\} ˆ{\ i =infty 1 comma} 2{ periodk = 1 } \ in c ( 3 . 9 ) $ It i s evident from f open parenthesis T toT˜ the powerK of * closingK parenthesisK . = overbar f sub open parenthesis(3.10) T closing parenthesis comma f ini Vs sub f a l si e comma . \quad i = 1In divided the by next 2 sub section comma( 1 on comma1 symmetric( 2 comma2 ( subidealsPDL3 open square we briefly bracket 1 comma describ infinity e the closing strictness parenthesis of comma the that the set s K sub i are1 closed under The place of K ( and KPDL[1,∞)) i s open . \noindentthe * hypheninclusions operation2 period in the .. It followingfollows that T schematic in K sub i arrowdblright-negationslash : bar T bar in K sub i comma i = 1 comma 2 period The strictness of the inclusion T˜ K and the result ( 3 . 5 ) for positive elements implies As a consequence comma we lack a basic decomposition( 1 result of integration theory period Suppose T = T to the power of * \ beginwith{ thea l istandard g n ∗} unique decomposition into positive and negative operators span (K+) K , i = 1, 2. \ tildeEquation:{T}\ opensubsetneq parenthesis 3 period K { 11 closing}\ subsetneq parenthesisC i ( i .. T K ={ T sub2 }\ plussubsetneq minus T sub minus K { comma3 } T. sub\ tag plusminux∗{$ ( : = 3 1 divided . 10 ) $} ∗ 1 by 2 openIt i parenthesis s evident bar from Tbar f (T plusminux) = f(T ), Tf closing∈ Vi, i parenthesis= 1, 2,PDL period[1, ∞), that the set s Ki are closed under \end{ a l i g n ∗} 2 , Thenthe comma∗− operation . It follows that T ∈ Kiarrowdblright − negationslash | T |∈ Ki, i = 1, 2. T = T to the power of * in K sub i negationslash-arrowdblright T sub plus comma T sub minus in K sub i comma i = 1 comma 2 period \noindentAsThe a consequence place of , $K we lack\ f r a c basic{ 1 }{ decomposition2 } ( $ and result $ of K integration{ PDL [ theory 1 . , Suppose\ infty ) } ) $ i s open . CompareT = T ∗ thiswith with the the standard result for unique a Borel measure decomposition space open into parenthesis positive X and comma negative mu closing operators parenthesis open parenthesis also true for the noncommutative \ centerlinevariant .. of{The tau hyphen strictness measurable of .. the operators inclusion .. on .. $ open\ tilde parenthesis{T}\ Nsubsetneq comma tau closing K { parenthesis1 }$ and .. for the .. a result .. semifinite ( 3 von . 5 ) for positive elements implies } 1 Nuemann .. algebra N with T = T+ − T−,T± := (| T | ±T ). (3.11) \ [faithful span normal{ C } semifinite( K trace ˆ{ + tau} closing{ i } parenthesis) \ subsetneq2 K { i } , i = 1 , 2 . \ ] f =Then overbar , f in L to the power of 0 open parenthesis X comma mu closing parenthesis double stroke right arrow exists ! 0 less f sub plus comma f sub minus in L to the power of 0 open parenthesis X comma mu closing parenthesis s period t period f = f sub plus minus f sub minus \noindent It i s evident from f $ ( Tˆ{ ∗ } ) = \ overline {\}{ f } { (T), }$ f $ \ in comma T = T ∗ ∈ K negationslash − arrowdblright T ,T ∈ K , i = 1, 2. V where{ i } L to, the power i = of 0\ openf r a c parenthesis{ i1 }{ 2 X} comma{ , } mu1 closing , parenthesis 2+ ,− denotes PDLi the [ set 1 of open , parenthesis\ infty equivalence) ,$ classes that closing the set s parenthesis$ K {Comparei of}$ mu arethis hyphen with closed measurable the under result functions for a Borel on X measure period space (X, µ)( also true for the noncommutative 3 periodvariant 2 .. Closed of τ symmetric− measurable subideals operators on (N , τ) for a semifinite von Nuemann \noindentDefinealgebra symmetrictheN with $ subspaces∗ faithful − $ of the operation normal symmetric semifinite .sequence\quad trace spaceIt followsτ m) sub 1 comma that infinity $ T by\ in K { i } arrowdblright −negationslash \midEquation:T open\mid parenthesis\ in 3K period{ i 1} 2 closing, iparenthesis = 1 .. u ,sub 1 2 comma . $ infinity = braceleftbigg x in m sub 1 comma infinity bar x 0 0 to the power of * open parenthesisf = f n∈ closingL (X, parenthesis µ) ⇒ ∃!0 < = f+ o, f parenleftbigg− ∈ L (X, µ) log s. nt.f divided= f+ − byf− n, parenrightbigg bracerightbigg comma Equation: As a consequence , we lack a basic decomposition result of integration theory . Suppose $ T = T ˆ{ ∗ }$ open parenthesis0 3 period 13 closing parenthesis .. l to the power of 1 comma w = braceleftbigg x in m sub 1 comma infinity bar x to the power withwhere the standardL (X, µ) denotes unique the decomposition set of ( equivalence into positive classes ) of andµ− negativemeasurable operators functions on X. of * open3 . parenthesis 2 Closed n closing symmetric parenthesis subideals= O parenleftbigg 1 divided by n parenrightbigg bracerightbigg period TheDefine ideal of symmetric compact operators subspaces U sub of 1 the comma symmetric infinity and sequence L to the space power ofm 1 commaby w comma defined by \ beginu sub{ 1a lcomma i g n ∗} infinity = j U sub 1 comma infinity comma l to the power of 11 comma,∞ w = j sub L 1 comma w comma Tare = quasi T hyphen{ + Banach} − symmetricT { − ideals } ,T period ..{\ Noticepm } also : = \ f r a c { 1 }{ 2 } ( \mid T \mid \pm T ).L to\ thetag power∗{$ of ( 1 comma 3 . w subsetneq 1 1 U sub) $} 1 comma infinity subsetneqlog n M sub 1 comma infinity period u = {x ∈ m | x∗(n) = o( )}, (3.12) \end{ a l i g n ∗} 1,∞ 1,∞ n 1 `1,w = {x ∈ m | x∗(n) = O( )}. (3.13) \noindent Then , 1,∞ n

1,w \ [ TThe = ideal T ˆ of{ compact∗ } \ in operatorsK { i U}1,∞ andnegationslashL , deﬁned−arrowdblright by T { + } ,T { − } \ in K { i } , i = 1 , 2 . \ ] 1,w u1,∞ = jU1,∞, ` = jL1, w, are quasi - Banach symmetric ideals . Notice also \noindent Compare this with the result for a Borel measure space $ ( X , \mu ) ( $ also true for the noncommutative v a r i a n t \quad o f $ \tau − $ measurable \quad o p e r a t o r s \quad on \quad $ ( N , \tau ) $ \quad f o r \quad a \quad s e m i f i n i t e von Nuemann \quad algebra L1,w U M . $ N $ with ( 1,∞ ( 1,∞ faithful normal semifinite trace $ \tau ) $

\ [ f = \ overline {\}{ f }\ in L ˆ{ 0 } (X, \mu ) \Rightarrow \ exists ! 0 < f { + } , f { − } \ in L ˆ{ 0 } (X, \mu ) s . t . f = f { + } − f { − } , \ ]

\noindent where $ L ˆ{ 0 } (X, \mu ) $ denotes the set of ( equivalence classes ) of $ \mu − $ measurable functions on $ X . $

\noindent 3 . 2 \quad Closed symmetric subideals

\noindent Define symmetric subspaces of the symmetric sequence space $ m { 1 , \ infty }$ by

\ begin { a l i g n ∗} u { 1 , \ infty } = \{ x \ in m { 1 , \ infty }\mid x ˆ{ ∗ } ( n ) = o ( \ f r a c {\ log n }{ n } ) \} , \ tag ∗{$ ( 3 . 1 2 ) $}\\\ e l l ˆ{ 1 , w } = \{ x \ in m { 1 , \ infty }\mid x ˆ{ ∗ } ( n ) = O ( \ f r a c { 1 }{ n } ) \} . \ tag ∗{$ ( 3 . 13 ) $} \end{ a l i g n ∗}

\noindent The ideal of compact operators $ U { 1 , \ infty }$ and $ L ˆ{ 1 , w } , $ defined by

\ [ u { 1 , \ infty } = j U { 1 , \ infty } , \ e l l ˆ{ 1 , w } = j { L } 1 , w , \ ]

\noindent are quasi − Banach symmetric ideals . \quad Notice a l s o

\ [ L ˆ{ 1 , w }\ subsetneq U { 1 , \ infty }\ subsetneq M { 1 , \ infty } . \ ] Measure Theory in Noncommutative Spaces .... 1 5 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 1 5 1 5 Define comma as b efore comma \ [ T-tilde\ r u l e :{3em = open}{0.4 brace pt }\ T in] M sub 1 comma infinity vextendsingle-vextendsingle-vextendsingle open brace 1 divided by log open parenthesis 1 plus NDefine closing , asparenthesis b efore sum, from k = 1 to N mu-tilde k open parenthesis T closing parenthesis closing brace sub k = 1 to the power of infinity in c closing brace period \noindent Define , as b efore , New results of N period Kalton comma A period Sedaev commak=1 and the second author comma state that 1 X Equation: open parenthesis 3 periodT˜ := 14{T closing∈ M parenthesis|{ .. T-tildeµk˜ cap(T )} U∞ sub∈ 1c} comma. infinity = K sub 1 cap U sub 1 comma infinity \ [ \ tilde {T} : = \{ T \ in 1,∞M log(1{ 1 + N ,) \ inftyk=1}\arrowvert \{\ f r a c { 1 }{\ log ( 1 and that U sub 1 comma infinity i s the maximal symmetric subidealN E subset M sub 1 comma infinity for which open parenthesis 3 period + N ) }\sum ˆ{ k = 1 } { N }\ tilde {\mu} k ( T ) \} ˆ{\ infty } { k = 1 }\ in 14 closingNew parenthesis results of holds N . period Kalton .. Thus , A . comma Sedaev , and the second author , state that c T-tilde\} =. tilde-T\ ] cap M sub 1 comma infinity negationslash-equal K sub 1 cap M sub 1 comma infinity = K sub 1 period Their results also show Equation: open parenthesis 3 period 1 5 closingT˜ parenthesis∩ U1,∞ = K ..1 T-tilde∩ U1,∞ cap U sub 1 comma infinity negationslash-equal(3.14) K sub 2 cap U sub 1 comma\noindent infinityNew period results of N . Kalton , A . Sedaev , and the second author , state that Henceand K that sub 1 equal-negationslashU1,∞ i s the maximal K sub symmetric 2 period .... It subideal is .... an openE ⊂ question M1,∞ for as whichto the existence ( 3 . 14 of a) maximal holds . closed symmetric \ beginsubidealThus{ a l Ui g , subn ∗} 2 such that T-tilde cap U sub 2 = K sub 2 cap U sub 2 period .. Is L to the power of 1 comma w subset equal U sub 2 ? \ tilde4 .. Origin{T}\ .. ofcap the .. noncommutativeU { 1 , ..\ integralinfty } = K { 1 }\cap U { 1 , \ infty }\ tag ∗{$ ( 3 . ˜ ˜ 14The ) origin $} of the formula open parenthesisT = T 1∩ period M1,∞ 16= closingK1 ∩ M parenthesis1,∞ = K1. as the integral in noncommutative geometry li es in Connes quoteright\end{Theira l i g trace n ∗} results also show theorem open square bracket 1 7 comma Theorem 1 closing square bracket comma open square bracket 4 2 comma p period 293 closing square\noindent bracketand : that $ U { 1 , \ infty }$ i s the maximal symmetric subideal $ E \subset M { 1 ˜ , Theorem\ infty ..} 4$ period for 1 which.. open parenthesis ( 3 . 14 Connes ) holdsT ∩ quoteright U1,∞ . 6=\quadK2 ..∩ trace UThus1,∞. .. theorem , closing parenthesis period(3 ... Let15) .. M .. be .. a .. compact n hyphen dimensional .. manifold comma \ [ \ tildeHence{T}K1 =6= K\ tilde2. It is{T an}\ opencap questionM { as1 to the , existence\ infty of}\ a maximalnot= K closed{ 1 symmetric}\cap M { 1 , \ infty } E .. a .. complex vector .. bundle .. on M comma .. and1 P,w .. a pseudo hyphen differential .. operator .. of order minus n .. acting subideal U2 such that T˜ ∩ U2 = K2 ∩ U2. Is L ⊆ U2? =on K s4 ections{ 1 Origin} of E. period\ ] .. of Then the the corresponding noncommutative operator P in H = L to integral the power of 2 open parenthesis M comma E closing parenthesis belongsThe to M origin sub 1 ofcomma the formula infinity open ( 1 parenthesis. 1 ) as the H integral closing parenthesis in noncommutative and geometry li es in Connes one has : \noindent’ traceTheir theorem results [ 1 7 , also Theorem show 1 ] , [ 4 2 , p . 293 ] : LineTheorem 1 Tr sub omega 4 open . 1 parenthesis ( Connes P closing ’ parenthesis trace = 1 divided theorem by n ) Res . open parenthesisLet M P closingbe parenthesis a open parenthesis 4 periodcompact 1 closing parenthesisn− dimensional Line 2 for any manifold Tr sub omega , E ina V sub complex 1 period vector bundle on M, and \ beginHere{ Resa l i ig s n the∗} restriction of the Adler endash Manin endash Wodzicki residue t o pseudo hyphen differential opera hyphen \ tildeP {Ta}\ pseudocap - differentialU { 1 , operator\ infty }\ ofnot order=− Kn { acting2 }\cap U { 1 , \ infty } . \ tag ∗{$ ( t ors of order minus n open square bracket 4 3 comma 1 7 closing square bracket2 period The monograph open square bracket 4 2 comma S 3 .on s 1 ections 5 of ) $}E. Then the corresponding operator P in H = L (M, E) belongs to M1,∞(H) 7 closingand square one bracket has : provides a detailed introduction t o the residue \endRes{ a and l i g nthe∗} origin and framework of the formula period We recall that comma for classical pseudo hyphen differential operators of order minus n acting on the trivial line bundle1 open parenthesis H = L to the power of 2 open parenthesis M closing parenthesis \noindent Hence $ K { 1 }\ne KTrω{(P2) =} Res(. $P )\ h (4 f.1) i l l I t i s \ h f i l l an open question as to the existence of a maximal closed symmetric closing parenthesis comma n Res open parenthesis P closing parenthesis = 1 dividedforany by openTrω parenthesis∈ V1. 2 pi closing parenthesis to the power of n integral sub S to the power\noindent of 1 Ms sigma u b i d sub e a l minus $ U n open{ 2 parenthesis}$ such P that closing $ parenthesis\ tilde {T open}\ parenthesiscap U x comma{ 2 } xi closing= K parenthesis{ 2 }\ dxdcap xi commaU { 2 } . $where\quad dxdHere xiI Ress is the $ is L Liouville the ˆ{ restriction1 measure , w of open}\ the parenthesissubseteq Adler – Manin volumeU – form{ Wodzicki2 closing} ? residue $parenthesis t o .. pseudo on the - cosphere differential bundle S to the power of 1 M −n[43, 17]. open parenthesisopera - t with ors fibres of order The monograph [ 4 2 , §7 ] provides a detailed introduction \noindentS tot the o the power4 residue\quad of 1 MOrigin Res sub and x thicksim\ thequad origin =o fopen the and brace\ frameworkquad x closingnoncommutative ofbrace the times formula S to\quad .the We power recalli n t of e g nthat r a minus l , for 1 .. classical over x in M where S to the power of n .. ispseudo the unit - differential .. n hyphen sphere closing parenthesis .. and sigma sub minus n .. i s the principal −n (H = L2(M)), \noindentsymboloperators ofThe P period of origin order .. On ofthe lefttheacting hand formula on side the of (open trivial 1 .parenthesis 1 line ) bundle as 4 the period integral 1 closing inparenthesis noncommutative one should read geometry for P the li unique es inextension Connes of ’ trace theorem [ 1 7 , Theorem 1 ] , [ 4 2 , p . 293 ] : the 1 Z negative order pseudo hyphen differentialRes( operatorP ) = Cn to theσ power−n(P )( ofx, infinity ξ)dxdξ, open parenthesis M closing parenthesis right arrow C to the (2π) 1 power\noindent of infinityTheorem open parenthesis\quad 4 M . closing 1 \quad parenthesis( Connes to aS compactM ’ \quad operatort r a c e L\ toquad the powertheorem of 2 open ) . parenthesis\quad Let M closing\quad parenthesis$ M $ right\quad be \quad a \quad compact 1 arrow$ n where− $ dimensionaldxdξ is the Liouville\quad manifold measure ( , volume form ) on the cosphere bundle S M( with $ E $ \quad1 a \quad complexn−1 vector \quad bundle \quadn on $ M , $ \quad and $ P $ \quad a pseudo − differential \quad operator \quad o f order Equation:fibres LS toM thex power∼= of{x 2} open × S parenthesisover Mx closing∈ M parenthesiswhere S .. openissquare the unit bracketn− 4 4sphere closing ) square and bracket period $ − n $ \quad a c t i n g Letσ Capital−n i s Delta the principaldenote the Hodge Laplacian on M and P sub a in B open parenthesis L to the power of 2 open parenthesis M closing parenthesissymbol closing of parenthesisP. On the denote left the hand extension side of of ( the 4 . 1 ) one should read for P the unique extension ∞ ∞ \noindentzerothof the orderons negative pseudo ections hyphen order ofdifferential pseudo $E - operator differential . $ \ associatedquad operatorThen to a the functionC corresponding(M) a→ inC C to(M the) to power operator a compact of infinity $ operator open P $ parenthesis in $H S to = the power L ˆ{ of2 } 2 1( M closing ML (M , parenthesis) → E )$ period belongsto .. Since $M { 1 , \ infty } ( H ) $ and onesigma has sub : minus n open parenthesis P sub a open parenthesis 1 plus Capital Delta closing parenthesis to the power of minus n slash 2 closing parenthesis bar S to the power of 1 M = a [44].L2(M) \ [ \formulabegin { opena l i g parenthesisn e d } Tr 4{\ periodomega 1 closing} ( parenthesis P ) provides = \ f r a c { 1 }{ n } Res (P) ( 4 . 1 ) \\ f o r any Tr {\omega }\ in V { 1 } . \end{ a l i g2 n e d }\ ] Equation:Let open∆ denote parenthesis the 4 Hodge period 2 Laplacian closing parenthesis on M ..and Tr subPa omega∈ B(L parenleftbig(M)) denote P sub the a extension open parenthesis of the 1 plus Capital Delta closing parenthesiszeroth to the order power pseudo of minus - differential n slash 2 parenrightbig operator associated = 1 divided to by a n function open parenthesisa ∈ C∞ 2( piS1 closingM). Since parenthesis to the power of n integral sub S to the power of 1 M a open parenthesis x comma xi closing parenthesis dxd xi period Here Res i s the restriction of theσ Adler(P (1 +−− ∆)−Maninn/2) | S−−1M =Wodzickia residue t o pseudo − differential opera − t ors of order $ − n [ 4− 3n a , 1 7 ] .$ Themonograph[42, \S 7 ] provides a detailed introduction t o the residue Resformula and the ( origin 4 . 1 ) provides and framework of the formula . We recall that , for classical pseudo − differential

\noindent operators of order $ − n$ acting1 onZ the trivial line bundle $ ( H = Lˆ{ 2 } ( Tr (P (1 + ∆)−n/2) = a(x, ξ)dxdξ. (4.2) M ) ) , $ ω a n n(2π) S1M \ [ Res ( P ) = \ f r a c { 1 }{ ( 2 \ pi ) ˆ{ n }}\ int { S ˆ{ 1 } M }\sigma { − n } ( P ) ( x , \ xi ) dxd \ xi , \ ]

\noindent where $ dxd \ xi $ is the Liouville measure ( volume form ) \quad on the cosphere bundle $ S ˆ{ 1 } M ( $ with fibres $ S ˆ{ 1 } M { x }\sim = \{ x \}\times S ˆ{ n − 1 }$ \quad over $ x \ in M $ where $ S ˆ{ n }$ \quad i s the unit \quad $ n − $ sphere ) \quad and $ \sigma { − n }$ \quad i s the principal

\noindent symbol of $P . $ \quad On the left hand side of ( 4 . 1 ) one should read for $ P $ the unique extension of the negative order pseudo − differential operator $ C ˆ{\ infty } (M) \rightarrow C ˆ{\ infty } ( M ) $ to a compact operator $Lˆ{ 2 } (M) \rightarrow $

\ begin { a l i g n ∗} \ tag ∗{$ L ˆ{ 2 } ( M ) $} [ 4 4 ] . \end{ a l i g n ∗}

Let $ \Delta $ denote the Hodge Laplacian on $M$ and $ P { a }\ in B ( L ˆ{ 2 } (M ) ) $ denote the extension of the zeroth order pseudo − differential operator associated to a function $ a \ in C ˆ{\ infty } ( S ˆ{ 1 } M ) . $ \quad Since

\ [ \sigma { − n } (P { a } ( 1 + \Delta ) ˆ{ − n / 2 } ) \mid S ˆ{ 1 } M = a \ ]

\noindent formula ( 4 . 1 ) provides

\ begin { a l i g n ∗} Tr {\omega } (P { a } ( 1 + \Delta ) ˆ{ − n / 2 } ) = \ f r a c { 1 }{ n ( 2 \ pi ) ˆ{ n }}\ int { S ˆ{ 1 } M } a ( x , \ xi ) dxd \ xi . \ tag ∗{$ ( 4 . 2 ) $} \end{ a l i g n ∗} 1 6 S . Lord and F . Sukochev

∞ ∞ 1 ∞ From the inclusion a ∈ C (M) → C (S M) the operator Pa, a ∈ C (M), i s the multiplier by the function a ∈ C∞(M) and [ 4 2 , Corollary 7 . 2 1 ] , [ 1 6 , §1 . 1 ] or [ 4 5 , p . 98 ] ,

n−1 Z Vol(S ) ∞ Trω(PaT∆) = n a(x)dx, a ∈ C (M), (4.3) n(2π) M where we set

−n/2 T∆ := (1 + ∆) ∈ M1,∞. Equation ( 4 . 3 ) originated the use of the term integral for the expression ( 1 . 1 ) . Why i s a singular trace needed in formula (4.1)? By the asymptotic expansion of classical symbols P − P0 ≡ σ−n(P ) for a classical pseudo - differential operator P of order −n where P0 i s of order < −n and σ−n(P ) is the principal symbol of P [46, XVIII ] . All pseudo - differential operators of order < −n on a compact n− dimensional manifold have trace class extensions [ 4 4 ] . Hence any singular functional f ∈ M1,∞ applied t o P, i . e .P 7→ f (P ), since it vanishes on L1, i s equivalent t o a functional on symbols of order −n. Moreover the vanishing on L1 p ermits that the statement need only b e proved locally . Hence M may b e taken to b e homogeneous . If f i s a positive trace this functional on symbols of order −n i s equivalent t o an invariant measure on S1M. Hence , up to a constant , it identifies with the Liouville integral . The above argument , from the proof of Connes ’ trace theorem in [ 1 7 ] , implies that any ∗ positive singular trace 0 < ρ ∈ M1,∞ can be substituted for the Dixmier trace in ( 4 . 1 ) . Therefore we may write , using the above notation : Theorem 4 . 2 ( Connes ’ trace theorem ) . Let M be a compact n− dimensional manifold , E a complex vector bundle on M, and P a pseudo - differential operator of order −n acting 2 on s ections of E. Then the corresponding operator P in H = L (M, E) belongs to M1,∞(H) and one has :

ρ(T )(2π)n ρ(P ) = ∆ Res(P ) (4.4) Vol(S1M)

∗ for any singular trace 0 < ρ ∈ M (H) 1,∞ . 2 ∗ Singular traces 0 < ρ ∈ M1,∞(L (M, E)) , which are not Dixmier traces and such that ρ(T∆) > 0, exist by a variant of the argument after ( 2 . 4 5 ) . Correspondingly

Z ρ(T∆) ∞ ρ(PaT∆) = a(x)dx, a ∈ C (M) (4.5) Vol(M) M provides the integral on a manifold for smooth functions up t o a non - zero constant . The result ( 4 . 5 ) raises the question about whether the class of Dixmier traces (V1) is t oo restrictive in the definition of the noncommutative integral ( 1 . 1 ) . If we define + ∗ Vsing := {0 < ρ ∈ M1,∞ | ρ i s a singular trace } (4.6) and then define as the ‘ integral ’

−n + −n ρ(ahDi ), a ∈ A, ρ ∈ Vsing, ρ(hDi ) > 0 (4.7) we have the same claim to formulas ( 4 . 1 ) and ( 4 . 3 ) . Of course , there are a great many formulas involving Dixmier traces b esides ( 4 . 1 ) and ( 4 . 3 ) , particularly in the 1 6 .... S period Lord and F period Sukochev \noindenthlinedevelopment1 6 \ h of f i l ( l loS cal . )Lord index and theory F . Sukochev in noncommutative geometry [ 4 7 , 1 6 , 1 9 , 4 8 From, the 4 inclusion9 , 5 a in 0 ,C to 5 the 1 , power 5 2 ] of. infinity There open are parenthesis geometric M conse closing - quencesparenthesis of right the formulation arrow C to the power of infinity open parenthesis\ [ \ rinvolving u l e { S3em to}{ the the0.4 power Dixmier pt }\ of 1] M trace closing and parenthesis associations the operator t o pseudo P sub - a differential comma a in Coperators to the power [ 1 7 of , infinity 5 3 open , parenthesis M closing parenthesis5 4 , comma 5 5 i ] s , the including multiplier extensions of Connes ’ trace theorem [ 5 6 , 5 7 , 5 3 ] . There by theare function a in C to the power of infinity open parenthesis M closing parenthesis and open square bracket 4 2 comma Corollary 7 period 2\noindent 1 closing squareFrom bracket the inclusion comma open square $ a bracket\ in 1C 6 commaˆ{\ infty S 1 period} (M) 1 closing square\rightarrow bracket or openC square ˆ{\ bracketinfty 4} 5 comma( S p ˆ{ 1 } Mperiod ) 98 $ closing the square operator bracket $P comma{ a } , a \ in C ˆ{\ infty } ( M ) ,$ i sthe multiplier byEquation: the function open parenthesis $ a 4\ periodin 3C closing ˆ{\ parenthesisinfty } ..( Tr subM omega )$ open and[42,Corollary7.21] parenthesis P sub a T sub Capital Delta ,[16,closing parenthesis\S 1.1]or[45,p.98] , = Vol open parenthesis S to the power of n minus 1 closing parenthesis divided by n open parenthesis 2 pi closing parenthesis to the power of n integral\ begin sub{ a l Mi g n a∗} open parenthesis x closing parenthesis dx comma a in C to the power of infinity open parenthesis M closing parenthesis comma Trwhere{\ weomega set } (P { a } T {\Delta } ) = \ f r a c { Vol ( S ˆ{ n − 1 } ) }{ n ( 2 \ piT sub) Capital ˆ{ n Delta}}\ :int = open{ parenthesisM } a 1 (plus Capital x ) Delta dx closing , parenthesis a \ in to theC power ˆ{\ infty of minus} n slash(M), 2 in M sub 1 comma\ tag ∗{ infinity$ ( period4 . 3 ) $} \endEquation{ a l i g n open∗} parenthesis 4 period 3 closing parenthesis originated the use of the term integral for the expression open parenthesis 1 period 1 closing parenthesis period \noindentWhy i s awhere singular we trace s e needed t in formula open parenthesis 4 period 1 closing parenthesis ? .. By the asymptotic expansion of classical symbols P minus P sub 0 equiv sigma sub minus n open parenthesis P closing parenthesis for a classical pseudo hyphen differential operator P\ [T of order{\ minusDelta n where} : P sub = 0 ( 1 + \Delta ) ˆ{ − n / 2 }\ in M { 1 , \ infty } . \ ] i s of order less minus n and sigma sub minus n open parenthesis P closing parenthesis is the principal symbol of P open square bracket 4 6 comma XVIII closing square bracket period All pseudo hyphen differential \noindentoperators ofEquation order less ( minus 4 . n 3 on ) a originated compact n hyphen the usedimensional of the manifold term integralhave trace class for extensions the expression open square ( 1bracket . 1 4) 4 . closing square bracket period WhyiHence s any a singular singular functional trace f needed in M sub in 1 comma formula infinity $ applied( 4 t o . P comma 1 i) period ?$ e period\quad P arrowright-mapstoBy the asymptotic f open expansion parenthesis of classical Psymbols closing parenthesis $ P − commaP since{ 0 it}\ vanishesequiv on L to\ thesigma power{ of − 1 comman } i s( P ) $ for a classical pseudo − differential operator $P$equivalent of order t o a functional $ − onn symbols $ where of order $ P minus{ n0 period}$ .... Moreover the vanishing on L to the power of 1 p ermits that ithe s o statement f order need $ < only b− e provedn $ locally and period $ \sigma .... Hence{ −M mayn b} e taken( toP b e )$ homogeneous is the period principal .... If f i symbol s of $P [ 4a positive 6 ,$ trace XVIII this functional ] . All on pseudo symbols− of orderdifferential minus n i s equivalent t o an invariant measure operatorson S to the powerof order of 1 M $ period< ..− Hencen $comma on upa compact to a constant $ comma n − it$ identifies dimensional with the Liouville manifold integral have period trace class extensions [ 4 4 ] . The above argument comma from the proof of Connes quoteright trace theorem in open square bracket 1 7 closing square bracket comma implies\noindent that anyHence positive any singular functional f $ \ in M { 1 , \ infty }$ appliedto $P ,$ i.e $ .singular P trace\mapsto 0 less rho$ in f M $( sub 1 comma P ) infinity ,$ to the since power itvanisheson of * can be substituted $Lˆ for the{ 1 Dixmier} , $trace i in s open parenthesis 4 period 1 closing parenthesis period .. Therefore we \noindentmay writeequivalent comma using the t o above a functional notation : on symbols of order $ − n . $ \ h f i l l Moreover the vanishing on $ LTheorem ˆ{ 1 } ....$ 4 p period ermits 2 .... that open parenthesis Connes quoteright .... trace .... theorem closing parenthesis period .... Let .... M .... be .... a .... compact n hyphen dimensional .... manifold comma \noindentE .... a ....the complex statement vector .... need bundle only .... b on e M proved comma .... locally and P .... . \ ah pseudo f i l l Hence hyphen differential $M$ may .... operator b e taken .... of to order b e minus homogeneous n .... . \ h f i l l I f f i s acting \noindenton s ectionsa of positive E period .. trace Then the this corresponding functional operator on symbols P in H = L of to orderthe power $ of− 2 openn $ parenthesis i s equivalent M comma E t closing o an parenthesis invariant measure belongson $ to S M ˆ{ sub1 1} commaM infinity . $ open\quad parenthesisHence , H up closing to parenthesisa constant and , it identifies with the Liouville integral . one has : TheEquation: above argument open parenthesis , from 4 period the 4 proof closing parenthesis of Connes .. rho ’ trace open parenthesis theorem P in closing [ 1 parenthesis 7 ] , implies = rho open that parenthesis any positive T sub Capital Deltasingular closing parenthesistrace $ open 0 < parenthesis\rho 2 pi\ in closingM parenthesis ˆ{ ∗ } { to1 the power , \ ofinfty n divided}$ by can Vol be open substituted parenthesis S to for the the power Dixmier of 1 M trace in ( 4 . 1 ) . \quad Therefore we closing parenthesis Res open parenthesis P closing parenthesis \noindentfor any singularmay write trace 0 less , using rho in Mthe sub above 1 comma notation infinity open : parenthesis H Case 1 * Case 2 period Singular traces .. 0 less rho in M sub 1 comma infinity open parenthesis L to the power of 2 open parenthesis M comma E closing parenthesis closing\noindent parenthesisTheorem to the\ powerh f i l l of4 * comma. 2 \ h .. f iwhich l l ( are Connes not .. Dixmier ’ \ h f i tracesl l t r a.. c and e \ ..h suchf i l l thattheorem ) . \ h f i l l Let \ h f i l l $ M $ \ hrho f i l l openbe parenthesis\ h f i l l a T\ subh f i Capital l l compact Delta closing $ n parenthesis− $ dimensional greater 0 comma\ h fexist i l l bymanifold a variant of , the argument after open parenthesis 2 period 4 5 closing parenthesis period .. Correspondingly \noindentEquation: open$ E parenthesis $ \ h f i l 4 l perioda \ h 5 f closingi l l complex parenthesis vector .. rho open\ h f i parenthesis l l bundle P sub\ h f a i T l l subon Capital $ M Delta , closing$ \ h parenthesis f i l l and = $ rho P open $ parenthesis\ h f i l l a T pseudo sub Capital− differential Delta closing parenthesis\ h f i l l dividedoperator by Vol\ h open f i l l parenthesiso f order M closing $ − parenthesisn $ \ h f integral i l l a c sub t i n gM a open parenthesis x closing parenthesis dx comma a in C to the power of infinity open parenthesis M closing parenthesis \noindentprovides theons integral ections on a manifold of $E for smooth . $ functions\quad Then up t o thea non corresponding hyphen zero constant operator period $ P $ in $H = L ˆ{ 2 } (The M result , .. E open )$ parenthesis belongsto 4 period $M5 closing{ 1 parenthesis , \ infty .. raises} the( question H about ) $whether and the class of Dixmier traces .. open parenthesisone has V : sub 1 closing parenthesis .. is t oo restrictive in the definition of the noncommutative integral open parenthesis 1 period 1 closing parenthesis period If we define \ beginV sub{ a sing l i g n to∗} the power of plus : = braceleftbig 0 less rho in M sub 1 comma infinity to the power of * bar rho .. i s a singular trace bracerightbig\rho ( open P parenthesis ) = 4\ periodf r a c {\ 6 closingrho parenthesis(T {\Delta } ) ( 2 \ pi ) ˆ{ n }}{ Vol ( S ˆ{ 1 } M)and then} Res define as ( the P quoteleft ) \ tag integral∗{$ quoteright ( 4 . 4 ) $} \endEquation:{ a l i g n ∗} open parenthesis 4 period 7 closing parenthesis .. rho open parenthesis a angbracketleft D right angbracket to the power of minus n closing parenthesis comma a in A comma rho in V sub sing to the power of comma to the power of plus rho open parenthesis angbracketleft D\noindent right angbracketfor any to the singular power of minus trace n closing $\ l e parenthesis f t . 0 < greater\rho 0 \ in M { 1 , \ infty } (H)\ begin { a l i g n e d } & ∗ \\we have the same claim to formulas open parenthesis 4 period 1 closing parenthesis and open parenthesis 4 period 3 closing parenthesis period&. .. Of\end course{ a lcomma i g n e d there}\ right are a. $ great many formulas involving Dixmier traces b esides open parenthesis 4 period 1 closing parenthesis and open parenthesis 4 period 3 closing parenthesis comma particularly\ hspace ∗{\ inf the i l l development} Singular of traces open parenthesis\quad $ lo cal0 closing< parenthesis\rho \ in indexM { 1 , \ infty } ( L ˆ{ 2 } (M ,theory E in ) noncommutative ) ˆ{ ∗ } geometry, $ \quad openwhich square bracket are not 4 7\ commaquad Dixmier 1 .. 6 comma traces 1 9 comma\quad .. 4and 8 comma\quad .. 4such 9 comma that .. 5 .. 0 comma .. 5 1 comma 5 2 closing square bracket period .. There are geometric conse hyphen \noindentquences of the$ \ formulationrho (T involving{\ theDelta Dixmier} trace) > and associations0 , $ t exist o pseudo by hyphen a variant differential of the argument after ( 2 . 4 5 ) . \quad Correspondingly operators open square bracket 1 7 comma .. 5 3 comma 5 4 comma .. 5 5 closing square bracket comma including extensions of Connes quoteright\ begin { a trace l i g n ∗} theorem open square bracket 5 6 comma .. 5 7 comma .. 5 3 closing square bracket period .. There are \rho (P { a } T {\Delta } ) = \ f r a c {\rho (T {\Delta } ) }{ Vol ( M ) }\ int { M } a ( x ) dx , a \ in C ˆ{\ infty } (M) \ tag ∗{$ ( 4 . 5 ) $} \end{ a l i g n ∗}

\noindent provides the integral on a manifold for smooth functions up t o a non − zero constant .

\ hspace ∗{\ f i l l }The r e s u l t \quad ( 4 . 5 ) \quad raises the question about whether the class of Dixmier traces \quad $ ( V { 1 } ) $ \quad i s t oo

\noindent restrictive in the definition of the noncommutative integral ( 1 . 1 ) . If we define

$ V ˆ{ + } { s i n g } : = \{ 0 < \rho \ in M ˆ{ ∗ } { 1 , \ infty }\mid \rho $ \quad i s a singular trace $ \} ( 4 . 6 ) $ and then define as the ‘ integral ’

\ begin { a l i g n ∗} \rho ( a \ langle D \rangle ˆ{ − n } ) , a \ in A, \rho \ in V ˆ{ + } { s i n g ˆ{ , }} \rho ( \ langle D \rangle ˆ{ − n } ) > 0 \ tag ∗{$ ( 4 . 7 ) $} \end{ a l i g n ∗}

\noindent we have the same claim to formulas ( 4 . 1 ) and ( 4 . 3 ) . \quad Of course , there are a great many formulas involving Dixmier traces b esides ( 4 . 1 ) and ( 4 . 3 ) , particularly in the development of ( lo cal ) index theory in noncommutative geometry [ 4 7 , 1 \quad 6 , 1 9 , \quad 4 8 , \quad 4 9 , \quad 5 \quad 0 , \quad 5 1 , 5 2 ] . \quad There are geometric conse − quences of the formulation involving the Dixmier trace and associations t o pseudo − differential operators [ 1 7 , \quad 5 3 , 5 4 , \quad 5 5 ] , including extensions of Connes ’ trace theorem [ 5 6 , \quad 5 7 , \quad 5 3 ] . \quad There are Measure Theory in Noncommutative Spaces .... 1 7 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 1 7 1 7 also many uses of the Dixmier trace in genuine noncommutative examples comma for example in the \ [ geometry\ r u l e {3em of t}{ otally0.4 pt disconnected}\ ] set s open square bracket 5 8 comma .. 5 9 comma 2 .. 0 closing square bracket period Arealso there many any of uses these of results the Dixmier for which trace the submajorisation in genuine noncommutative property of the Dixmier examples trace , i for s example in the a necessitygeometry open of parenthesis t otally disconnected recall Dixmier settraces s [ correspond 5 8 , 5 t 9 o , the 2 fully 0 ] symmetric . singular functionals on m sub 1 comma infinity \noindentfrom Sectionalso 2 period many 1 closing usesof parenthesis the Dixmier ? .. Several trace current in proofs genuine depend noncommutative on submajorisation examples comma e period , for g period example residue in and the heat geometryAre of there t otally any of disconnectedthese results for set which s [ the 5 8 submajorisation , \quad 5 9 , property 2 \quad of0 the ] .Dixmier trace kernel i s a necessity ( recall Dixmier traces correspond t o the fully symmetric singular functionals results are dependent on submajorisation open parenthesis cf period proofs in open square bracket 1 .. 5 closing square bracket and open Are thereon m1 any,∞ of these results for which the submajorisation property of the Dixmier trace i s squarefrom bracket Section 3 3 closing2.1)? squareSeveral bracket current closing parenthesis proofs depend period on.. It submajorisation i s open whether this , e . g . residue and adependence necessity is a( necessity recall for Dixmier the results traces or an artifact correspond of the proofs t o theperiod fully symmetric singular functionals on $ m { 1 , \heatinfty kernel}$ results are dependent on submajorisation ( cf . proofs in [ 1 5 ] and [ 3 3 ] ) . 4 periodIt i s 1 open .. Normal whether extension this dependence is a necessity for the results or an artifact of the proofs . To make4 . 1 things Normal simpler comma extension we dispense with the constants for the moment period .. Define the noncom hyphen \noindentmutative integralfromSection as $2 . 1 ) ?$ \quad Several current proofs depend on submajorisation , e . g . residue and heat kernel resultsTo make are dependent things simpler on submajorisation , we dispense with ( the cf . constants proofs in for [ the 1 \ momentquad 5]and[33]). . Define the \quad It i s open whether this Equation:noncom open - parenthesismutative integral 4 period 8 as closing parenthesis .. Capital Phi sub rho open parenthesis a closing parenthesis = rho open parenthesis adependence angbracketleft D is right a necessity angbracket to for the powerthe results of minus n or closing an artifactparenthesis comma of the a in proofs A comma . rho in V sub sing to the power of comma to the power of plus rho open parenthesis angbracketleft D right angbracket to the power of minus n closing parenthesis = 1 period \noindent 4 . 1 \quad Normal extension−n + −n By the discussion in the introductionΦρ(a) = commaρ(ahDi the), same a ∈ formula A, ρ ∈ Capital Vsing, Phiρ(hD subi rho) = defines 1. a state of A and(4 A. to8) the power of prime prime period By the discussion in the introduction , the same formula Φ defines a state of A and A00. \noindentLet M b eTo a compact make things n hyphen simpler dimensional , we Riemannian dispense manifold with the commaρ constants A = C to for the the power moment of infinity . \ openquad parenthesisDefine the M closing noncom − Let M b e a compact n− dimensional Riemannian manifold , A = C∞(M) acting by multi- parenthesismutative acting integral by multipliers as pliers on L2(M), and d + d∗ the Hodge – Dirac operator acting ( densely ) on square integrable on L to the power of 2 open parenthesis M closing parenthesis comma and d plus d to the power of * the Hodge endash Dirac operator \ beginsections{ a l i g n ∗} acting openof the parenthesis exterior densely bundle closing . Then parenthesis on square integrable sections \Phiof the{\ exteriorrho bundle} ( period a .. ) Then = \rho ( a \ langle D \rangle ˆ{ − n } ) , a \ in A , angbracketleft\rho \ in d plusV d ˆ to{ the+ power} { s of i n * g right ˆ{ angbracket, }}\rho to the power( \ oflangle minus n =D open parenthesis\rangle ˆ 1{ plus − Capitaln } Delta) closing = 1 parenthesis . \ tag ∗{$ ( hd + d∗i−n = (1 + ∆)−n/2 =: T to4 the . power 8 of minus ) $} n slash 2 = : T sub Capital Delta ∆ 2 \endwhich{whicha l i we g n take∗} we as take acting as actingas a compact as a compactoperator on operator L to the power on L of(M 2) open. By parenthesis ( 4 . 5 ) M closing parenthesis period .. By open parenthesis 4 period 5 closing parenthesis \noindent By the discussion in the introductionZ , the same formula $ \Phi {\rho }$ defines a state of Equation: open parenthesis 4 period 9 closing parenthesis .. Capital∞ Phi sub rho open parenthesis f closing parenthesis = integral sub M fd $ A $ and $ A ˆ{\prime \primeΦρ}(f). = $ fdxˆ, f ∈ C (M) (4.9) x-circumflex sub comma f in C to the power of infinity openM parenthesis M closing parenthesis where d x-circumflex i s the normalised volume form period .. Recall Capital Phi sub rho∞ i s a state on L to the power of infinity open Letwhere $M$d bxˆ i e s a the compact normalised $n volume− $ form dimensional . Recall RiemannianΦρ i s a state manifold on L (M $). ,Hence A = , by C ˆ{\ infty } ( parenthesisuniform M closing continuity parenthesis of both period the .. Hence left hand comma side by anduniform right hand side of ( 4 . 9 ) Mcontinuity ) $ acting of both bythe left multipliers hand side and right hand side of open parenthesis 4 period 9 closing parenthesis onCapital $ L Phi ˆ{ sub2 } rho( open M parenthesis ) ,$and$d f closing parenthesisZ + = integral dˆ{ sub ∗ } M$ fd the x-circumflex Hodge −− subDirac comma operatorf in C open actingparenthesis ( M densely closing ) on square integrable sections parenthesis comma Φρ(f) = fdxˆ, f ∈ C(M), M \noindentwhich identifiesof the Capital exterior Phi sub bundle rho bar . C\ openquad parenthesisThen M closing parenthesis with the normalised integral on C open parenthesis which identifies Φ | C(M) with the normalised integral on C(M). Hence Φ | C(M) ∈ C(M)∗ M closing parenthesis periodρ .. Hence Capital Phi sub rho bar C open parenthesis M closingρ parenthesis in C open parenthesis M closing has an extension t o L∞(M) given by the normalised integral parenthesis\ [ \ langle to thed power + of * d has ˆ{ ∗ } \rangle ˆ{ − n } = ( 1 + \Delta ) ˆ{ − n / 2 } = : T an{\ extensionDelta t}\ o] L to the power of infinity open parenthesisZ M closing parenthesis given by the normalised integral ∞ phi open parenthesis f closing parenthesis =φ( integralf) = subfdxˆ M, fdf ∈ x-circumflexL (M). sub comma f in L to the power of infinity open parenthesis M M closing parenthesis period M, \noindentIf oneIf one .. considerswhich considers we.. for take the .. for as moment the acting .. moment a ashomogeneous a compact a homogeneous .. compact operator .. manifold on compact $ .. L M ˆ{ comma2 manifold} ..( by inspection M ) ..by . Capital $ \quad Phi subBy rho ( in 4 . 5 ) Φ ∈ L∞(M)∗ L toinspection the power of infinityρ open parenthesisi s a invariant M closing parenthesisstate , and to one the mightpower of think * i sa it invariant i s obvious state that comma we and one might think it i s Φ = φ M. obvious\ beginmust that{ a l i weg have n must∗} haveρ Capitalsince Phi the sub volume rho = form phi since provides the unique invariant measure class on \PhitheWhile volume{\rho this form} last provides( phrase f the uniquei ) s = invariant\ int measure{ M } classfd on M\ periodhat{x ..} While{ , this} lastf phrase\ in i sC ˆ{\ infty } (M) \ tag ∗{$ ( Φ ∈ L∞(M)∗ 4true .true comma 9 , there there ) $} is no no reason reason a prioria priori why thewhy state the Capital state Phi subρ rho in L tocorresp the power onds of infinity t o a open measure parenthesis . M closing parenthesis to\end the{ powerWea l i g n of∗} * corresp onds t o a measure period .... We L∞(M) knowknow that that there there can b can e an b infinitude e an infinitude of invariant of invariantstates on L states to the on power of infinitywhich open all parenthesis agree with M closing the parenthesis .. which all C(M) [60, 6 agree\noindent withLebesgue thewhere integral $ d on \hat{x} $Theorem i s the 3 normalised . 4 ] ( and first volume Baire form class . functions\quad R e c a). l l $ \Phi {\rho }$ i s a state on argument7 Φ = φ $ LLebesgue ˆ{\Itinfty wasintegral not} on until( C open M recently parenthesis ) that . $ M an\ closingquad elementaryHence parenthesis , by open uniform squarewas bracket found 6 0 proving comma Theoremρ for 3 period the 4 closing square bracket n− opencontinuity parenthesist orus of and . both first An Baire the earlier class left claim functions hand [ 4 side 2 6 ,closing p and. 297 parenthesis right ] , applied hand period monotone side of ( convergence 4 . 9 ) t o both sides C∞− L∞− It wasof ( not 4 .until 9 ) recently t o extend that an from elementaryfunctions argument to t the o powerfunctions of 7 .. was . found This proving method Capital of Phi proof sub i rho s = phi for the n hyphen \ [ t\ orusPhicircular period{\rho , .. monotone An} earlier( claim convergence f open ) square = can\ bracketint b e applied{ 4M 2 comma} tfd o p the period\ righthat 297{x hand} closing{ side, square} , sincef bracket\ thein comma integralC(M), applied is monotone convergence\ ] L∞(M). t o botha sides normal of open linear parenthesis function 4 periodon 9 closingTo parenthesis apply monotone t o extend convergence t o the left hand side it Φ ∈ L∞(M) . frommust C to b the e powerknown of infinityρ hyphen∗ functions t o L to the power of infinity hyphen functions period .. This method of proof i s circular comma\noindent monotoneIn [which 6 1 convergence ] we identifies used a different $ \Phi method{\rho than}\ thatmid for tC orii to ( show M the ) $following with the result normalised . We integral on $C (can Mremark b e ) applied . that $ t o the\quad [ right 6 1Hence hand ] concentratedside $ comma\Phi since{\ onlyrho the onintegral}\ Dixmiermid is a normal tracesC(M) linear . function We state on\ in L the to theC result power ( for of M infinity ) ˆ open{ ∗ parenthesis }$ has Man closing extensiongeneral parenthesis singular t operiod $traces L ˆ{\ hereinfty . } ( M ) $ given by the normalised integral To apply monotone convergence t o the left hand side it must b e known Capital Phi sub rho in L to the power of infinity open parenthesis M\ [ closing\phi parenthesis( f sub ) * period = \ int { M } fd \hat{x} { , } f \ in L ˆ{\ infty } (M). \ ] In open square bracket 6 1 closing square bracket we used a different method than that for t orii to show the following result period We 6 are indebted to B . de Pagter for pointing this out and bringing Rudin ’ s paper to our attention . We also remark We thank P . Dodds for additional explanation . \noindentthat .. openI f square one \ bracketquad c 6 o n1 s closing i d e r s square\quad bracketf o r the .. concentrated\quad moment only on\quad Dixmiera homogeneous traces period ..\quad We statecompact the result\quad for generalmanifold \quad 7 generalisation of this argument is discussed in our final section , Section 6 . singular$ M traces, $ \quad byThe inspection \quad $ \Phi {\rho }\ in $ $here L ˆperiod{\ infty } ( M ) ˆ{ ∗ }$ i s a invariant state , and one might think it i s obvious that we must have $ \hlinePhi {\rho } = \phi $ s i n c e the6 sub volume We are form indebted provides to B period the de unique Pagter for invariant pointing this measure out and bringing class onRudin $ quoteright M . $ s paper\quad toWhile our attention this period last We phrase also i s thank P period Dodds for additional explanation period \noindent7 sub Thetrue generalisation , there of this is noargument reason is discussed a priori in our why final the section state comma $ \ SectionPhi {\ 6 periodrho }\ in L ˆ{\ infty } (M ) ˆ{ ∗ }$ corresp onds t o a measure . \ h f i l l We

\noindent know that there can b e an infinitude of invariant states on $ L ˆ{\ infty } ( M ) $ \quad which all agree with the Lebesgue integral on $C ( M ) [ 6 0 , $ Theorem3 . 4 ] ( and first Baire class functions 6 ) .

It was not until recently that an elementary $ argument ˆ{ 7 }$ \quad was found proving $ \Phi {\rho } = \phi $ f o r the $ n − $ t orus . \quad An earlier claim [ 4 2 , p . 297 ] , applied monotone convergence t o both sides of ( 4 . 9 ) t o extend from $ C ˆ{\ infty } − $ functions t o $Lˆ{\ infty } − $ functions . \quad This method of proof i s circular , monotone convergence can b e applied t o the right hand side , since the integral is a normal linear function on $ L ˆ{\ infty } ( M ) . $ To apply monotone convergence t o the left hand side it must b e known $ \Phi {\rho }\ in L ˆ{\ infty } (M) { ∗ } . $

In [ 6 1 ] we used a different method than that for t orii to show the following result . We remark that \quad [ 6 1 ] \quad concentrated only on Dixmier traces . \quad We state the result for general singular traces here .

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

$ 6 { We }$ are indebted to B . de Pagter for pointing this out and bringing Rudin ’ s paper to our attention . We also thank P . Dodds for additional explanation .

\ centerline { $ 7 { The }$ generalisation of this argument is discussed in our final section , Section 6 . } 1 8 .... S period Lord and F period Sukochev \noindenthline1 81 8 \ h f i l l S . Lord and F . Sukochev S . Lord and F . Sukochev Theorem 4 period 3 period .. Let M be a compact n hyphen dimensional Riemannian manifold comma Capital Delta the Hodge endash Lapla\ [ \ r hyphen u l e {3em}{0.4 pt }\ ] cian on L to the power of 2 open parenthesis M closing parenthesis comma .. and T sub Capital Delta .. as above period .. Then fT Capital Theorem 4 . 3 . Let M be a compact n− dimensional Riemannian manifold , ∆ the Delta in M sub 1 comma infinity open parenthesis2 L to the power of 2 open parenthesis M closing parenthesis2 closing parenthesis .. if and only \noindentHodgeTheorem – Lapla -4 . cian 3 . on\quadL (MLet), and $M$T∆ beas a above compact . $ Then n −fT$∆ dimensional∈ M1,∞(L (M)) Riemannianif manifold $ , if f in L to the power of 2 open2 parenthesis M closing parenthesis .. and \DeltaEquation:and$ only the open if Hodge parenthesisf ∈ L−−(MLapla 4) periodand− 10 closing parenthesis .. Capital Phi sub rho open parenthesis f closing parenthesis : = rho open parenthesiscian on fT $ Capital L ˆ{ Delta2 } closing( M parenthesis ) , = $ integral\quad suband M fd $ x-circumflex T {\Delta sub comma}$ \quad forall fas in L above to the . power\quad of 2Then open parenthesis $ fT \Delta \ in M { 1 , \ infty } ( L ˆZ{ 2 } ( M ) ) $ \quad if and only if $ f \ in L ˆ{ 2 } M closing parenthesis 2 (for M any ) singular $ \quad trace 0and less rho in MΦρ sub(f) :=1 commaρ(fT ∆) infinity = fd openxˆ, parenthesis∀f ∈ L (M L) to the power of 2 open parenthesis (4.10) M closing parenthesis M closing parenthesis to the power of * such that rho open parenthesis T sub Capital Delta closing parenthesis = 1 period \ begin { a l i g n ∗} 2 ∗ Prooffor period any singular .. We adapt trace only0 those< ρ ∈ parts M1, of∞( openL (M square)) such bracket that 6 1ρ( commaT∆) = 1 Theorem. Proof 2 . periodWe 5 closing adapt square only bracket concerning open parenthesis\Phithose{\ 4rho periodparts} of1 0( [ closing 6 1 f , Theorem parenthesis ) : 2 period =. 5 ]\ concerning..rho Define( the (sequence fT 4 . 1\ 0Delta ) . Define) = the sequence\ int { M } fd \hat{x} { , } \ f ospace r aspace l l f \ in L ˆ{ 2 } (M) \ tag ∗{$ ( 4 . 10 ) $} \endhline{ a l i g n ∗} m sub 0 = l to the power of 1 .. in the norm of .. m sub 1 comma infinity open parenthesis 4 period 1 1 closing parenthesis \noindentand the idealfor M any sub 0 singular given by m trace sub 0 = j$ sub 0 M< sub 0\ periodrho ..\ Thein RieszM seminorm{ 1 , on M\ infty sub 1 comma} ( infinity L ˆ{ is defined2 } (M) by m = `1 in the norm of m (4.11) and the ideal M given by m = j . The Riesz ) Êquation:{ ∗ }$0 open such parenthesis that $ 4\ periodrho 1 2(T closing1,∞ parenthesis{\Delta ..} bar) T bar = sub0 10 : = .inf $ Q in0 M subM0 0 bar T minus Q bar sub 1 comma infinity seminorm on M is defined by periodProof . \quad We adapt1,∞ only those parts of [ 6 1 , Theorem 2 . 5 ] concerning ( 4 . 1 0 ) . \quad Define the sequence By construction every singular functional 0 less rho in M sub 1 comma infinity to the power of * vanishes on M sub 0 period .. Hence \noindentbar rho openspace parenthesis T closing parenthesis bar less or equal bar rho bar bar T bar sub 0 comma T in M sub 1 comma infinity period k T k0:= inf 0 k T − Q k1,∞ . (4.12) The proof in open square bracket 6 1 closing squareQ∈M bracket involved comma firstly comma showing that \ [ bar\ r u fT l e Capital{3em}{ Delta0.4 pt bar}\ 0] less or equal C bar f bar sub 2 bar T sub Capital Delta bar sub 1 comma infinity ∗ forBy a constant construction C greater every 0 open singular parenthesis functional achieved0 by< open ρ ∈ M square1,∞ vanishes bracket 6 on 1 commaM0. CorollaryHence 4 period 5 closing square bracket and open square bracket 6 1 comma Examples 4 period 6 comma 4 period 7 closing square bracket closing parenthesis period .. Hence comma from $the m preceeding{ 0 } = two equations\ e l l ˆ{ comma1 }$ fT\quad Capital| ρ(T ) in|≤k Delta theρ kk inT norm Mk0 sub,T of 1∈ comma\ Mquad1,∞ infinity. $ m and{ 1 , \ infty } ( 4 . 1 1 ) $ barThe rho open proof parenthesis in [ 6 1 ] fT involved Capital Delta , firstly closing , showing parenthesis that bar less or equal C bar rho bar bar f bar sub 2 bar T sub Capital Delta bar suband 1 comma the ideal infinity $M { 0 }$ given by $ m { 0 } = j { M { 0 }} . $ \quad The Riesz seminorm on $ M { 1 , \ infty }$ is defined by for all f in C to the power of infinity open parenthesisk fT ∆ k 0 ≤ MC closingk f k2k parenthesisT∆ k1,∞ period .. Moreover comma .. if C to the power of infinity open parenthesis M closing parenthesis ni f sub n right arrow f in L to the power of 2 open parenthesis M closing parenthesis .. in the L to the power of\ begin 2 hyphenfor{ a a l inorm constantg n ∗} then fTC Capital > 0( achieved Delta i s by [ 6 1 , Corollary 4 . 5 ] and [ 6 1 , Examples 4 . 6 , 4 . 7 ] ) \ parallelbounded. Hence commaT , indeed from\ parallel fT the Capital preceeding{ Delta0 } in two M: sub equations = 1 comma\ inf , infinity fT{ ∆Q∈ comma M\1in,∞ and andM } { 0 }\ parallel T − Q \ parallel { 1 , bar\ infty rho open} parenthesis. \ tag ∗{ open$ ( parenthesis 4 . f minus 1 2f sub ) n closing$} parenthesis T sub Capital Delta closing parenthesis bar less or equal C bar\end rho{ a bar l i g nbar∗} f minus f sub n bar 2 bar T| subρ(fT Capital∆) |≤ C Deltak ρ kk barf 1k2 commak T∆ k1 infinity,∞ right arrow 0 period The result openf parenthesis∈ C∞(M 4). period 1 0 closing parenthesisC∞(M now) follows3 f from→ Connesf ∈ quoterightL2(M) trace theoremL2 open− parenthesis specifically \noindentfor allBy construction everyMoreover singular , if functional $n 0 < \rho \ in inM the ˆ{ ∗ } { 1 , \ infty }$ commanorm from open then parenthesisfT ∆ i s4 bounded period 9 closing , indeed parenthesisfT ∆ ∈ M closing1,∞, and parenthesis period blacksquare vanishesAs a corollary on $M comma{ 0 } . $ \quad Hence

Capital Phi sub rho open parenthesis| ρ((f − ff closingn)T∆) |≤ parenthesisC k ρ kk f =− integralfn k 2 k subT∆ Mk 1 fd, ∞ x-circumflex → 0. sub comma f in L to the power of infinity open parenthesis\ [ \mid M\ closingrho parenthesis(T) \mid \ leq \ parallel \rho \ parallel \ parallel T \ parallel { 0 } ,TandThe we\ have resultin anM example ( 4{ .1 0where ) , now\ followsinfty } from. Connes\ ] ’ trace theorem ( specifically , from (4.9)). CapitalAs a Phi corollary sub rho in, L to the power of infinity open parenthesis M closing parenthesis sub * open parenthesis i period e period the noncommutativeZ integral i s normal closing parenthesis period \noindent The proof in [ 6 1 ] involved , firstly ,∞ showing that Φρ(f) = fdxˆ, f ∈ L (M) M \ [ \ paralleland we havefT an example\Delta where\ parallel 0 \ leq C \ parallel f \ parallel { 2 }\ parallel T {\Delta } \ parallel { 1 , \ infty }\ ] ∞ Φρ ∈ L (M)∗ \noindent( i . efor . the a noncommutative constant $ C integral> 0 i s normal ( $ achieved ) . by [ 61 , Corollary 4 . 5 ] and [ 61 , Examples 4 . 6 , 4 . 7 ] ) . \quad Hence , from the preceeding two equations $ , fT \Delta \ in M { 1 , \ infty }$ and

\ [ \mid \rho ( fT \Delta ) \mid \ leq C \ parallel \rho \ parallel \ parallel f \ parallel { 2 }\ parallel T {\Delta }\ parallel { 1 , \ infty }\ ]

\noindent f o r a l l $ f \ in C ˆ{\ infty } ( M ) . $ \quad Moreover , \quad i f $ C ˆ{\ infty } (M) \ ni f { n }\rightarrow f \ in L ˆ{ 2 } ( M ) $ \quad in the $ L ˆ{ 2 } − $ norm then $ fT \Delta $ i s bounded , indeed $ fT \Delta \ in M { 1 , \ infty } , $ and

\ [ \mid \rho ( ( f − f { n } )T {\Delta } ) \mid \ leq C \ parallel \rho \ parallel \ parallel f − f { n }\ parallel 2 \ parallel T {\Delta }\ parallel 1 , \ infty \rightarrow 0 . \ ]

\noindent The result ( 4 . 1 0 ) now follows from Connes ’ trace theorem ( specifically , from $ ( 4 . 9 ) ) . \ blacksquare $ As a corollary ,

\ [ \Phi {\rho } ( f ) = \ int { M } fd \hat{x} { , } f \ in L ˆ{\ infty } (M) \ ]

\noindent and we have an example where

\ [ \Phi {\rho }\ in L ˆ{\ infty } (M) { ∗ } \ ]

\noindent ( i . e . the noncommutative integral i s normal ) . Measure Theory in Noncommutative Spaces .... 1 9 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 1 9 1 9 4 period 2 .. Tools for the noncommutative integral \ [ As\ r mentioned u l e {3em}{ in0.4 the pt introduction}\ ] comma when Equation: open parenthesis 4 period 13 closing parenthesis .. Capital Phi sub rho open parenthesis a closing parenthesis = rho open parenthesis4 . 2a angbracketleft Tools for D rightthe angbracketnoncommutative to the power integral of minus n closing parenthesis comma a in A to the power of prime prime comma rho\noindent in VAs sub mentioned sing4 . to 2 the\quad in power theTools of introduction comma for to the the , powerwhen noncommutative of plus rho open integral parenthesis angbracketleft D right angbracket to the power of minus n closing parenthesis = 1 \noindent As mentioned in the introduction , when defines a normal state on the von Neumann algebra−n A to the00 power+ of prime prime−n subset L open parenthesis H closing parenthesis comma Φ (a) = ρ(ahDi ), a ∈ A , ρ ∈ V , ρ(hDi ) = 1 (4.13) .. there are Radon endash Nikodymρ sing \ begin { a l i g n ∗} theoremsdefines comma a normal dominated state convergence on the von theorems Neumann comma algebra etc periodA00 ⊂ for L this(H), statethere period are Radon – Nikodym \PhiVery lit{\ trho le i s} known( about a equivalent ) = measure\rho hyphen( a theoretic\ langle results whenD \ openrangle parenthesisˆ{ − 4 periodn } 1) 3 closing , a parenthesis\ in iA s not ˆ{\prime \primetheorems} , ,\ dominatedrho \ in convergenceV ˆ{ + } theorems{ s i n g ˆ ,{ etc, .}}\ for thisrho state( . \ langle D \rangle ˆ{ − n } ) = known t o Very lit t le i s known about equivalent measure - theoretic results when ( 4 . 1 3 ) i s not 1 \btag e normal∗{$ ( open 4 parenthesis . 13 and hence ) $} the emphasis at present centres on characterising normality in open parenthesis 4 period 1 3 closing \end{knowna l i g n ∗} t o b e normal ( and hence the emphasis at present centres on characterising normality parenthesisin ( comma4 . 1 3 see ) , see Section 6 closing parenthesis period \noindentSectiondefines 6 ) . a normal state on the von Neumann algebra $ A ˆ{\prime \prime }\subset L( Define the classical weak l to the power of p hyphenp spaces .. open parenthesis or equivalently the fully symmetric Lorentz sequence H ) , $ Define\quad thethere classical are Radon weak −−` − Nikodymspaces ( or equivalently the fully symmetric Lorentz Linesequence 1 spaces l to the power of p comma infinity closing parenthesis : Line 2 l to the power of p comma w = open brace x in c sub 0 bar xtheorems to the power , of dominated * open parenthesis convergence n closing theorems parenthesis , = etc O open . for parenthesis this state n to the . power of minus 1 divided by p closing parenthesis closing brace comma 1 less or equal p less infinity period open parenthesis 4 periodspaces` 14p,∞ closing): parenthesis Very lit t le i s known about equivalent measure − theoretic results when ( 4 . 1 3 ) i s not known t o Denote the corresp onding closedp,w fully symmetric∗ ideals of− compact1 operators by L to the power of p comma w open parenthesis i period e periodb e normall to the power ( and of p hence comma the` w == emphasis{x ∈ c0 | x at(n) present = O(n p ) centres}, 1 ≤ p 1, p known+ q $= 1 ,, and\ e a l l singularˆ{ p trace , w }$ i s the $ p Theorem 4 period∗ 4 open parenthesis open square bracket 1 4 comma Lemma 1 period 4 closing square bracket comma open square bracket 0 < ρ ∈ M1,∞, we have TV ∈ M1,∞ and the H o¨ lder inequality : −2 ..$ 0 comma convexification Theorem 5 period of 1 $ closing\ e l l squareˆ{ 1 bracket , comma w } open, square 1 \ bracketleq 1p 5 comma< \ Lemmainfty 6 period[ 2 6 open 2 parenthesis ] . i$ closing\quad S i m i l a r l y , from \quad [ 1 5 ] \quad or \quad $ [ 3 8 ] , L ˆ{ p , w }$ \quad i s the $ p − $ convexification of parenthesis closing square bracket closing parenthesis period .. For T in M subp p comma1 infinity | ρ(TV ) |≤ ρ(| TV |) ≤ C ρ(| T |p) q) , $ Land ˆ{ V1 in M ,sub q w comma} . infinity $ \quad commaThe p comma $ p q greater− $p 1 convexification comma p to1ρ the(|V | powerq $M of minus{ p 1 plus , q to\ theinfty power}$ of minus\quad 1 =o f 1 comma the andi d ea a singular l $ M trace{ 1 0 less , rho in\ infty M sub 1} comma, $ infinity denoted√ to the $Z power{ ofp comma}$ to in the [ power 1 \quad of * we5 ]have , iTV s in strictly M sub 1 comma larger infinity that $ L ˆ{wherep ,Cp w:=1} if($see$[ρ ∈ V C := 1 + 2 5p−1 ] )ˆ{ 8 } . $ and the HandV o-dieresis∈L (H lder) : inequality1 and : p p o therwise . The inequality also holds for barT rho∈ Mopen1,∞ parenthesis TV closing parenthesis bar less or equal rho open parenthesis bar TV bar closing parenthesis less or equal C sub p rho\ hspace open parenthesis∗{\ f i l l }We bar\ Tquad bar todo the\ powerquad ofknow p closing , \quad parenthesisfrom hline\quad fromF. p to\ 1quad sub rhoC iopen p r i a parenthesis n i , \quad barD. V bar\quad q closingGuido parenthesis , \quad and \quad S. \quad S c a r l a t t i \quad [ 1 \quad 4 ] , \quad and \quad l a t e r \quad Guido \quad and 1 divided by q comma | ρ(TV ) |≤ ρ(| TV |) ≤k V k ρ(| T |) \noindentwhere subT and . V I s C o inl a to [ the 2 power\ h f i l of l p0 sub ] L , sub that open Dixmier parenthesis traces H to the satisfy power of a : H= sub $ closing\ddot{ parenthesiso} $ lder to the inequality power of 1 if .: rho\ h f i l l The result of [ 2 \ h f i l l 0 , Theorem 5 . 1 ] in V sub 1 and C sub p : = 1 plus 2 square root of p minus(thep 1= divided 1, q = ∞ bycase p o). therwise period .... The inequality also holds for T in M sub 1 comma\noindent infinityapplies to all singular traces $ \rho $ usedin(4.13) . The H o¨ lder inequality i s used t o prove that ρ(ahDi−n) i s a trace on A when (A,H,D) bar rho open parenthesis TV closing parenthesis bar less or equal rho open parenthesis bar TV bar closing parenthesis less or equal bar V i s a spectral triple and ρ ∈ V [14], [1 5, Theorem 6 . 1 ] . In [ 2 0 ] it was noted the bar\noindent rho open parenthesisTheorem4 bar . T 4 bar ( [closing 1 41 parenthesis ,Lemma1 open . 4 parenthesis ] , [ 2 the\quad p = 10 comma ,Theorem5 q = infinity . case 1 ] closing , [ 15 parenthesis ,Lemma6 period . 2( i ) ] ) . \quad For $ T same\ in proofM { appliesp , t o\ anyinfty singular}$ trace . The H dieresis-o lder inequality i s used t o prove that rho open parenthesis a angbracketleft D− rightn angbracket to the power of minus n andCorollary $ V \ in 4M . 1{ . qLet , (A\,H,Dinfty) be} a, spectral p , triple q > such1 that , phD ˆ{i − ∈1 } M1,+∞. q ˆ{ − 1 } = closingThen parenthesis .. i s a trace on A when open parenthesis A comma H comma D closing parenthesis ..( i s 4 . 7 ) 1a spectral , $ and triple a and singular rho in V sub trace 1 open $ square 0 < bracket\rho 1 4 closing\ in squareM ˆ bracket{ ∗ } comma{ 1 open , square\ infty bracketˆ{ 1, 5}} comma$ we Theorem have 6 $ period TV ( resp . ( 4 . 1 3 ) ) defines a finite positive trace ( resp . trace state ) on A. 1\ in closingM square{ 1 bracket , period\ infty .. In} open$ square bracket 2 .. 0 closing square bracket it was noted the same proof applies andt o anythe singular H $ \ traceddot period{o} $ lder inequality : Corollary .... 4 period 1 period .... Let .... open parenthesis A comma H comma D closing parenthesis .... be .... a .... spectral .... triple .... such\ [ \ ....mid that ....\rho angbracketleft( TV D right ) angbracket\mid \ toleq the power\rho of minus( n\ inmid M subTV 1 comma\mid infinity) period\ leq .... ThenC ....{ p open}\ parenthesisrho ( 4 \mid T \mid ˆ{ p } ) \ r u l e {3em}{0.4 pt } ˆ{ p } { 1 } {\rho ( \mid −1V p,w\mid } q ) \ f r a c { 1 }{ q } period 7 closing8Note that parenthesis the (p, ∞)− summable condition for K - cycles ( spectral triples ) is sometimes stated as hDi ∈ L , , \ ] −p 1,w −1 open1 ≤ parenthesisp < ∞[2, respp . 546 period ] , [ 3 .. , open p . 1 parenthesis59 ] ( equivalent 4 period to hD 1i 3 closing∈ L parenthesis) and sometimes closing as parenthesishDi ∈ M ..1,∞ defines(p = 1) a finite positive trace open −1 p,w −p 1,w −p parenthesisand h respDi period∈ L ..( tracep > 1)[2 state, IV closing . 2 ] ( notparenthesis equivalent .. to onh AD periodi ∈ L or hDi ∈ M1,∞). To distinguish the 1,w hlinestrictly smaller ideal L from the usually quoted domain of the Dixmier trace ( the ideal M1,∞) we have avoided \noindent $ where1,∞ { and V } C {\ in }ˆ{ p } { L }ˆ{ : = } { (H }ˆ{ 1 } { ) } i f { : }\rho 8 subthe Note notation that theL openfor parenthesisM1,∞. p comma infinity closing parenthesis hyphen summable condition for K hyphen cycles open parenthesis spectral\ in V triples{ 1 closing}$ andparenthesis $ C is{ sometimesp } : stated = as 1 angbracketleft + 2 \ Df r aright c {\ angbracketsqrt { p to− the power1 }}{ ofp minus}$ 1 o in therwise L to the power . \ h of f i p l l The inequality also holds for comma$ T w\ in commaM { 1 , \ infty }$ 1 less or equal p less infinity open square bracket 2 comma p period 546 closing square bracket comma open square bracket 3 comma p period\ begin 1{ 59a l closing i g n ∗} square bracket open parenthesis equivalent to angbracketleft D right angbracket to the power of minus p in L to the power of\ 1mid comma\ wrho closing( parenthesis TV ) and sometimes\mid \ asleq angbracketleft\rho D( right\mid angbracketTV to the\mid power) of minus\ leq 1 in M\ parallel sub 1 comma infinityV \ parallel open parenthesis\rho ( p =\mid 1 closingT parenthesis\mid and) \\ ( the p = 1 , q = \ infty case ) . \endangbracketleft{ a l i g n ∗} D right angbracket to the power of minus 1 in L to the power of p comma w open parenthesis p greater 1 closing parenthesis open square bracket 2 comma IV period 2 closing square bracket open parenthesis not equivalent to angbracketleft D right angbracket to the powerThe H of minus $ \ddot p in{o L} to$ the lder power inequality of 1 comma w i or s angbracketleft used t o prove D right that angbracket $ \rho to the power( a of minus\ langle p in M subD 1 comma\rangle infinityˆ{ − closingn } ) parenthesis $ \quad periodisatraceon .. To distinguish the $A$ strictly when $( A , H , D )$ \quad i s asmaller spectral ideal Ltriple to the power and of $ 1\ commarho w\ fromin theV usually{ 1 } quoted[ domain 1 4 of the ] Dixmier , trace [open 1 parenthesis 5 ,$Theorem6.1]. the ideal M sub 1 comma\quad In [ 2 \quad 0 ] it was noted the same proof applies infinityt o any closing singular parenthesis trace we have . avoided the notation L to the power of 1 comma infinity for M sub 1 comma infinity period \noindent C o r o l l a r y \ h f i l l 4 . 1 . \ h f i l l Let \ h f i l l $ ( A , H , D ) $ \ h f i l l be \ h f i l l a \ h f i l l s p e c t r a l \ h f i l l t r i p l e \ h f i l l such \ h f i l l that \ h f i l l $ \ langle D \rangle ˆ{ − n }\ in M { 1 , \ infty } . $ \ h f i l l Then \ h f i l l ( 4 . 7 )

\noindent ( resp . \quad ( 4 . 1 3 ) ) \quad defines a finite positive trace ( resp . \quad trace state ) \quad on $ A . $

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

$ 8 { Note }$ thatthe $( p , \ infty ) − $ summable condition for K − cycles ( spectral triples ) is sometimes stated as $ \ langle D \rangle ˆ{ − 1 }\ in L ˆ{ p , w } , $ $ 1 \ leq p < \ infty [ 2 ,$ p.546] ,[3,p.159](equivalentto $ \ langle D \rangle ˆ{ − p }\ in L ˆ{ 1 , w } ) $ and sometimes as $ \ langle D \rangle ˆ{ − 1 }\ in M { 1 , \ infty } ( p = 1 ) $ and $ \ langle D \rangle ˆ{ − 1 }\ in L ˆ{ p , w } ( p > 1 ) [ 2 ,$ IV.2](notequivalentto $ \ langle D \rangle ˆ{ − p }\ in L ˆ{ 1 , w }$ or $ \ langle D \rangle ˆ{ − p }\ in M { 1 , \ infty } ) . $ \quad To distinguish the strictly smaller ideal $ L ˆ{ 1 , w }$ from the usually quoted domain of the Dixmier trace ( the ideal $ M { 1 , \ infty } ) $ we have avoided the notation $ L ˆ{ 1 , \ infty }$ f o r $ M { 1 , \ infty } . $ 20 .... S period Lord and F period Sukochev \noindenthline2020 \ h f i l l S . Lord and F . Sukochev S . Lord and F . Sukochev 5 .. Zeta functions .. and .. heat .. kernels \ [ Zeta\ r u functionsl e {3em}{ and0.4 heat pt }\ kernel] asymptotics are alternative ways t o measure quoteleft the log divergence of the trace quoteright period The5 Wodzicki Zeta residue functions .. Res .. on .. classical and pseudo heat hyphen differential kernels operators .. of order .. minus n .. on .. a n hyphen \noindentdimensionalZeta functions5 compact\quad andZeta Riemannian heat functions kernel manifold asymptotics\quad M derivesand are it\ squad alternative nameheat as the\ waysnoncommutativequad t ok emeasure r n e l s residue ‘ the log divergence fromof the the zeta trace function ’ . formulation of the residue for positive elliptic pseudo hyphen differential opera hyphen \noindentt ors openTheZeta square Wodzicki functions bracket residue 4 3 comma and heat Res .. 6 3 kernel closing on square classicalasymptotics bracket pseudo period are - differential alternative operators ways t o measureof order ‘ the log divergence ofExplicitely the−n traceon comma ’ a . ifn 0− lessdimensional Q in Op sub compact cl to the power Riemannian of d open manifoldparenthesis MM closingderives parenthesis it s name comma as thed greater 0 comma i s elliptic commanoncommutative then Q to the power residue of minus from s comma the zeta s greater function n slash formulation d comma i s of trace the class residue open for square positive bracket elliptic 6 4 comma .. 4 4 comma 4 .. 6 closingThe Wodzickipseudo square bracket - differential residue period\quad operaRes - t\ orsquad [ 4on 3 ,\quad 6 3 ]classical . pseudo − differential operators \quad o f order \quad $ − n $ \quad on \quad a $ n d − $ −s It i s knownExplicitely the function , if 0 < Q ∈ Opcl(M), d > 0, i s elliptic , then Q , s > n/d, i s trace class [ 6 4 , dimensionalzeta4 sub 4 , Q 4 open compact 6 parenthesis] . Riemannian s closing parenthesis manifold : = Tr$ M open $ parenthesis derives Q it to s the name power as of theminus noncommutative s closing parenthesis residue comma s greater n slashfrom dIt comma the i s known zeta function the function formulation of the residue for positive elliptic pseudo − differential opera − tcalled ors [the 4 zeta 3 , function\quad and6 3 initially ] . introduced for the Laplacian open square bracket 6 5 closing square bracket has a meromorphic con −s hyphen ζQ(s) := Tr(Q ), s > n/d, \ hspacet inuation∗{\ withf i l l } simpleExplicitely pole at s = , n if slash $ d 0 open< squareQ bracket\ in 6 4Op comma ˆ{ d 6 6} closing{ c l square} ( bracket M and) from , open d square> 0 bracket , $ 4 3 i s ellipticcalled the , zeta then function $Qˆ{ and − initiallys } introduced, s > forn the / Laplacian d ,$ [ 6 5 istraceclass[64, ] has a meromorphic \quad 4 4 , 4 \quad 6 ] . closingcon square - t bracket inuation with simple pole at s = n/d[6 4, 66] and from [ 4 3 ] Equation: open parenthesis 5 period 1 closing parenthesis .. res sub s = n slash d zeta sub Q open parenthesis s closing parenthesis = limint right\noindent arrow s openIt i parenthesis s known n the slash function d closing parenthesis sub plus to the power of open parenthesis s minus n slash d closing parenthesis Tr open parenthesis Q to the power of minus s closing parenthesis(s = minus−s 1 divided1 by− dn/d Res open parenthesis Q to the power of minus n slash \ [ \zeta { Q } ( sress )=n/dζ :Q(s) = = lim Tr+ (− n/d Q)Tr( ˆ{Q − ) =s− }Res()Q ,). s > n /(5.1) d , \ ] d closing parenthesis period →s(n/d) d

The Wodzicki residue has a similar identification involving the heat kernel operator e to the power of minus−tQ tQ comma t greater 0 period TheThe heat Wodzicki kernel operator residue is so has named a similar since identificationthe kernel K open involving parenthesis the t heat comma kernel x comma operator y closinge parenthesis, t > 0. in C to the power of \noindentThe heatcalled kernel the operator zeta function is so named and since initially the kernel introducedK(t, x, y) ∈ C for∞((0 the, ∞)× LaplacianM ×M) associated [ 6 5 ] has a meromorphic con − infinity open parenthesis open parenthesis−t∆ 0 comma infinity closing parenthesis times M times M closing parenthesis associated tinuationwithsimplepoleatt ot the o thetrace trace hyphen - class class family ee to the, t$s power> 0, ∆ = ofthe minus Hodge n t / Capital Laplacian d Delta [ ,comma 6 4 t greater , 0 6 comma 6 Capital ]$ Deltaandfrom[43] the Hodge Laplacian comma Z \ beginopen{ parenthesisa l i g n ∗} e to the power of(e minus−t∆h)( tx Capital) = K Delta(t, x, h y) closingh(y)dy, parenthesis h ∈ L2(M open) parenthesis x closing parenthesis = integral sub M K openr e s parenthesis{ s = t comma n x/ comma d }\ y closingzeta parenthesis{ QM} ( h open s parenthesis ) = y\ closinglim parenthesis{\rightarrow dy comma{ hs in} L to( the n power / of 2d open ) } { + }ˆ{ ( parenthesiss } −i s a Mn solution closing / parenthesis to d the ) ( local Tr ) heat ( Q equation ˆ{ − [s 6} 5 ] ) = − \ f r a c { 1 }{ d } Res ( Q ˆ{ − n / d } ).i s a\ solutiontag ∗{$ to ( the open 5 parenthesis . 1 local) $} closing parenthesis heat equation open square bracket 6 5 closing square bracket \end{ a l i g n ∗} open parenthesis partialdiff sub t( minus∂t − ∆ Capitalx)K(t, x, Delta y) = sub 0, x closinglim K( parenthesist, x, y) = δy( Kx) open, parenthesis t comma x comma y closing parenthesis + = 0 comma limint right arrow t 0 to the power of plus K open parenthesis→t0 t comma x comma y closing parenthesis = delta sub y open parenthesis x\noindent closingwhere parenthesisTheδ is Wodzicki the comma Dirac residuedelta function has a and similar the limit identification i s in the weak involving sense . The the zeta heat function kernel operator $ e ˆ{ − tQ where} and, delta the t is trace the> Dirac of0 the delta . heat $ function kernel and operator the limit arei s in related the weak by sense the period Mellin .. transform The zeta function and Theheatthe trace of kernel the heat operator kernel operator is sonamed are related by since the Mellin the kerneltransform $K ( t , x , y ) \ in C ˆ{\ infty } ( ( 0 , \ infty ) \times M \Ztimes∞ M ) $ associated Capital Gamma open parenthesis s closing parenthesis zetas sub−1 Q open−tQ parenthesis s closing parenthesis = integral sub 0 to the power of Γ(s)ζQ(s) = t Tr(e )dt. infinity t to the power of s minus 1 Tr open parenthesis e to0 the power of minus tQ closing parenthesis dt period \noindent t o the t r a c e − class family $ e ˆ{ − t \Delta } , t > 0 , \Delta $ the Hodge Laplacian , UsingUsing the asymptotic the asymptotic expansion expansion of the heat of thekernel heat operator kernel open operator square bracket[ 6 5 , 6 5 6 comma 7 , 6 .. 8 6 ] 7 the comma Wodzicki .. 6 8 closing square bracket the Wodzickiresidue residue i is s associated t o the heat kernel by \ [associated ( e ˆ{ t o− the heatt kernel\Delta by } h ) ( x ) = \ int { M } K ( t , x , y ) h ( yEquation: ) dy open , parenthesis h \ in 5 periodL ˆ{ 2 closing2 } (M) parenthesis ..\ ] limint right arrow t 0 to the power of plus t to the power of n slash d Tr ) ) open parenthesis e to the power ofn/d minus tQ−tQ closingn parenthesis = Capital1 Gamman open−n/d parenthesis n divided by d to the power of closing lim t Tr(e ) = Γ( ress=n/dζQ(s) = − Γ( Res(Q ). (5.2) + parenthesis res sub s = n slash→ dt0 zeta sub Q open parenthesisd s closing parenthesisd d = minus 1 divided by d Capital Gamma open parenthesis n divided\noindent by d toi the s a power solution of closing to parenthesis the ( local Res open ) heat parenthesis equation Q to the [ 6power 5 ] of minus n slash d closing parenthesis period FromFrom Connes Connes quoteright ’ trace trace theorem theorem ( open Theorem parenthesis 4 . 1 Theorem and 4 4 period . 2 ) the 1 and Dixmier .... 4 period trace 2 of closinghDi− parenthesisn has the Dixmier trace of angbracketleft\ [( claim\ partial to D rightb e angbracket{ t } − to the \Delta power of{ minusx } n)K(t,x,y)=0, has claim to b e \lim {\rightarrow { t } 0 ˆ{ + }} K ( t , x , y ) = \ delta { −yn } ( x1 ) , \ ] thethe noncommutative noncommutative version version of the Wodzicki of the residueWodzicki Res residue open parenthesis Res (Q Q to), 0 the< Q power∈ Op ofcl( minusM). nDo closing residue parenthesis comma 0 less Q in Op suband cl to heat the power kernel of formulas 1 open parenthesis similar t M o closing ( 5 . 1 parenthesis ) and ( 5 period . 2 ) hold .... Do for residue Dixmier traces ? Sections and5 heat . 1 kernel and formulas 5 . 2 similarlist what t o open is known parenthesis . 5 period 1 closing parenthesis and open parenthesis 5 period 2 closing parenthesis hold for\noindent DixmierWe traceswhere detail ? .. only Sections$ \ delta the 5 lat period$ est is 1 .. the known and Dirac 5 period identifications delta 2 function between and Dixmier the limit traces i s and in theresidues weak sense . \quad The zeta function and thelistand tracewhat heatis knownof kernels the period heat . kernel For applications operator of are the related residue and by the heat Mellin kernel to transform index formulations in WeNCG detail onlysee the [ 4 lat7 , est 1 .. known9 , 4identifications 9 , 5 0 , between 5 2 ] . Dixmier traces .. and residues and \ [ heat\Gamma5 kernels . 1 period( Residues s .. For ) applications of\zeta functions of{ theQ } residue( and s heat ) kernel = to\ int indexˆ formulations{\ infty } in{ NCG0 } see t ˆ{ s − 1 } Tr ( e ôpen{ −A square . ConnestQ bracket} ) introduced 4 7dt comma . the 1\ ] .. association 9 comma .. 4 b 9 etween comma a5 0 generalised comma .. 5 2zeta closing function square , bracket period 5 period 1 .. Residues of zeta functions A period Connes introduced the association b etween a generalised zeta function comma \noindentinfinity Equation:Using open the parenthesis asymptotic 5 period expansion 3 closing of parenthesis the heat .. zeta kernel T open operator∞ parenthesis [ s closing6 5 , parenthesis\quad 6 :7 = , Tr\quad open parenthesis6 8 ] the T Wodzicki residue i s associated t o the heat kernel by s X s to the power of s closing parenthesis =ζT sum(s) := mu Tr( nT open) = parenthesisµn(T ) , T closing0 < T ∈ parenthesis M1,∞ to the power of s comma(5.3) 0 less T in M sub 1 comma infinity n = 1 \ begin { a l i g n ∗} n = 1 \lim {\rightarrow { t } 0 ˆ{ + }} t ˆ{ n / d } Tr ( e ˆ{ − tQ } ) = \Gamma ( \ f r a c { n }{ d }ˆ{ ) } r e s { s = n / d }\zeta { Q } ( s ) = − \ f r a c { 1 }{ d }\Gamma ( \ f r a c { n }{ d }ˆ{ ) } Res ( Q ˆ{ − n / d } ). \ tag ∗{$ ( 5 . 2 ) $} \end{ a l i g n ∗}

\noindent From Connes ’ trace theorem ( Theorem 4 . 1 and \ h f i l l 4 . 2 ) the Dixmier trace of $ \ langle D \rangle ˆ{ − n }$ has claim to b e

\noindent the noncommutative version of the Wodzicki residue Res $ ( Q ˆ{ − n } ) , 0 < Q \ in Op ˆ{ 1 } { c l } ( M ) . $ \ h f i l l Do r e s i d u e

\noindent and heat kernel formulas similar t o ( 5 . 1 ) and ( 5 . 2 ) hold for Dixmier traces $ ? $ \quad Sections 5 . 1 \quad and 5 . 2 list what is known .

We detail only the lat est \quad known identifications between Dixmier traces \quad and residues and heat kernels . \quad For applications of the residue and heat kernel to index formulations in NCG see [ 4 7 , 1 \quad 9 , \quad 4 9 , 5 0 , \quad 5 2 ] .

\noindent 5 . 1 \quad Residues of zeta functions

\noindent A . Connes introduced the association b etween a generalised zeta function ,

\ begin { a l i g n ∗} \ infty \\\zeta T( s ) :=Tr(Tˆ{ s } ) = \sum \mu n ( T ) ˆ{ s } , 0 < T \ in M { 1 , \ infty }\ tag ∗{$ ( 5 . 3 ) $}\\ n = 1 \end{ a l i g n ∗} Measure Theory in Noncommutative Spaces .... 2 1 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 2 1 2 1 and the calculation of a Dixmier trace with the result that \ [ Equation:\ r u l e {3em open}{0.4 parenthesis pt }\ ] 5 period 4 closing parenthesis .. limint s right arrow 1 to the power of plus to the power of open parenthesis s minus 1and closing the parenthesis calculation zeta of Ta Dixmieropen parenthesis trace with s closing the parenthesis result that = limint N right arrow infinity 1 divided by log open parenthesis 1 plus N closing parenthesis sum from n = 1 to N mu n open parenthesis T closing parenthesis \noindentif either limitand exists the opencalculation square bracket of a 2 Dixmier comma p periodtrace 306 with closing the square result bracket that period .. Meromorphicity of the formula open n=1 parenthesis 5 period 3 closing parenthesis(s i s discussed in open square1 bracketX 4 7 comma .. 6 9 comma .. 6 6 closing square bracket period \ begin { a l i g n ∗} lim −1)ζT (s) = lim µn(T ) (5.4) Generalisations of open parenthesiss 5→ period1+ 4 closingN parenthesis→∞ log(1 + appearedN) in open square bracket 2 .. 5 closing square bracket and later open\lim square{ s bracket\rightarrow 1 5 closing square1 bracket ˆ{ + }} periodˆ{ ( s } − 1N ) \zeta T ( s ) = \lim { N \rightarrow \ infty }\ f r a c { 1 }{\ log ( 1 + N ) }\sum ˆ{ n = 1 } { N }\mu n ( T ) \ tag ∗{$ ( In openif either square limit bracket exists 7 0 [closing 2 , p square . 306 ] bracket . Meromorphicity we translated the resultsof the open formula square ( 5 bracket . 3 ) i 1 s .. discussed 5 comma Theorem in 4 period 1 1 closing square5 . bracket 4 and ) $ open} square bracket 2 5 comma Theorem 3 period 8 closing square bracket to l to the power of infinity comma see Theo \end{[a 4l ig 7 n ,∗} 6 9 , 6 6 ] . Generalisations of ( 5 . 4 ) appeared in [ 2 5 ] and later [ 1 5 ] . hyphen In [ 7 0 ] we translated the results [ 1 5 , Theorem 4 . 1 1 ] and [ 2 5 , Theorem 3 . 8 ] rem .. 5∞ period 1 and Corollary 5 period 1 below period .. Recall T to the power of plus = K sub 1 to the power of plus = K sub 2 to the \noindentto ` ,ifsee either Theo - limit exists [ 2 , p . 306 ] . \quad Meromorphicity of the formula ( 5 . 3 ) i s discussed in [ 4 7 , \quad 6 9 , \quad 6 6 ] . power of plus open parenthesis 3 period 5 closing parenthesis comma+ so + + rem 5 . 1 and Corollary 5 . 1 below . Recall T = K1 = K2 (3.5), so GeneralisationsEquation: open parenthesis of ( 5 5 . period 4 ) 5 appeared closing parenthesis in [ 2 ..\quad Tr sub5 omega ] and open later parenthesis [ 15 T closing] . parenthesis = limint s right arrow 1 to the power of plus to the power of open parenthesis s minus 1 closing parenthesis zeta T open parenthesis s closing parenthesis \ hspacecalculates∗{\ thef i l l Dixmier} In [ or 7 Connes0 ] we endash translated Dixmier the trace results( ofs any open [ 1 parenthesis\quad 5 Dixmier , Theorem or Connes 4 . endash 1 1 ] Dixmier and [ 2closing 5 , parenthesis Theorem 3 . 8 ] to $ \ e l l ˆ{\ infty } , $ see Theo − Trω(T ) = lim −1)ζT (s) (5.5) measurable s→1+ positivecalculates operator the 0 less Dixmier T in M or sub Connes 1 comma – infinity Dixmier as the trace residue of anyat s = ( 1 Dixmier of the zeta or function Connes zeta – Dixmier T period .. ) What about \noindentgeneral 0 lessrem T\ inquad M sub5 1 . comma 1 and infinity Corollary ? 5 . 1 below . \quad R e c a l l $ T ˆ{ + } = K ˆ{ + } { 1 } = K ˆ{ +measurable} { 2 } positive( 3 operator . 50 )< T ,∈ $ M1 so,∞ as the residue at s = 1 of the zeta function ζT. TheWhat zeta function about of a positive compact operator T given by open parenthesis 5 period 3 closing parenthesis relies on the assumption that there exists some s sub 0 for which T to the power of s i s trace class s greater s sub 0 open parenthesis equally T in L to the power of s\ begin for s greater{ a l i g s n sub∗} 0 closing parenthesis period .. For Tr {\omega } ( T ) = \lim { s \rightarrow 1 ˆ{ + }}ˆ{ ( s } − 1 ) \zeta T convenience we assume s sub 0 = 1 period .. Thegeneral0 space< of T compact∈ M1,∞ operators? for which the zeta function (exists s and ) \ opentag ∗{ parenthesis$ ( 5 s minus . 1 5 closing ) $ parenthesis} zeta T open parenthesis s closing parenthesis in L to the power of infinity open parenthesis\end{ a l iThe g openn ∗} zeta parenthesis function 1 of comma a positive 2 closing compact square operator bracket closingT given parenthesis by ( 5 . was 3 ) relies studied on in the open assumption square bracket 1 .. 5 closing square s s bracketthat period there .. Define exists the some norm s0 for which T i s trace class s > s0 ( equally T ∈ L for s > s0). \noindent calculates the Dixmier or Connes −− Dixmier trace of any ( Dixmier or Connes −− Dixmier ) measurable barFor T bar convenience Z sub 1 : = limint we assume s right arrows0 sub= 1 to 1. theThe power space of plus of to compact the power operators of supremum for open which parenthesis the zeta s minus 1 closing parenthesis barpositive T barfunction sub operator s comma exists and $ 0(s −<1)ζTT(s) ∈ \Lin∞((1, 2])M was{ 1 studied , in\ infty [ 1 5}$ ] . as Define the residue the norm at $ s = 1 $ of the zeta function $ \wherezeta barT T bar . sub $ s\ =quad Tr openWhat parenthesis about bar T bar to the power of s closing parenthesis 1 divided by s period .. Then open square bracket 1 5 comma Theorem 4 period 5 closing square bracketsup(s identified that k T k Z1 := lim 1+ − 1) k T ks, \ beginZ sub{ 1a l = i g open n ∗} brace T in L to the power of infinity bars→ bar T bar Z sub 1 less infinity closing brace equiv M sub 1 comma infinity g eand n e r that a l 0 < T \ ins 1 M { 1 , \ infty } ? where k T ks= Tr (| T | ) s . Then [ 1 5 , Theorem 4 . 5 ] identified that \ende to{ a the l i g powern ∗} of minus 1 bar T bar sub 0 less or equal bar T bar Z sub 1 less or equal bar T bar sub 1 comma infinity comma where .. bar times bar sub 0 is the RieszZ seminorm= {T ∈ L∞ open|k T parenthesisk Z < ∞} 4 ≡ period M 1 2 closing parenthesis period The zeta function of a positive1 compact operator1 $ T $ given1,∞ by ( 5 . 3 ) relies on the assumption Henceand comma that open parenthesis s minus 1 closing parenthesis zeta sub bar T bar open parenthesis s closing parenthesis in L to the power ofthat infinity there open parenthesis exists some open parenthesis $ s { 0 1} comma$ for 2 closing which square $Tˆ bracket{ s } closing$ i parenthesis s trace class if and only $ ifs T in> M subs 1{ comma0 } infinity( $ commae q u a l l which y $ we T rewrite\ in as openL ˆ{ parenthesiss }$ f o defining r $ s the > s { 0 } ) . $ \quad For e−1 k T k ≤ k T k Z ≤k T k , conveniencefollowing function we tassume o be 0 for $ r ins open{ 0 square} = bracket0 1 0 . comma $ 1\quad 1 closing1The,∞ parenthesis space of closing compact parenthesis operators for which the zeta function exists and $ ( s − 1 ) \zeta T ( s ) \ in L ˆ{\ infty } ( ( 1 , 2 ] Equation:where openk · parenthesis k0 is the 5 Riesz period seminorm 6 closing parenthesis ( 4 . 1 2 .. ) 1 . divided by r zeta sub bar T bar Row 1 1 plus 1 underbar Row 2 r . in L to ) $ was studied in [ 1 \quad∞ 5 ] . \quad Define the norm the powerHence of infinity, ( opens − 1) parenthesisζ|T |(s) ∈ L open((1, 2]) squareif and bracket only 0 if commaT ∈ M infinity1,∞, which closing we parenthesis rewrite closingas ( defining parenthesis the Leftrightarrow T in M sub 1 commafollowing infinity period function t o be 0 for r ∈ [0, 1)) \ [ This\ parallel is known commaT from\ parallel open squareZ bracket{ 1 } 7 0 closing: = square\lim bracket{ s comma\rightarrow t o be equivalent}ˆ{\ t osup ( s } { 1 ˆ{ + }} − Equation:1 ) open\ parallel parenthesisT 5 period\ parallel 7 closing parenthesis{ s } , ..\ ] 1 divided by k zeta sub bar T bar parenleftbigg 1 plus 1 divided by k 1 1 + 1 ∞ parenrightbigg in l to the power of infinityζ|T | Leftrightarrow∈ L T in([0 M, ∞ sub)) 1⇔ commaT ∈ infinity M1,∞. period (5.6) r r We can obtain positive unitarily invariant singular functionals on M sub 1 comma infinity comma that will equate t o open parenthesis 5 period\noindentThis 5 closing iswhere known parenthesis $ , from\ parallel [ 7 0 ] , tT o be\ equivalentparallel t{ o s } = $ Tr $ ( \mid T \mid ˆ{ s } ) \ f r a c { 1 }{ s } . $when\quad .. T ..Then i s .. a [ .. 1 Tauberian 5 , Theorem .. operator 4 . comma 5 ] identified .. by .. applying that .. a .. generalised .. limit .. xi in S sub infinity open parenthesis l to the power of infinity closing parenthesis open1 parenthesis1 resp period phi in \ [ZS sub{ infinity1 } = open\{ parenthesisT \ openin squareζL ˆ(1{\ +bracketinfty) ∈ 0` comma}\∞ ⇔mid infinityT ∈\ Mparallel closing. parenthesisT \ closingparallel parenthesis(5Z.7) closing{ 1 } parenthesis< \ infty to the k |T | k 1,∞ sequence\}\equiv open parenthesisM { 1 5 period , 7 closing\ infty parenthesis}\ ] .. open parenthesis resp period function open parenthesis 5 period 6 closing parenthesis closingWe parenthesis can obtain period positive .. Exactly unitarily which generalised invariant limits singular pro functionalshyphen on M1,∞, that will equate t o ( 5 duce. linear5 ) when functionalsT openi s parenthesis a Tauberian and hence operator singular traces , by closing applying parenthesis ai s an generalised open question period limit We know that choosing \noindent and that∞ xi inξ BL∈ capS∞ DL(` open)( resp parenthesis. φ resp∈ S∞ period([0, ∞ phi))) to in theBL open sequence square ( 5bracket . 7 ) 0 comma ( resp . infinity function closing ( 5 .parenthesis 6 ) ) cap DL open square bracket. 0 comma Exactly infinity which closing generalised parenthesis limits closing pro parenthesis - duce linear results infunctionals a Dixmier trace ( and period hence singular traces \ [We e) ˆ summarise{ i −s an1 open the}\ results questionparallel of open . We squareT know bracket\ parallel that 7 choosing 0 closing{ 0 square}\ξ ∈ BLleq bracket∩ DL\ comma(parallelresp based.φ ∈ onBLT open[0, ∞\ square)parallel∩ DL[0 bracket, ∞)) 1Z ..{ 5 comma1 }\ Theoremleq \ 4 parallel Tperiod\resultsparallel 1 1 closing in asquare{ Dixmier1 bracket , trace comma\ infty . see} open, \ square] bracket 2 .. 5 closing square bracket and open square bracket 1 .. 9 closing square bracket forWe additional summarise the results of [ 7 0 ] , based on [ 1 5 , Theorem 4 . 1 1 ] , see [ 2 5 ] and information[ 1 9 ]period for additional .. Define the information averaging sequence . Define E : L theto the averaging power of infinitysequence openE parenthesis: L∞([0, ∞)) open→ ` square∞ by bracket 0 comma infinity closing\noindent parenthesiswhere closing\quad parenthesis$ \ parallel right arrow l\ tocdot the power\ parallel of infinity by{ 0 }$ is the Riesz seminorm ( 4 . 1 2 ) . E sub k open parenthesis f closing parenthesis : =Z integralk sub k minus 1 to the power of k f open parenthesis t closing parenthesis dt comma E (f) := f(t)dt, f ∈ L∞([0, ∞)). f\ inhspace L to the∗{\ powerf i l l } ofHence infinity open$ , parenthesis (k s open− square1 bracket ) \zeta 0 comma{\ infinitymid closingT \ parenthesismid } ( closing s parenthesis ) \ in periodL ˆ{\ infty } ( ( 1 , 2 ] )$ ifandonlyifk−1 $T \ in M { 1 , \ infty } , $ which we rewrite as ( defining the

\noindent following function t o be 0 for $ r \ in [ 0 , 1 ) ) $

\ begin { a l i g n ∗} \ f r a c { 1 }{ r }\zeta {\mid T \mid }\ l e f t (\ begin { array }{ c} 1 + 1 {\underline {\}}\\ r \end{ array }\ right ) \ in L ˆ{\ infty } ( [ 0 , \ infty )) \Leftrightarrow T \ in M { 1 , \ infty } . \ tag ∗{$ ( 5 . 6 ) $} \end{ a l i g n ∗}

\noindent This is known , from [ 7 0 ] , t o be equivalent t o

\ begin { a l i g n ∗} \ f r a c { 1 }{ k }\zeta {\mid T \mid } ( 1 + \ f r a c { 1 }{ k } ) \ in \ e l l ˆ{\ infty }\Leftrightarrow T \ in M { 1 , \ infty } . \ tag ∗{$ ( 5 . 7 ) $} \end{ a l i g n ∗}

\noindent We can obtain positive unitarily invariant singular functionals on $ M { 1 , \ infty } , $ that will equate t o ( 5 . 5 ) when \quad $ T $ \quad i s \quad a \quad Tauberian \quad operator , \quad by \quad applying \quad a \quad generalised \quad l i m i t \quad $ \ xi \ in S {\ infty } ( \ e l l ˆ{\ infty } ) ( $ resp $ . \phi \ in $ $ S {\ infty } ( [ 0 , \ infty ) ) )$ tothesequence(5.7) \quad ( resp . function ( 5 . 6 ) ) . \quad Exactly which generalised limits pro − duce linear functionals ( and hence singular traces ) i s an open question . We know that choosing $ \ xi \ in BL \cap DL ( $ resp $ . \phi \ in BL [ 0 , \ infty ) \cap DL [ 0 , \ infty ) ) $ results in a Dixmier trace .

We summarise the results of [ 7 0 ] , based on [ 1 \quad 5 , Theorem4 . 11 ] , see [ 2 \quad 5 ] and [ 1 \quad 9 ] for additional information . \quad Define the averaging sequence $ E : L ˆ{\ infty } ( [ 0 , \ infty ) ) \rightarrow \ e l l ˆ{\ infty }$ by

\ [E { k } ( f ) : = \ int ˆ{ k } { k − 1 } f ( t ) dt , f \ in L ˆ{\ infty } ( [ 0 , \ infty )). \ ] 22 .... S period Lord and F period Sukochev \noindenthline2222 \ h f i l l S . Lord and F . Sukochev S . Lord and F . Sukochev Define the map L to the power of minus 1 : L to the power of infinity open parenthesis open square bracket 1 comma infinity closing parenthesis\ [ \ r u l e { closing3em}{ parenthesis0.4 pt }\ ] right arrow L to the power of infinity open parenthesis open square bracket 0 comma infinity closing parenthesis closing parenthesis by −1 ∞ ∞ L toDefine the power the of map minusL 1 open: L parenthesis([1, ∞)) → gL closing([0, ∞ parenthesis)) by open parenthesis t closing parenthesis = g open parenthesis e to the power \noindent Define the map $ L ˆ{ − 1 } : L ˆ{\ infty } ( [ 1 , \ infty )) \rightarrow of t closing parenthesis comma g in L to the power−1 of infinityt open parenthesis∞ open square bracket 1 comma infinity closing parenthesis closing Lparenthesis ˆ{\ infty period} ( [ 0 , \Linfty(g)(t) =)g(e ) ), $ g ∈ byL ([1, ∞)). DefineDefine the floor the floor map p map : l top the: `∞ power→ L∞ of([1 infinity, ∞)) by right arrow L to the power of infinity open parenthesis open square bracket 1 comma infinity\ [ L ˆ closing{ − parenthesis1 } (g)(t)=g(eˆ closing parenthesis by { t } ) , g \ in L ˆ{\ infty } ([ 1Line , 1 infinity\ infty Line 2)). p open parenthesis\ ] open brace a sub k Row 1 infinity Row 2 k =∞ 1 closing brace. open parenthesis t closing parenthesis : = sum a sub k chi sub open square bracket∞ k comma kX plus 1 closing parenthesis open parenthesis t closing parenthesis comma open brace a p({a } )(t) := a χ (t), {a }∞ ∈ `∞. sub k closing brace sub k = 1 to the powerk k of= 1infinity in l tok the[k,k power+1) of infinityk k=1 period Line 3 k = 1 \noindentDefine commaDefine finally the comma floor the map mapping $ pL : open : parenthesis\ e l l ˆ{\ l toinfty the power}\ ofrightarrow infinity closing parenthesisL ˆ{\ infty to the} power( of [ * right 1 arrow , k = 1 open\ infty parenthesis) l ) to $ the by power of infinity closing parenthesis to the power of * by L openDefine parenthesis , finally omega , the closing mapping parenthesisL :(`∞ :) =∗ → omega(`∞)∗ circby E circ L to the power of minus 1 circ p comma omega in open parenthesis l to the\ [ \ powerbegin of{ a infinity l i g n e d Case}\ infty 1 * Case\\ 2 period pIn the ( following\{ statementsa { k }\}\ we usebegin the notation{ array }{ establishedc}\ infty in open\\ parenthesisk =∗ 21 period\end{ 8array closing}) parenthesis ( t endash ) open: = parenthesis\sum 2 a { k }\ chi { [ k , kL(ω) + := ω 1◦ E ◦ )L−}1 ◦ p,( ω t∈ (`∞ )) , \{ a { k }\} ˆ{\ infty } { k = 1 } period 1 2 closing parenthesis and open parenthesis 2 period 2 1 closing parenthesis. endash open parenthesis 2 period 2 9 closing parenthesis period\ in \ e l l ˆ{\ infty } . \\ kTheoremIn = the .. 1 following 5\ periodend{ a 1l i statements ..g n open e d }\ parenthesis] we use open the square notation bracket established 7 0 comma in .. ( Theorems 2 . 8 ) – .. ( 23 period. 1 2 ) 1 and comma ( 2 .. . 3 period 3 closing square bracket21)–(2.29). closing parenthesis periodTheorem .. Let P to the 5 . power 1 of ( * [ =7 P0 =, P to Theorems the power of 2 in 3 L . open 1 , parenthesis 3 . 3 ] H) . closing parenthesis .. be a ∗ 2 projectionLet andP = P = P ∈ L(H) be a projection and \noindent0 less T inDefine M sub 1 comma, finally infinity , period the mapping $L : ( \ e l l ˆ{\ infty } ) ˆ{ ∗ } \rightarrow ( \ e l l ˆ{\ infty } ) ˆThen{ ∗ comma}$ by for any phi in BL open square bracket 0 comma infinity closing parenthesis cap DL open square bracket 0 comma infinity closing parenthesis comma 0 < T ∈ M1,∞. \ [ \trl e sub f t L.L( open parenthesis\omega phi closing) parenthesis : = open\omega parenthesis\ circ PTP closingE \ parenthesiscirc L = ˆ phi{ − parenleftbigg1 }\ circ 1 dividedp by r Tr , parenleftbig\omega Then , for any φ ∈ BL[0, ∞) ∩ DL[0, ∞), PT\ in to the( power\ e l of l 1ˆ plus{\ infty hline sub} r) to\ begin the power{ a l i ofg n 1 e dP} parenrightbig& ∗ \\ parenrightbigg period &. \end{ a l i g n e d }\ right . \ ] Similarly comma for any xi in BL cap DL comma L open1 parenthesis xi closing parenthesis in D sub 2 .. and tr (PTP ) = φ( Tr(PT 1+ 1P )). Tr sub L open parenthesis xi closing parenthesisL(φ) open parenthesisr PTP closingr parenthesis = xi parenleftbigg 1 divided by k Tr parenleftbig PT to the power of 1 plus to the power of hline k to the power of 1 P parenrightbig parenrightbigg period Similarly , for any ξ ∈ BL ∩ DL, L(ξ) ∈ D and \noindentMoreover commaIn the limint following s right arrow statements 1 to the power we of use plus theto the notation power of open established2 parenthesis s minus in ( 1 2 closing . 8 )parenthesis−− (2.12)and(2.21) Tr open parenthesis −− ( 2 . 2 9 ) . Theorem \quad 5 . 1 \quad ( [ 7 0 , \quad Theorems \quad 3 . 1 , \quad 3 . 3 ] ) . \quad Let $ P ˆ{ ∗ } PT to the power of s P closing parenthesis .. exists iff PTP1 is Tauberian and in either case = P = P ˆ{ 2 }\ in LTr ((PTP H) = )ξ $( Tr(\quadPT 1+ be ak projection1P )). and tr sub upsilon open parenthesis PTP closingL(ξ) parenthesisk = Tr sub omega open parenthesis PTP closing parenthesis = limint s right arrow 1 to the power of plus to the power of open parenthesis s minus 1 closing parenthesis Tr open parenthesis PT to the power of s P closing \ begin { a l i g n ∗} (s s parenthesisMoreover for all upsilon, lim ins DL→1+ open−1) squareTr (PT bracketP ) 1exists comma iff infinityPTP closingis Tauberian parenthesis and comma in either omega case in D sub 2 period 0 Corollary< T .... 5\ periodin M 1 ....{ open1 parenthesis , \ infty open} square. bracket 7 0 comma .... Corollaries .... 3 period 2 comma .... 3 period 4 closing \end{ a l i g n ∗} square bracket closing parenthesis period .... Let A in L open parenthesis(s H closing parenthesis .... and 0 less T in M sub 1 comma infinity s period .... Then comma for trυ(PTP ) = Trω(PTP ) = lim −1)Tr(PT P ) + \ centerlineany phi in BL{Then open ,square for bracket any $ 0\ commaphi infinity\ in closingBL parenthesiss [→1 0 cap , DL\ infty open square) bracket\cap 0 commaDL infinity [ 0 closing , parenthesis\ infty comma) , tr$ sub} L open parenthesis phi closing parenthesisforallυ open∈ parenthesisDL[1, ∞), ω AT∈ D closing2. parenthesis = phi parenleftbigg 1 divided by r Tr parenleftbig AT to the power of 1 plus hline sub r to the power of 1 parenrightbig parenrightbigg period \ [Similarly t r Corollary{ L( comma for 5\ anyphi . 1 xi in) ( BL [} 7 cap( 0 DL , PTP comma Corollaries ) = \ 3phi . 2 ,( 3\ .f r4 a c ]{ )1 . }{ Letr } ATr∈ L(H () and PT ˆ{ 1 + \ r u l e {3em}{0.4 pt }}ˆ{ 1 } { r } P)).Tr sub0 < L T open∈ M1 parenthesis,\∞]. xi closing parenthesis open parenthesis AT closing parenthesis = xi parenleftbiggThen , for 1 divided by k Tr parenleftbig AT to the power of 1 plus to the power of hline k to the power of 1 parenrightbig parenrightbigg period Moreover comma .. if PTP is Tauberian for all projections P in the von Neumann algebra generated anyφ ∈ BL[0, ∞) ∩ DL[0, ∞), \ centerlineby A and A{ toSimilarly the power of , * for comma any $ \ xi \ in BL \cap DL , L ( \ xi ) \ in D { 2 }$ \quad and } tr sub upsilon open parenthesis AT closing parenthesis =1 Tr sub1+ omega open1 parenthesis AT closing parenthesis = limint s right arrow 1 to trL(φ)(AT ) = φ( Tr(AT r)). the\ [ power Tr { ofL( plus to the\ xi power of) open} ( parenthesis PTP s ) minus = r 1 closing\ xi parenthesis( \ f r a c Tr{ open1 }{ parenthesisk } Tr AT ( to the PT power ˆ{ 1 of s closing + ˆ{\ parenthesisr u l e {3em}{0.4 pt }}} k ˆ{ 1 } P)). \ ] for all upsilon in DL open square bracket 1Similarly comma infinity , for closingany ξ parenthesis∈ BL ∩ DL, comma omega in D sub 2 period For the situation s sub 0 = p comma see open square bracket 1 5 closing square bracket period 1 \ centerline {Moreover $ , \limTr { (ATs ) =\rightarrowξ( Tr(AT 1+ 1k ˆ1{)).+ }}ˆ{ ( s } − 1 ) $ Tr $ ( PT ˆ{ s } L(ξ) k P ) $ \quad exists iff $ PTP $ is Tauberian and in either case } Moreover , if PTP is Tauberian for all projections P in the von Neumann algebra ∗ \ begingenerated{ a l i g n ∗} by A and A , t r {\upsilon } ( PTP ) = Tr {\omega } ( PTP ) = \lim { s \rightarrow 1 ˆ{ + }}ˆ{ (

s } − 1 ) Tr ( PT ˆ{ s } P) \\ f o r(s a l l \upsilon \ in DL [ 1 , \ infty ) s , \omega \ in D { 2 } . trυ(AT ) = Trω(AT ) = lim −1)Tr(AT ) + \end{ a l i g n ∗} s→1 forallυ ∈ DL[1, ∞), ω ∈ D2. \noindent C o r o l l a r y \ h f i l l 5 . 1 \ h f i l l ( [ 7 0 , \ h f i l l Corollaries \ h f i l l 3 . 2 , \ h f i l l 3 . 4 ] ) . \ h f i l l Let $ A \ in L ( H ) $ For\ h f thei l l situationand $ 0 s0 =

\ begin { a l i g n ∗} any \phi \ in BL [ 0 , \ infty ) \cap DL [ 0 , \ infty ), \\ t r { L( \phi ) } ( AT ) = \phi ( \ f r a c { 1 }{ r } Tr ( AT ˆ{ 1 + \ r u l e {3em}{0.4 pt }}ˆ{ 1 } { r } )). \end{ a l i g n ∗}

\ centerline { Similarly , for any $ \ xi \ in BL \cap DL , $ }

\ [ Tr { L( \ xi ) } ( AT ) = \ xi ( \ f r a c { 1 }{ k } Tr ( AT ˆ{ 1 + ˆ{\ r u l e {3em}{0.4 pt }}} k ˆ{ 1 } )). \ ]

Moreover , \quad if $ PTP $ is Tauberian for all projections $ P $ in the von Neumann algebra generated by $A$ and $Aˆ{ ∗ } , $

\ begin { a l i g n ∗} t r {\upsilon } ( AT ) = Tr {\omega } ( AT ) = \lim { s \rightarrow 1 ˆ{ + }}ˆ{ ( s } − 1 ) Tr ( AT ˆ{ s } ) \\ f o r a l l \upsilon \ in DL [ 1 , \ infty ), \omega \ in D { 2 } . \end{ a l i g n ∗}

\ centerline {For the situation $ s { 0 } = p ,$ see[15]. } Measure Theory in Noncommutative Spaces .... 23 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 23 23 5 period 2 .. Heat kernel asymptotics \ [ We\ r u follow l e {3em the}{ exposition0.4 pt }\ of] open square bracket 3 2 closing square bracket comma following open square bracket 2 5 closing square bracket and open square bracket 1 5 closing square bracket period .. From open parenthesis 2 period 7 closing parenthesis and open parenthesis 2 period 1 9 closing5 . parenthesis 2 Heat kernel asymptotics \noindentL openWe parenthesisfollow5 . 2the\quad C exposition closingHeat parenthesis of kernel [ 3 2 open ] asymptotics , following parenthesis [ 2 f 5 closing ] and parenthesis [ 1 5 ] . open From parenthesis ( 2 . 7 )t closingand ( 2 parenthesis . 1 = 1 divided by log t integral9 ) sub 1 to the power of t f open parenthesis s closing parenthesis ds divided by s sub comma f in L to the power of infinity open \noindent We follow the exposition of [ 3 2 ] , following [ 2 5 ] and [ 1 5 ] . \quad From(2 . 7)and(2 . 19) parenthesis open square bracket 1 comma infinity closing1 Z parenthesist ds closing parenthesis period It was noted in open square bracketL(C 1)( 5f comma)(t) = Lemma 5f( periods) 1f closing∈ L∞([1 square, ∞)). bracket that the function log t s , \ [L(C)(f)(t)=g T comma alpha open parenthesis t closing parenthesis1 = 1\ dividedf r a c { 1 by}{\ t Tr openlog parenthesist }\ int e to theˆ{ powert } { of1 minus} f open ( parenthesis s ) tT closing\ f r a c It{ parenthesisds was}{ noteds to} the in{ [ power, 1} 5 , off Lemma minus\ in alpha 5 .L 1 closing ]ˆ{\ thatinfty parenthesis the function} ( comma [ alpha 1 greater , \ 0infty )). \ ] b elongs t o L to the power of infinity open parenthesis open square bracket 1 comma infinity closing parenthesis closing parenthesis only 1 −α for 0 less T in L to the power of 1 comma wgT, comma α(t) = whereTr( Le− to(tT the) ) power, α > of0 1 comma w is the symmetric subspace from open parenthesis 3 period\noindent 1 3 closingIt was parenthesis noted in [ 1 5 , Lemma 5t . 1 ] that the function or openb elongs parenthesis t o L 4∞ period([1, ∞)) 1only 4 closing for parenthesis0 < T ∈ L1 period,w, where .. In generalL1,w is we the have symmetric only subspace from ( 3 . \ [L g open1 3 T parenthesis ) or , ( 4\ .alpha C 1 closing 4 ) . parenthesis( In generalt open) we = parenthesis have\ f r a only c { g T1 comma}{ t } alphaTr closing ( parenthesis e ˆ{ − in L( to the tT power ) of ˆ{ infinity −open \alpha parenthesis}} ) open, square\alpha bracket> 1 comma0 \ ] infinity closing parenthesis closing parenthesis comma 0 less T in M sub 1 comma infinity period ∞ For this reason heat kernel results areL(C likely)(gT, to α) b∈ eL restricted([1, ∞)) only, 0 t< o T Connes∈ M1, endash∞. Dixmier traces open parenthesis V sub 2 closing parenthesis or Ces grave-a ro invariant Dixmier traces open parenthesis V sub 3 closing parenthesis outside of\noindent TauberianFor this operatorsb elongsreason in heat tL to o kernel the $Lˆ power results{\ ofinfty 1 comma are likely} w( period to b[ .. e Fromrestricted 1 open , square only\ infty t bracket o Connes) 3 2 )$ closing – Dixmier onlyfor square traces bracket $0 < T \ in 1,w L ând{ (1V open2) or , square Ces w bracket}a` ro, invariant $ 3 3 where closing Dixmier square $ L ˆ bracket traces{ 1 : ( ,V3) outside w }$ is of Tauberian the symmetric operators subspace in L from. From ( 3 . 1 3 ) orTheorem ([ 4 3 2 . ].. 1 5 4 period ) . \ 2quad .. openIn parenthesis general open we have square only bracket 3 2 .. comma .. Theorems .. 3 endash 6 closing square bracket .. and .. open square bracket 3 3 comma .. Theorem .. 33 closing square bracket closing parenthesis period .. Let omega in CDL open square bracket 1 comma\ [ L infinity ( C closing ) parenthesis ( g .. T and , \alpha ) \ in L ˆ{\ infty } ( [ 1 , \ infty )), and[33] : 0 0< less TT in M\ subin 1 commaM { infinity1 , have\ infty trivial kernel} . period\ ] .. Then tr subTheorem omega open parenthesis5 . 2 ( T [ closing3 2 parenthesis , Theorems = 1 divided 3 by – Capital 6 ] Gamma and open [ 3 3parenthesis , Theorem 1 divided by alpha plus 1 closing parenthesis omega circ L open parenthesis C closing parenthesis parenleftbigg 1 divided by t Tr parenleftbig e to the power of minus open 33 ] ) . Let ω ∈ CDL[1, ∞) and 0 < T ∈ M have trivial kernel . Then parenthesis\noindent tTFor closing this parenthesis reason to heat the power kernel of minus results alpha areparenrightbig likely1,∞ parenrightbigg to b e restricted comma alpha only greater t o 0 Connes period −− Dixmier traces $ ( V { 2 } ) $ or Ces $ \grave{a} $ ro invariant Dixmier traces $ ( V { 3 } ) $ outside of Tauberian operators in Let omega in CDL open square bracket 1 comma1 infinity closing1 parenthesis−α .. and 0 less T in L to the power of 1 comma w have trivial $ L ˆ{ 1 , w } . $ \trquad(T ) =From [ 3ω 2◦ L ] (C)( Tr(e−(tT ) )), α > 0. kernel period .. Then ω Γ( 1 + 1) t tr sub omega open parenthesis T closing parenthesisα = 1 divided by Capital Gamma open parenthesis 1 divided by alpha plus 1 closing 1,w parenthesis\ beginLet{ a l omega i gω n∈∗}CDL parenleftbigg[1, ∞) and 1 divided0 < by T ∈ t Tr L parenleftbighave trivial e to kernel the power . of minus Then open parenthesis tT closing parenthesis to the power of minusand alpha [ parenrightbig 3 3 ] parenrightbigg : comma alpha greater 0 period 1 1 −α \endLet{ a omega l i g n ∗} in D open parenthesis Ctr closing(T ) = parenthesisω open( Tr( squaree−(tT ) bracket)), α 1 > comma0. infinity closing parenthesis .. and 0 less T in M sub 1 ω Γ( 1 + 1) t comma infinity have trivial kernel period .. Then α \noindent Theorem \quad 5 . 2 \quad ( [ 3 2 \quad , \quad Theorems \quad 3 −− 6 ] \quad and \quad [ 3 3 , \quad Theorem \quad 33 ] ) . \quad Let tr subLet omegaω ∈ openD(C)[1 parenthesis, ∞) and T closing0 < T parenthesis∈ M1,∞ have = 1 trivial divided kernel by Capital . Gamma Then open parenthesis 1 divided by alpha plus 1 closing parenthesis$ \omega omega\ in parenleftbiggCDL [ 1 divided 1 by , t Tr\ infty parenleftbig) e$ to\ thequad powerand of minus open parenthesis tT closing parenthesis to the power of $ 0 < T \ in M { 1 , \ infty }$ have trivial kernel . \quad Then minus alpha parenrightbig parenrightbigg comma alpha1 greater1 0− period(tT )−α trω(T ) = 1 ω( Tr(e )), α > 0. Finally comma if 0 less T in L to the power ofΓ( 1α comma+ 1) wt .. is Tauberian with trivial kernel comma then \ [tr t r sub{\ omegaomega open} parenthesis( T T ) closing = parenthesis\ f r a c { 1 =}{\ 1 dividedGamma by Capital( \ f Gammar a c { 1 open}{\ parenthesisalpha } 1+ divided 1 by ) alpha}\ plusomega 1 closing\ circ Finally , if 0 < T ∈ L1,w is Tauberian with trivial kernel , then L(C)(parenthesis limint t right arrow\ f r a c infinity{ 1 }{ 1 dividedt } Tr by t Tr ( parenleftbig e ˆ{ − e to( the tT power )of ˆminus{ − open \alpha parenthesis}} tT)), closing parenthesis\alpha to the >power0 of minus . \ ] alpha parenrightbig = 1 divided by Capital Gamma open parenthesis 1 divided by alpha plus 1 closing parenthesis limint t right arrow 0 to the power of plus t1 1 divided1 by alpha−(tT Tr)−α parenleftbig1 e to the1 power−tT of− minusα tT to the power of minus alpha parenrightbig trω(T ) = 1 lim Tr(e ) = 1 lim t Tr(e ), α > 0 comma alpha greater 0 for all omega in DLt→∞ open square bracket 1 commat→ infinity0+ closing parenthesis period Γ( α + 1) t Γ( α + 1) α \noindentFor exampleLet comma $ \omega if angbracketleft\ in DCDL right angbracket [ 1 to , the\ powerinfty of minus) $ n i\ squad knownand t o b $ e Tauberian0 < T and b\ elongsin L t o ˆ L{ to1 the forallω ∈ DL[1, ∞). power, w of} 1$ comma have w trivial then kernel . \quad Then tr sub omegaFor open example parenthesis , if angbracketlefthDi−n i s known D right t o angbracket b e Tauberian to the power and b of elongs minus n t closing o L1,w parenthesisthen = 1 divided by Capital Gamma open\ [ t parenthesisr {\omega n divided} ( by 2 T plus ) 1 closing = \ parenthesisf r a c { 1 limint}{\Gamma t right arrow( 0\ tof r a the c { power1 }{\ of plusalpha t n} divided+ by 1 2 Tr ) parenleftbig}\omega e to the( \ f r a c { 1 }{ t } Tr ( e ˆ{ − ( tT ) ˆ{ − \alpha }} )), \alpha > 0 . \ ] power of minus tD to the power of 2 parenrightbig−n period 1 n −tD2 trω(hDi ) = n lim t Tr(e ). From Weyl quoteright s formula for the eigenvalues of the Hodge+ Laplacian on a n hyphen dimensional compact Γ( 2 + 1) t→0 2 Riemannian manifold open parenthesis and also using comments from open square bracket 1 5 comma p period 2 78 closing square bracket closing\noindentFrom parenthesis WeylLet comma ’ $ s\ formulaomega we have for\ that thein eigenvaluesD ( C of the ) Hodge [ Laplacian 1 , \ oninfty a n− dimensional) $ \quad compactand $ 0 < T \ in M 0{ lessRiemannian1 open , parenthesis\ infty manifold 1}$ plus (have Q and to the alsotrivial power using of kernel 2 comments closing . parenthesis\quad fromThen [ 1 to 5 the , p power . 2 78 of ]minus ) , we n slash have 2 that d in L to the power of 1 comma w

and i s Tauberian for any positive elliptic operator in Q in2 Op−n/2 subd cl1 to,w the power of d open parenthesis M closing parenthesis period .. This fact\ [ tcomma r {\ combinedomega } with( T ) = \ f0 r a< c(1{ +1 Q}{\) Gamma∈ L ( \ f r a c { 1 }{\alpha } + 1 ) }\omega ( \ f rthe a c { above1 }{ equationst } Tr comma ( reconstructs e ˆ{ − open( parenthesis tT ) 5 ˆ period{ − 2 \ closingalphad parenthesis}} )), period \alpha > 0 . \ ] and i s Tauberian for any positive elliptic operator in Q ∈ Opcl(M). This fact , combined with Forthe heat above kernel formulas equations involving , reconstructs tr sub omega ( 5 open . 2 parenthesis ) . AT closing parenthesis comma A in L open parenthesis H closing parenthesis comma 0For less heat T in kernel M sub 1formulas comma inﬁnity involving commatrω see(AT open),A ∈ square L(H), bracket0 < T ∈ 2 M 51 closing,∞, see square [ 2 5 ] bracket and [ 1and 5 ]open . square bracket 1 5 closing square\noindent bracketFinally period , if $ 0 < T \ in L ˆ{ 1 , w }$ \quad is Tauberian with trivial kernel , then \ begin { a l i g n ∗} t r {\omega } ( T ) = \ f r a c { 1 }{\Gamma ( \ f r a c { 1 }{\alpha } + 1 ) }\lim { t \rightarrow \ infty }\ f r a c { 1 }{ t } Tr ( e ˆ{ − ( tT ) ˆ{ − \alpha }} ) = \ f r a c { 1 }{\Gamma ( \ f r a c { 1 }{\alpha } + 1 ) }\lim { t \rightarrow 0 ˆ{ + }} t \ f r a c { 1 }{\alpha } Tr ( e ˆ{ − tT ˆ{ − \alpha }} ), \alpha > 0 \\ f o r a l l \omega \ in DL [ 1 , \ infty ). \end{ a l i g n ∗}

\ centerline {For example , if $ \ langle D \rangle ˆ{ − n }$ i s known t o b e Tauberian and b elongs t o $ L ˆ{ 1 , w }$ then }

\ [ t r {\omega } ( \ langle D \rangle ˆ{ − n } ) = \ f r a c { 1 }{\Gamma ( \ f r a c { n }{ 2 } + 1 ) }\lim { t \rightarrow 0 ˆ{ + }} t \ f r a c { n }{ 2 } Tr ( e ˆ{ − tD ˆ{ 2 }} ) . \ ]

\noindent From Weyl ’ s formula for the eigenvalues of the Hodge Laplacian on a $ n − $ dimensional compact Riemannian manifold ( and also using comments from [ 1 5 , p . 2 78 ] ) , we have that

\ [ 0 < ( 1 + Q ˆ{ 2 } ) ˆ{ − n / 2 d }\ in L ˆ{ 1 , w }\ ]

\noindent and i s Tauberian for any positive elliptic operator in $ Q \ in Op ˆ{ d } { c l } (M ) . $ \quad This fact , combined with the above equations , reconstructs ( 5 . 2 ) .

\ centerline {For heat kernel formulas involving $ tr {\omega } ( AT ) , A \ in L(H ) , 0 < T \ in M { 1 , \ infty } ,$ see [25]and[15] . } 24 .... S period Lord and F period Sukochev \noindenthline2424 \ h f i l l S . Lord and F . Sukochev S . Lord and F . Sukochev 6 .. Characterising the .. noncommutative .. integral \ [ The\ r u functional l e {3em}{0.4 pt }\ ] Equation:6 Characterisingopen parenthesis 6 period the 1 closing noncommutative parenthesis .. Capital Phi sub f integral comma T open parenthesis a closing parenthesis : = f open parenthesisThe aT functional closing parenthesis comma a in L open parenthesis H closing parenthesis comma \noindentwhere f i s6 a trace\quad onCharacterising a two hyphen sided idealthe \ ofquad compactnoncommutative operators J and 0\quad less T ini n J t ecomma g r a l i s the general format of noncommutative integration introduced by Connes period .. We are concerned in this section \noindent The functional with the characterisation of linear functionalsΦf on,T ( La) open := f(aT parenthesis), a ∈ L H(H closing), parenthesis constructed by open(6 parenthesis.1) 6 period 1 closing parenthesis period \ beginThewhere functional{ a l i g f n i∗} s Capital a trace Phi on sub a two f comma - sided T i ideal s normal of compact open parenthesis operators containedJ and in L0 < open T ∈ parenthesis J , i s the H general closing parenthesis sub * closing parenthesis\Phiformat{ iff and of only , noncommutative if T Capital} (a):=f(aT),a Phi sub integration f comma T open introduced parenthesis by a Connes closing parenthesis . We = are\ Trin open concernedL(H), parenthesis in this aT closing parenthesis\ tag ∗{$ comma ( 6 forall. 1 asection in ) $} with the characterisation of linear functionals on L(H) constructed by ( 6 . 1 ) . \endL open{ a l iThe g parenthesis n ∗} functional H closingΦf,T parenthesisi s normal comma ( contained for a positive in L trace(H)∗) classif and quoteleft only if densityΦf,T ( quoterighta) = Tr T(aT period), ∀a ..∈ From the definition of the canonicalL(H trace), for Tr a : positive trace class ‘ density ’ T. From the definition of the canonical trace Tr : \noindent where f i s a trace on a two − sided ideal of compact operators $ J $ and $ 0 < T \ in Line 1 infinity Line 2 Tr open parenthesis aT closing parenthesis =∞ sum lambda sub m angbracketleft h sub m comma ah sub m right angbracketJ ,$ comma i s the Line general 3 m = 1 format of noncommutative integration introducedX by Connes . \quad We are concerned in this section where open brace h sub m closing brace sub mTr( =aT 1) to = the powerλmhhm of, ah infinitymi, is an orthonormal basis of eigenvectors for the positive compact with the characterisation of linear functionals on $ L ( H ) $ constructed by ( 6 . 1 ) . operator T with m = 1 eigenvalues 0 less open brace lambda sub m closing brace sub m = 1 to the power of infinity in l to the power of 1 period .. Define the \ hspace ∗{\ f i l l }∞The functional $ \Phi { f , T }$ i snormal( contained in $L ( H ) { ∗ } normalwhere linear functional{hm}m=1 onis an l to orthonormal the power of infinity basis of comma eigenvectors sigma sub for T the in open positive parenthesis compact l to operator the powerT ofwith infinity closing parenthesis sub ) $ if and only if $ ∞\Phi 1{ f , T } ( a ) =$Tr$(∞ aT∞ ) ,1 \ f o r a l l a \ in $ * thicksimeigenvalues = l to the0 power< {λm of} 1m=1 comma∈ ` . byDefine the normal linear functional on ` , σT ∈ (` )∗ ∼= ` , by Line 1 infinity Line 2 sigma sub T open parenthesis open brace a sub m closing brace sub m = 1 to the power of infinity closing parenthesis :\noindent = sum lambda$L sub m ( a sub H m period ) Line ,$ 3 mforapositive = 1 trace∞ class ‘ density ’ $T .$ \quad From the definition of the canonical trace Tr : ∞ X Then σT ({am}m=1) := λmam. \ [ \Equation:begin { a l open i g n e parenthesis d }\ infty 6 period\\ 2 closing parenthesis .. Tr open parenthesis aT closing parenthesis = sigma sub T open parenthesis openTr brace ( angbracketleft aT ) h = sub\ msum comma\lambda ah sub m right{ m angbracket}\ langlem Row= 1 1h infinity{ m Row} 2, m = ah 1 closing{ m }\ brace.rangle comma forall, \\ a in L open parenthesismThen = H 1 closing\end parenthesis{ a l i g n e d }\ period] Conversely comma sigma open parenthesis open brace angbracketleft h sub m comma ah sub m right angbracket Row 1 infinity Row 2 m = 1 closing brace. ∞ comma\noindent 0 less sigmawhere in open $ \{ parenthesish Tr({ maT l to}\}) the = σ power({hˆ{\h of,infty ah infinityi} } closing{ m), parenthesis =∀a ∈ L 1(H} sub$). * is comma anorthonormal a in L open parenthesis(6. basis2) H of closing eigenvectors parenthesis for the positive compact operator T m m m = 1 $ Tdefines $ with a positive normal linear functional on L open parenthesis H closing parenthesis for any choice of orthonormal basis open brace h sub meigenvalues closingConversely brace sub $ m, 0 = 1< to the\{\ power oflambda infinity { m }\} ˆ{\ infty } { m = 1 }\ in \ e l l ˆ{ 1 } . $ \quad Define the normal linear functional on $ \ofe lH l periodˆ{\ infty .. If e sub} m, i s\ thesigma sequence{ T with}\ 1 inin the m( to the\ e l power l ˆ{\ ofinfty th place} and) 0{ otherwise ∗ } \ commasim set= T sub\ e l sigmal ˆ{ t1 o} b e, the $ by ∞ operator σ({hh , ah i} ), 0 < σ ∈ (`∞) , a ∈ L(H) with eigenvalues 0 less open brace sigmam openm parenthesism = 1 e sub m closing∗ parenthesis closing brace sub m = 1 to the power of infinity in l \ [ \ begin { a l i g n e d }\ infty \\ sub 1 associated to the basis open brace h sub m closing brace sub m = 1 to the power of period to the power of infinity∞ .. By this construction \sigmadefines{ aT positive} ( normal\{ a linear{ m functional}\} ˆ{\ on inftyL(H) for} any{ m choice = of 1 orthonormal} ) : basis = \{sumhm}m=1\lambda { m } a { m } Equation: open parenthesis 6 period 3 closing parenthesis .. Trth open parenthesis aT sub sigma closing parenthesis = sigma open parenthesis . \\ of H. If em i s the sequence with 1 in the m place and 0 otherwise , set Tσ t o b e the open brace angbracketleft h sub m comma ah sub m right∞ angbracket closing brace sub m = 1 to the power∞ of infinity closing parenthesis comma moperator = 1 \end with{ a l eigenvalues i g n e d }\ ] 0 < {σ(em)}m=1 ∈ `1 associated to the basis {hm}m=1. By this forall aconstruction in L open parenthesis H closing parenthesis period By open parenthesis 6 period 2 closing parenthesis and open parenthesis 6 period 3 closing parenthesis the positive normal linear functionals Capital Phi sub Tr comma T open parenthesis a closing parenthesis = Tr open parenthesis aT closing parenthesis comma 0 less T in L to \noindent Then ∞ the power of 1 Tr(aTσ) = σ({hhm, ahmi}m=1), ∀a ∈ L(H). (6.3) are exactly characterised by \ begin0 less{ sigmaa l i g n in∗} openBy parenthesis ( 6 . 2 l ) to and the ( power 6 . 3 of ) infinity the positive closing parenthesisnormal linear sub * functionals .. applied t o the sequence of .. quotedblleft expectation Tr ( aT ) = \sigma { T } ( \{\ langle h { m } , ah { m }\rangle \}\ begin { array }{ c}\ infty \\ values quotedblright .. open brace angbracketleft h sub m comma ah sub m right1 angbracket Case 1 infinity Case 2 m = 1 mfor = some 1 orthonormal\end{ array basis}), open brace\ f o r h aΦ sub lTr l ,T m(aa closing) = Tr(\ in braceaT ), L(H). sub0 < m T =∈ 1L to the power\ oftag period∗{$ to ( the 6 power . of infinity 2 ) .... $} We now know that\end a{ similararea l i g exactly n ∗} characterisation characterised exists by for any functional Capital Phi sub f comma T as defined in open parenthesis 6 period 1 closing parenthesis∞ period \noindent0 < σConversely∈ (`∞) applied , t o the sequence of “ expectation values ” {hh , ah i} 6 period 1 .. Characterisation∗ and singular traces m m m = 1 Let J b e a open parenthesis two sided closing∞ parenthesis ideal contained in the compact operators L to the power of infinity of the separable for some orthonormal basis {hm}m=1. We now know that a similar characterisation exists for complex\ [ \sigma ( \{\ langle h { m } , ah { m }\rangle \}\ begin { array }{ c}\ infty \\ m = 1 \end{ array }) , 0any< functional\sigma Φf,T\ inas defined( \ ine l ( l 6ˆ .{\ 1 )infty . } ) { ∗ } , a \ in L(H) \ ] Hilb6 ert . 1space HCharacterisation period .. By a trace on and J we singular mean a linear traces functional f : J right arrow C such that f open parenthesis open square bracket a commaLet T closingJ b e square a ( two bracket sided closing ) ideal parenthesis contained = 0 in the compact operators L∞ of the separable for all a in L open parenthesis H closing parenthesis comma T in J comma i period e period f vanishes on the commutator subspace Com J complex Hilb ert space H. By a trace on J we mean a linear functional f : J → C such that of\noindent J period .. Notedefines it i s a positive normal linear functional on $ L ( H ) $ for any choice of orthonormal basis $ \{ f ([ha, T{])m =}\} 0 for allˆ{\a ∈infty L(H),T} ∈{ Jm, i . = e . f 1 vanishes}$ on the commutator subspace Com J of notJ assumed. Note that it fi i s s notpositive assumed or continuous that f period i s positive or continuous . The .. characterisationThe characterisation for .. open parenthesis for 6 period( 6 . 1 1 ) closing in parenthesis Theorem .. in .. 6 Theorem . 1 b .. elow 6 period was 1 .. first b elow was .. first .. shown for f\ ..noindent in V sub PDLo f open $ H square . $ bracket\quad 1 commaI f $ infinity e { closingm }$ parenthesis i s the comma sequence with 1 in the $mˆ{ th }$ place and 0 otherwise , set $ T {\shownsigma for f}$∈ t o VPDL b e[1, the∞), operator 0 less0 T< in M T sub∈ 1 comma M , infinityin [ 7 comma 0 , ....Theorem in .... open 3 . square 6 ] using bracket residues 7 .... 0 comma ( Corollary .... Theorem 5 . 1 3 ) period . The 6 closing square bracket .... usingwith residues eigenvalues .... open parenthesis $1, 0∞ < Corollary\{\ 5 periodsigma 1 closing( parenthesis e { m } period) ....\} Theˆ{\ resultinfty .... below} { m .... open = parenthesis 1 }\ in t o \ e l l { 1 }$ associatedresult to the basis $ \{ h { m }\} belowˆ{\ infty } { m = 1 ˆ{ . }}$ \quad ( t o By this construction appearappear closing ) i parenthesis s due to collaboration i s due to collaboration with N . with Kalton N period and Kaltonuses results and uses on results the commutator on the commutator subspace subspace and sums of commutators open square bracket 7 1 comma .. 7 2 closing square bracket period \ beginand{ a l sums i g n ∗} of commutators [ 7 1 , 7 2 ] . Tr ( aT {\sigma } ) = \sigma ( \{\ langle h { m } , ah { m }\rangle \} ˆ{\ infty } { m = 1 } ), \ f o r a l l a \ in L(H). \ tag ∗{$ ( 6 . 3 ) $} \end{ a l i g n ∗}

\ centerline {By ( 6 . 2 ) and ( 6 . 3 ) the positive normal linear functionals }

\ [ \Phi { Tr , T } ( a )=Tr(aT) , 0 < T \ in L ˆ{ 1 }\ ]

\noindent are exactly characterised by

\ centerline { $ 0 < \sigma \ in ( \ e l l ˆ{\ infty } ) { ∗ }$ \quad applied t o the sequence of \quad ‘‘ expectation values ’’ \quad $\ l e f t . \{\ langle h { m } , ah { m }\rangle \}\ begin { a l i g n e d } & \ infty \\ & m = 1 \end{ a l i g n e d }\ right . $ }

\noindent for some orthonormal basis $ \{ h { m }\} ˆ{\ infty } { m = 1 ˆ{ . }}$ \ h f i l l We now know that a similar characterisation exists for

\noindent any functional $ \Phi { f , T }$ as defined in ( 6 . 1 ) .

\noindent 6 . 1 \quad Characterisation and singular traces

\noindent Let $ J $ b e a ( two sided ) ideal contained in the compact operators $ L ˆ{\ infty }$ of the separable complex Hilb ert space $H . $ \quad By a trace on $ J $ wemean a linear functional f $ : J \rightarrow C$suchthatf $( [ a , T ] ) = 0$ f o r a l l $ a \ in L(H),T \ in J , $ i . e . f vanishes on the commutator subspace Com $ J $ o f $ J . $ \quad Note i t i s not assumed that f i s positive or continuous .

\ hspace ∗{\ f i l l }The \quad characterisation for \quad ( 6 . 1 ) \quad in \quad Theorem \quad 6 . 1 \quad b elow was \quad f i r s t \quad shown f o r f \quad $ \ in V { PDL [ 1 , \ infty ) } , $

\noindent $ 0 < T \ in M { 1 , \ infty } , $ \ h f i l l in \ h f i l l [ 7 \ h f i l l 0 , \ h f i l l Theorem 3 . 6 ] \ h f i l l using residues \ h f i l l ( Corollary 5 . 1 ) . \ h f i l l The r e s u l t \ h f i l l below \ h f i l l ( t o

\noindent appear ) i s due to collaboration with N . Kalton and uses results on the commutator subspace and sums of commutators [ 7 1 , \quad 7 2 ] . Measure Theory in Noncommutative Spaces .... 25 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 25 25 Theorem 6 period 1 period .. Let 0 less T in J and f be a trace on J period .. Then there exists an orthonormal basis \ [ open\ r u l brace e {3em h}{ sub0.4 m pt closing}\ ] brace sub m = 1 to the power of infinity .. of H open parenthesis consisting of eigenvectors for T closing parenthesis .. and a linear functional sigma : l to the power of infinity right arrow C such that Theorem 6 . 1 . Let 0 < T ∈ J and f be a trace on J . Then there exists Line 1 f open parenthesis aT closing∞ parenthesis = sigma open parenthesis open brace angbracketleft h sub m comma ah sub m right \noindentan orthonormalTheorem 6 basis . 1 . {\hquadm}m=1Letof $ 0H( consisting< T \ ofin eigenvectorsJ$ andfbeatraceon for T ) and a linear $J .$ \quad Then there exists an orthonormal basis angbracket Row 1 infinity∞ Row 2 m = 1 closing brace. open parenthesis 6 period 4 closing parenthesis Line 2 for all a in L open parenthesis H closing$ \{functional parenthesish { m periodσ}\}: ` →ˆC{\suchinfty that} { m = 1 }$ \quad of $H ( $ consisting of eigenvectors for $T ) $ \quad and a linear functional $ \sigma : \ e l l ˆ{\ infty }\rightarrow C $ such that Straightforward corollaries show comma when f greater 0 comma∞ that f has the property f(aT ) = σ({hh , ah i} ) (6.4) bar f open parenthesis aT closing parenthesis bar lessm or equalm m C bar= 1 a bar some C greater 0 comma \ [ \ifbegin and only{ a l ifi g 0 n lesse d } sigmaf in ( open aT parenthesis ) = l to\ thesigma power of( infinity\{\ closinglangle parenthesish to{ them power} , of * ah period{ m .. The}\ basisrangle open brace\}\ begin { array }{ c}\ infty \\ mh sub = m closing 1 \end brace{ array sub m}) = 1 ( to the 6 power . of 4 infinity ) in\\ the theoremforalla i∈ s L any(H) basis. of eigenvectors of T f o r a l l a \ in L(H). \end{ a l i g n e d }\ ] rearranged so thatStraightforward the sequence of eigenvalues corollaries lambda show sub, when m associated f > 0, that t o h f sub has m the i s propertya decreasing sequence period A singular trace i s a trace on a two sided ideal J comma containing the finite rank operators F comma that vanishes on F subset J period .. The| theoremf(aT ) |≤ hasC k thea k followingsome C corollary > 0, period \ centerlineCorollary ..{ 6Straightforward period 1 period .. Let corollaries 0 less T in J .. show and f .. , be when .. a positive f $ > singular0 trace , $ .. thaton J period f has .. Then the thereproperty .. exists} ∞ ∗ ∞ an ..if andorthonormal only if basis0 < σ open∈ (` brace) . hThe sub m basis closing{hm brace}m=1 subin mthe = theorem 1 to the power i s any of basis infinity of .. eigenvectors of H open parenthesis consisting .. of \ [ \mid f ( aT ) \mid \ leq C \ parallel a \ parallel some C > 0 , \ ] eigenvectorsof T forrearranged T closing parenthesis so that the .. and sequence .. a singular of eigenvalues state λm associated t o hm i s a decreasing opensequence parenthesis . generalised limit closing parenthesis L : l to the power of infinity right arrow C such that Line 1A f open singular parenthesis trace aT i s closing a trace parenthesis on a two = sided f open ideal parenthesisJ , containing T closing parenthesis the finite L rank open operators parenthesisF, open brace angbracketleft h sub\noindent m commathat vanishesif ah suband m on only rightF angbracket⊂ if J . $The 0 Row< theorem 1 infinity\sigma has Row the\ 2in followingm = 1( closing corollary\ e brace. l l ˆ{\ open . infty parenthesis} ) 6 ˆ{ period ∗ } 5 closing. $ \ parenthesisquad The Line b a s2 i fors all$ \{ a inCorollary Lh open{ parenthesism }\} 6 H.ˆ{\ closing1 .infty parenthesisLet} { m0 period< = T 1 ∈}$ J inand the theoremf be i a s positive any basis singular of eigenvectors trace of $ T $ rearranged so that the sequence of eigenvalues $ \lambda ∞{ m }$ associated t o $ h { m }$ i s a decreasing sequence . In particularon J . commaThen therefor the noncommutative exists an integral orthonormal from open basis parenthesis{hm}m 4=1 periodof 1 3H closing( consisting parenthesis comma of Capitaleigenvectors Phi sub rho for openT parenthesis) and a closing a singular parenthesis state = rho open parenthesis a angbracketleft D right angbracket to the power of minus ∞ An closing singular( generalised parenthesis trace comma limit i s a a)L intrace: A` to→ ontheC such apower two that of sided prime prime ideal comma $ J rho , in $ V sub containing sing to the thepower finite of comma rank to the operators power of plus $ rho F open, $ parenthesis angbracketleft D right angbracket to the power of minus n closing parenthesis = 1 that vanishes on $ F \subset J . $ \quad The∞ theorem has the following corollary . the corollary implies that there existsf( aaT basis) = f( ofT eigenvectors)L({hh , ah openi} brace h) sub (6.5) m closing brace sub m = 1 to the power of infinity of D such m m m = 1 that \noindentrho open parenthesisC o r o l l a r y a angbracketleft\quad 6 . 1 D . right\quad angbracketLet$ to 0 theforalla power< T∈ of L minus(H\)in. n closingJ $ parenthesis\quad and = L f sub\quad rho openbe \ parenthesisquad a positive open singular trace \quad on $ J . $ \quad Then ther e \quad e x i s t s brace angbracketleft h subIn particular m comma ah , for sub the m right noncommutative angbracket closing integral brace sub from m= ( 4 1 .to 1 the 3 ) power , of infinity closing parenthesis anwhere\quad L suborthonormal rho is a generalised basis limit $ period\{ h { m }\} ˆ{\ infty } { m = 1 }$ \quad of $H ( $ consisting \quad of eigenvectors for $ T ) $ \quad and \quad a singular state Example 6 period 1 period .... Let T to the power−n of n b e00 the n hyphen+ t orus comma−n R to the power of n slash Z to the power of n comma Φ (a) = ρ(ahDi ), a ∈ A , ρ ∈ V , ρ(hDi ) = 1 with Laplacian Capital Delta periodρ .... Let J be any two hyphen sidedsing ideal \noindentwith trace( f and generalised g b e a bounded limit Borel $ function ) L g such : that\ e g l open l ˆ{\ parenthesisinfty }\ Capital∞ rightarrow Delta closing parenthesisC $ such in J that period .. Let f in L to the corollary implies that there exists a basis of eigenvectors {hm}m=1 of D such that the power of infinity open parenthesis T to the power of n closing parenthesis and \ [ \denotebegin by{ a l i g n e d } f(aT)=f(T)L(−n ∞ \{\ langle h { m } , ah { m }\rangle ρ(ahDi ) = Lρ({hhm, ahmi}m=1) \}\fgbegin open{ parenthesisarray }{ c}\ Capitalinfty Delta\\ closingm parenthesis = 1 \end{ array }) ( 6 . 5 ) \\ fthe o rwhere action a l lofL fρ onis a L a to generalised\ thein powerL(H). of limit 2 open . parenthesis\ Tend to{ thea l i power g n e d of}\ n] closing parenthesis by pointwise multiplication coupled with the n n n compactExample operator g 6 open . 1 parenthesis . Let T Capitalb e the Deltan− closingt orus parenthesis, R /Z , commawith Laplacian ∆. Let J be any two - c periodsided f periodideal open square bracket 1 1 comma S 4 closing square bracket period .. Then comma by Theorem 6 period 1 comma ∞ n \ centerlinef openwith parenthesis trace{ In f particular and fg openg b parenthesis e a , bounded for Capital the Borel noncommutative Delta function closing parenthesisg such integral that closingg from(∆) parenthesis (∈ 4 J . . = 1Let sigma 3 )f open,∈ }L parenthesis(T ) angbracketleft e to the powerand of denote i m times by x comma fe to the power of i m times x right angbracket closing parenthesis \ [ where\Phi m{\ in Zrho to the} power( of a n open ) parenthesis = \rho ordered( by a Cantor\ langle enumerationD closing\rangle parenthesisˆ{ − andn sigma} ) i s a , linear a functional\ in onA l ˆ{\prime to\prime the power} of, infinity\rho period\ in .. As V ˆ{ + } { s i n gfg ˆ{(∆), }}\rho ( \ langle D \rangle ˆ{ − n } ) = 1 \ ] angbracketleftthe action e of tof theon powerL2(T ofn) iby m pointwise times x comma multiplication fe to the power coupled of i m with times the x right compact angbracket operator = integralg(∆), sub T n e to the power of minus ic m . ftimes . [ 1 x 1 f ,open§4 ] parenthesis . Then x , closing by Theorem parenthesis 6 . e 1 to , the power of i m times x dx = integral sub T n f open parenthesis x closing parenthesis dx \noindent the corollary implies that there exists a basis of eigenvectors $ \{ h { m }\} ˆ{\ infty } { m and f(fg(∆)) = σ(heim·x, feim·xi) =sigma 1 }$ open of parenthesis $D$ 1 such closing that parenthesis = f open parenthesis g open parenthesis Capital Delta closing parenthesis closing parenthesis n ∞ we obtainwhere m ∈ Z ( ordered by Cantor enumeration ) and σ i s a linear functional on ` . As \ [ \rho ( a \ langle D \rangle ˆ{ − n } ) = L {\rho } ( \{\ langle h { m } , Equation: open parenthesis 6 period 6 closing parenthesisZ .. f open parenthesisZ fg open parenthesis Capital Delta closing parenthesis closing parenthesisah { m }\ = f openrangle parenthesis\} ˆ g{\h openeiminfty·x parenthesis, feim·}xi{=m Capitale− =im Delta·xf 1(x}) closingeim)·xdx\ parenthesis] = f(x) closingdx parenthesis integral sub T n f open parenthesis x closing parenthesis dx period Tn Tn and \noindent where $ L {\rho }$ is a generalised limit . σ(1) = f(g(∆)) \noindent Example 6 . 1 . \ h f i l l Let $ T ˆ{ n }$ b e the $ n − $ t orus $ , R ˆ{ n } / Z ˆ{ n } , $we with obtain Laplacian $ \Delta . $ \ h f i l l Let $ J $ be any two − sided i d e a l

\noindent with trace f and $ g $ b e a boundedZ Borel function $ g $ such that $ g ( \Delta ) \ in J . $ \quad Let $ f \ inf(fg(∆))L ˆ ={\ f(ginfty(∆)) }f(x()dx. T ˆ{ n } ) $ and (6.6) denote by Tn

\ [ f g ( \Delta ) \ ]

\noindent the action of $ f $ on $Lˆ{ 2 } ( T ˆ{ n } ) $ by pointwise multiplication coupled with the compact operator $ g ( \Delta ) , $ c . f . [ 1 1 , \S 4 ] . \quad Then , by Theorem 6 . 1 ,

\ [ f ( f g ( \Delta ) ) = \sigma ( \ langle e ˆ{ i m \cdot x } , f e ˆ{ i m \cdot x }\rangle ) \ ]

\noindent where m $ \ in Z ˆ{ n } ( $ ordered by Cantor enumeration ) and $ \sigma $ i s a linear functional on $ \ e l l ˆ{\ infty } . $ \quad As

\ [ \ langle e ˆ{ i m \cdot x } , f e ˆ{ i m \cdot x }\rangle = \ int { T n } e ˆ{ − i m \cdot x } f ( x ) e ˆ{ i m \cdot x } dx = \ int { T n } f ( x ) dx \ ]

\noindent and

\ [ \sigma ( 1 ) = f ( g ( \Delta )) \ ]

\noindent we obtain

\ begin { a l i g n ∗} f ( f g ( \Delta ) ) = f ( g ( \Delta )) \ int { T n } f ( x ) dx . \ tag ∗{$ ( 6 . 6 ) $} \end{ a l i g n ∗} 26 .... S period Lord and F period Sukochev \noindenthline2626 \ h f i l l S . Lord and F . Sukochev S . Lord and F . Sukochev Therefore comma for any ideal J with a non hyphen trivial trace f comma by choosing the appropriate bounded Borel \ [ function\ r u l e { g3em comma}{0.4 the pt expression}\ ] f open parenthesis times g open parenthesis Capital Delta closing parenthesis closing parenthesis mayTherefore serve as the , for integral any on ideal the nJ hyphenwith a t non orus - open trivial parenthesis trace f up, by t o choosing a constant the closing appropriate parenthesis bounded period .... Note the identification open\noindent parenthesisBorelTherefore function 6 periodg, 6 , closingthe for expression anyparenthesis ideal holds $ J $ with a non − trivial trace f , by choosing the appropriate bounded Borel functionfor all f in L $ to g the power , $ of the infinity expression open parenthesis T to the power of n closing parenthesis period Can the same statement as for the n hyphen t orus inf( the·g(∆)) example above be made for the Lebesgue \ [ f ( \cdot g ( \Delta )) \ ] integralmay of serve a compact as the Riemannian integral on manifold the n M− t comma orus ( considering up t o a constantnon hyphen ) trivial . Note positive the identification singular traces ( 6 instead. 6 ) of holds arbitrary traces ? We certainly know comma from Corollary .. 6 period 1 and Theorem .. 4 period 3 comma that if the sequence \noindentEquation:may open serve parenthesis as the6 period integral 7 closing on parenthesis the $ .. n open− brace$ t angbracketleft orus ( up h t sub o a m constant comma fh sub ) . m\ righth f i l angbracket l Note the closing identification ( 6 . 6 ) holds ∞ n brace sub m = 1 to the power of infinity in c commaforall forallf f∈ inL C( toT the). power of infinity open parenthesis M closing parenthesis \ beginfor a{ basisa l i g of n ∗} eigenvectors open brace h sub m closing brace sub m = 1 to the power of infinity of the Hodge endash Laplacian Capital Delta openf o r parenthesis aCan l l the ordered f same\ in so statement thatL the ˆ{\ eigenvalues asinfty for the} n(− t orus T ˆ{ inn the} example). above be made for the Lebesgue \endassociated{integrala l i g n ∗} t o of the a compact eigenvectors Riemannian are decreasing manifold closing parenthesisM, considering comma non then - trivial positive singular traces f openinstead parenthesis of arbitrary fg open parenthesis traces ? Capital Delta closing parenthesis closing parenthesis = f open parenthesis g open parenthesis Capital DeltaCan the closingWe same parenthesis certainly statement closing know as parenthesis, from for Corollary the integral $ n sub 6− M . fdx$ 1 and comma t orus Theorem forall in f inthe C 4 to example . the 3 , power that above of if infinity the be sequence open made parenthesis for the M Lebesgue closing parenthesis integralfor any positive of a non compact hyphen Riemannian trivial singular manifold trace f on an $ ideal M J and , $ with considering an appropriate non choice− oftrivial positive singular traces instead of arbitrary traces $ ? $ bounded Borel function g open parenthesis so that g∞ open parenthesis∞ Capital Delta closing parenthesis in J closing parenthesis period The property .. open parenthesis 6 period{hh 7m closing, fhmi} parenthesism=1 ∈ c, ∀ ..f is∈ ..C a restricted(M) form of a property for compact (6.7) .. Riemannian manifolds \ centerline {We certainly know , from Corollary \quad 6 . 1 and Theorem \quad 4 . 3 , that if the sequence } known as Quantum Unique Ergodicity open∞ parenthesis QUE closing parenthesis .. open square bracket 7 3 comma .. 7 .. 4 closing square for a basis of eigenvectors {hm}m=1 of the Hodge – Laplacian ∆( ordered so that the eigenvalues bracketassociated period .. It ti s o not the well eigenvectors known which are manifolds decreasing ) , then \ beginhave{ thea l i QUE g n ∗} property period \{\ langle h { m } , fh { m }\rangleZ \} ˆ{\ infty } { m = 1 }\ in c , \ f o r a l l The same result as for the n hyphen torus applies t o anything quoteleft flatRow∞ 1 quoteright Row 2 period flat next example involving the f non\ in hyphenC commutative ˆ{\ infty t} orus(M) firstf( appearedfg(∆))\ =tag open f(g∗{(∆)) parenthesis$ (fdx, 6 for∀ .ff in∈ V 7C sub(M PDL )) $} open square bracket 1 comma infinity closing parenthesis M closing\end{ a parenthesis l i g n ∗} in open square bracket 7 0 closing square bracket period Examplefor any .. 6 positive period 2 non period - trivial .. Consider singular .. two trace .. unitaries f on an .. u ideal commaJ vand .. such with .. anthat appropriate .. uv = lambda choice vu comma of .. for .. lambda : = e to\noindent the powerbounded offor 2 Borelpi a i theta basis function in ofS open eigenvectorsg( parenthesisso that g the(∆) $∈\{ J ). h { m }\} ˆ{\ infty } { m = 1 }$ of the Hodge −− Laplacian $ \unitDelta ..The circle( property closing $ ordered parenthesis ( 6 so . 7 period that ) is .. the Denote a eigenvalues restricted .. by .. Fform sub of theta a property open parenthesis for compact u comma Riemannianv closing parenthesis .. the .. * hyphen algebraassociatedmanifolds .. of linear t o.. known combinations the eigenvectors as Quantum .. sum sub Unique are open decreasing parenthesis Ergodicity m ) ( comma QUE , then )n closing [ 7 3 parenthesis , 7 4 in ] J . a sub It m i comma s not n u to the power of m v to the powerwell known of n comma which manifolds have the QUE property . \ [ f ( f g ( \Delta ) ) = f ( g ( \Delta0 )) \ int { M } fdx , \ f o r a l l J subsetThe Z to same the power result of as 2 for.... ithe s ....n− a finittorus eset applies comma t o .... anything with product ‘ [ ....T he abnext = sum example sub r comma involving s open parenthesis sum sub m commaf \ in n a subC rˆ{\ minusinfty m comma} (M) n lambda to\ the] power of mn b sub m comma. s minus n closing parenthesis u to the power of r v to the power ofthe s .... non and - involution commutative t orus first appeared ( for f ∈ VPDL[1,∞)) in [ 7 0 ] . a toExample the power of * = 6 sum . 2 sub . r commaConsider s open parenthesis two unitaries lambda to theu, powerv such of rs to that the poweruv of hline= λvu, a sub minus r comma minus s \noindent for any2 positiveπiθ non − trivial singular trace f on an ideal $ J $ and with an appropriate choice of closingfor parenthesisλ := u toe the power∈ ofS r( vthe to the unit power circle of s comma ) . a Denote comma b in by F subFθ theta(u, v) openthe parenthesis∗− algebra u comma v closing parenthesis bounded Borel function $ gP ( $ so thatm n $ g ( \Delta ) \ in J ) . $ periodof .... linear The assignment combinations tau sub 0 open( parenthesism,n)∈J am,nu a closingv , parenthesis = a sub 0 comma 0 i s a faithful trace on 2 P P mn r s F subJ theta⊂ openZ parenthesisi s a finit u comma e set , v closing with parenthesis product periodab = Let openr,s( parenthesism,n ar−m,n Hλ subb thetam,s−n comma)u v piand sub theta closing parenthesis denoteThe propertyinvolution the cyclic\ representationquad ( 6 . associated 7 ) \quad t oi tau s \ subquad 0 perioda restricted .. The closure form comma of a C property sub theta for open compact parenthesis\quad u commaRiemannian v closing manifolds parenthesisknown∗ as comma QuantumP ofrs Unique Ergodicityr s ( QUE ) \quad [ 7 3 , \quad 7 \quad 4 ] . \quad It i s not well known which manifolds a = r,s(λ a−r,−s)u v , a, b ∈ Fθ(u, v). The assignment τ0(a) = a0,0 i s a faithful havepi subtrace the theta QUEon open property parenthesis . F sub theta open parenthesis u comma v closing parenthesis closing parenthesis in the operator norm i s called a rotationF (u, C v to). theLet power(H , πof) *denote hyphen algebrathe cyclic comma representation .. open square associated bracket 7 .. t 5 o closingτ . squareThe closure bracket comma,C (u, v or), the noncommutative The sameθ result asθ forθ the $ n − $ torus applies t o anything0 ‘ $\ l e f t . \ f l aθ t \ begin { array }{ c} ’ \\ t orusof openπ (F parenthesis(u, v)) in thelambda operator equal-negationslash norm i s called 1 closing a rotation parenthesisC open∗− algebra square bracket , [ 2 7 comma 5 ] 3 , comma or the 4 2 closing square bracket . \end{ arrayθ θ}The\ right .$ next example involving the periodnoncommutative .. Canonically comma t finit orus e linear(λ 6= combinations 1) [2, 3, of42] u. to theCanonically power of m v, tofinit the e power linear of n combinations right arrow H sub of theta are dense in H sub non −mcommutativen t orus first appeared ( for f $ \ in V { PDL [ 1 , \ infty ) } ) $ in [ 7 0 ] . theta periodu v → Hθ are dense in Hθ. DefineDefine Capital∆ Delta(umvn sub) = theta (m2 + openn2)u parenthesismvn. It u can to the b e power shown of m that v to the the power‘ noncommutative of n closing parenthesis Laplacian = open parenthesis m to the \noindent Exampleθ \quad 6 . 2 . \quad Consider \quad two \quad u n i t a r i e s \quad $ u , v $ \quad such \quad that \quad power of’ 2∆ plusθ has n toa unique the power positive of 2 closing extension parenthesis ( also u to denoted the power∆ ofθ)∆ mθ v: toDom the power(∆θ) of→ nH periodθ with .. compact It can b e shown that the quoteleft $ uv = \lambda vu , $ \quad f o r \quad $ \lambda :m =n e ˆ{ 2 \ pi i \theta }\ in S noncommutativeresolvent Laplacian , see the quoteright previous Capital citations Delta . sub The theta eigenvectors hm,n = u v ∈ Hθ form a complete ( $hasorthonormal the a unique positive system extension . Note open that parenthesis also denoted Capital Delta sub theta closing parenthesis Capital Delta sub theta : Dom openunit parenthesis\quad c Capital i r c l e Delta ) . \ subquad thetaDenote closing parenthesis\quad by right\quad arrow$ H F sub{\ thetatheta with} compact( uresolvent , comma v ) $ \quad the \quad $ ∗ − $ algebra \quad o f l i n e a r \quad combinations \quad $ \sum { ( m , n ) \ in J } a { m see the previous citationsh periodh , π ..(a The)h eigenvectorsi =: τ ((um hvn sub)∗au mm commavn) = τ n(a =), u to∀a the∈ F power(u, v) of m v to the power of n in H sub theta form a , n } u ˆ{ m } v ˆ{ m,nn } θ , $m,n 0 0 θ complete orthonormal 2 2 systemfor periodany (m, Note n) that∈ Z . Using the Cantor enumeration of Z , from Theorem 6 . 1 we obtain \noindentangbracketleft$ J h sub\ msubset comma n commaZ ˆ{ 2 pi} sub$ theta\ h f iopen l l i parenthesis s \ h f i l la closinga finit parenthesis e set h , sub\ h m f i comma l l with n right product angbracket\ h f i= l l : tau$ sub ab =0 open\sum parenthesis{ r open , parenthesis s } ( u to\f(sum theag(∆ powerθ{))m = of f(g m(∆ , vθ)) toτ n0 the(a}), powera∀a{∈ ofFr nθ( closingu,− v) parenthesism , n to}\ the powerlambda of *ˆ au{ mn to the} powerb { ofm m v ,to s − n } ) u ˆ{ r } v ˆ{ s }$ \ h f i l l and involution the powerfor of any n closing ideal J parenthesiswith a non = tau - trivial sub 0 opentrace parenthesis f , by choosing a closing the parenthesis appropriate comma bounded forall a Borel in F sub function theta open parenthesis u comma v closingg such parenthesis that g(∆ ) ∈ J . In fact , hh , bh i = τ(b) for any b in the weak closure C (u, v)00 \noindent $ a ˆ{ ∗θ } = \sum { rm,n , sm,n} ( \lambda ˆ{ r s ˆ{\ r u l e {3em}{θ0.4 pt }}} a { − r forof anyπ openθ(Fθ( parenthesisu, v)) andwhere m commaτ denotes n closingthe parenthesis normal in extension Z to the power of τ0 of. 2Hence period .. Using the Cantor enumeration of Z to the power of, 2 comma− s from} Theorem) u ˆ 6{ periodr } 1v we ˆ{ obtains } , a , b \ in F {\theta } ( u , v ) . $ \ h f i l l The assignment $ \tau { 0 } ( a ) = a { 0 , 0 }$ i s a faithful00 trace on f open parenthesis ag open parenthesisf(ag Capital(∆θ)) Delta = f(g(∆ subθ)) thetaτ(a), closing∀a ∈ parenthesisCθ(u, v) . closing parenthesis = f open parenthesis g open parenthesis Capital Delta sub theta closing parenthesis closing parenthesis tau sub 0 open parenthesis a closing parenthesis comma forall a in F sub theta open\noindent parenthesis$ u F comma{\theta v closing} parenthesis( u , v ) .$Let$( H {\theta } , \ pi {\theta } ) $ denotefor any the ideal cyclic J with a representation non hyphen trivial trace associated f comma by t ochoosing $ \tau the appropriate{ 0 } . bounded $ \quad BorelThe function closure g $ , C {\theta } (such u that , g open v parenthesis ) , $ Capital o f Delta sub theta closing parenthesis in J period .. In fact comma angbracketleft h sub m comma n comma$ \ pi bh sub{\ mtheta comma} n(F right angbracket{\theta = tau} open( parenthesis u , vb closing ) parenthesis ) $ in thefor any operatornorm b in the weak closure i s called C sub theta a rotation open parenthesis$ C ˆ{ ∗ u } comma − $ v algebraclosing parenthesis , \quad to[ the 7 power\quad of5 prime ] , prime or the of noncommutative tpi orus sub theta $ ( open\lambda parenthesis F\ne sub theta1)[2,3,42].$ open parenthesis u comma v closing parenthesis closing parenthesis\quad Canonically and where tau , denotes finit the e linear combinations of normal$ u ˆ{ extensionm } v of ˆ tau{ n sub}\ 0 periodrightarrow .. Hence H {\theta }$ are dense in $H {\theta } . $ f open parenthesis ag open parenthesis Capital Delta sub theta closing parenthesis closing parenthesis = f open parenthesis g open parenthesis Capital\noindent DeltaDefine sub theta closing$ \Delta parenthesis{\theta closing} parenthesis( u ˆ tau{ m open} parenthesisv ˆ{ n } a closing) = parenthesis ( m comma ˆ{ 2 } forall+ a in n C ˆ sub{ 2 theta} ) open parenthesisu ˆ{ m } uv comma ˆ{ n v} closing. $ parenthesis\quad It to the can power b e of shown prime prime that period the ‘ noncommutative Laplacian ’ $ \Delta {\theta }$ has a unique positive extension ( also denoted $ \Delta {\theta } ) \Delta {\theta } : $ Dom $ ( \Delta {\theta } ) \rightarrow H {\theta }$ with compact resolvent , see the previous citations . \quad The eigenvectors $ h { m , n } = u ˆ{ m } v ˆ{ n }\ in H {\theta }$ form a complete orthonormal system . Note that

\ [ \ langle h { m , n } , \ pi {\theta } ( a ) h { m , n }\rangle = : \tau { 0 } ( ( u ˆ{ m } v ˆ{ n } ) ˆ{ ∗ } au ˆ{ m } v ˆ{ n } ) = \tau { 0 } ( a ) , \ f o r a l l a \ in F {\theta } ( u , v ) \ ]

\noindent forany $( m , n ) \ in Z ˆ{ 2 } . $ \quad Using the Cantor enumeration of $ Z ˆ{ 2 } , $ from Theorem 6 . 1 we obtain

\ [ f ( ag ( \Delta {\theta } ) ) = f ( g ( \Delta {\theta } )) \tau { 0 } ( a ) , \ f o r a l l a \ in F {\theta } ( u , v ) \ ]

\noindent for any ideal $ J $ with a non − trivial trace f , by choosing the appropriate bounded Borel function $ g $ such that $g ( \Delta {\theta } ) \ in J . $ \quad In f a c t $ , \ langle h { m , n } , bh { m , n }\rangle = \tau ( b )$ for any $b$ in theweak closure $C {\theta } ( u , v ) ˆ{\prime \prime }$ o f $ \ pi {\theta } (F {\theta } ( u , v ) )$ andwhere $ \tau $ denotes the normal extension of $ \tau { 0 } . $ \quad Hence

\ [ f ( ag ( \Delta {\theta } ) ) = f ( g ( \Delta {\theta } )) \tau ( a ) , \ f o r a l l a \ in C {\theta } ( u , v ) ˆ{\prime \prime } . \ ] Measure Theory in Noncommutative Spaces .... 2 7 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 2 7 2 7 6 period 2 .. A dominated convergence theorem for the noncommutative integral \ [ The\ r u following l e {3em}{ ..0.4 dominated pt }\ ] .. convergence theorem .. appeared in .. open square bracket 7 0 closing square bracket .. as .. a .. statement .. for .. Dixmier6 . 2 A dominated convergence theorem for the noncommutative integral traces period We provide the general statement here period \noindentThe following6 . 2 \quad dominatedA dominated convergence convergence theorem theorem appeared for the in noncommutative [ 7 0 ] as a integral state- Letment N b e a weaklyfor Dixmier closed * hyphen traces subalgebra . We provide of B open the general parenthesis statement H closing here parenthesis . period We recall that N sub * denotes the predual of N comma Let N b e a weakly closed ∗− subalgebra of B(H). We recall that N denotes the predual \noindentor the set ofThe all normalfollowing linear\ functionalsquad dominated on N period\quad convergence theorem \∗quad appeared in \quad [ 7 0 ] \quad as \quad a \quad statement \quad f o r \quad Dixmier of N , tracesWe say a . positive We provide compact the operator general T i s open statement parenthesis here N comma . h closing parenthesis hyphen dominated if comma for some orthonormal or the set of all normal linear functionals on N . basis \ hspace ∗{\Wef say i l l } aLet positive $N$ compact b e operator a weaklyT closedi s (N , h)− $ dominated∗ − $ subalgebraif , for some orthonormal of $B ( basis H ) . $ Werecall that open brace∞ h sub m closing brace sub m = 1 to the power of infinity of eigenvectors of T comma there exists h in H such that .. bar Ph sub $ N {hm ∗}m }$=1 of denotes eigenvectors the predual of T, there of exists $Nh ,∈ $ H such that k P hm k ≤k P h k for all proj m bar lessections or equal bar Ph bar for all proj ections P in N period \noindentTheorem 6or period the 2 setperiod of .... all Let 0 normal less T in linear J and f be functionals a positive singular on trace $ N on J . period $ .... If T is open parenthesis N comma h closing parenthesis hyphen dominated comma P ∈ N . Wesayathen Capital positive Phi sub compact f comma T operator = f open parenthesis $T$ times i s T $ closing ( N parenthesis , h in N ) sub *− period$ dominated if , for some orthonormal basis $Proof\{Theorem periodh { ..m Without 6}\} . 2 . lossˆ{\ openinftyLet brace0 h}< sub{ T ∈m m J closingand = bracef 1 be}$ a sub positive of m =eigenvectors 1 tosingular the power trace of of infinity on $TJ can. ,b e$If rearranged thereT is exists so that the $ eigenvalues h \ in Hlambda $( suchN sub, h m)− that associateddominated\quad t o , h$ sub\ parallel m Ph { m }\ parallel \ leq \ parallel Ph \ parallel $ for all proj ections form a decreasing sequence period .. By hypothesis .. angbracketleft h sub m comma Ph sub m right angbracket less or equal angbracketleft h\ begin comma{ Pha l i gright n ∗} angbracket .. for all proj ections P in N period thenΦ = f(·T ) ∈ N . P Then\ in angbracketleftN. h sub m comma ah sub m rightf,T angbracket less∗ or equal angbracketleft h comma ah right angbracket comma 0 less a \end{ a l i g n ∗} in N comma as a i s a uniform limit of finit e∞ linear positive spans of Proof . Without loss {hm}m=1 can b e rearranged so that the eigenvalues λm associated proj ectionsh open square bracket 1 0 comma p period 23 closing square brackethh , P hperiodi ≤ .. For hh, any P h generalisedi limit L and 0 less a in N comma \noindentt o mTheoremform a decreasing6 . 2 . \ h sequence f i l l Let . $ By 0 hypothesis< T \ in m J $m and f be a positivefor all proj singular trace on $ J Equation: openP parenthesis∈ N . 6 periodhh , ah8 closingi ≤ hh, parenthesis ahi, 0 < a ..∈ L N open, parenthesisa angbracketleft h sub m comma ah sub m right angbracket . $ ections\ h f i l l If$T$is$(Then m m N , h ) − as$ dominatedi s a uniform , limit of finit e linear closingpositive parenthesis spans less ofor equal proj limint ections supremum [ 1 0 , p angbracketleft . 23 ] . For h sub any m generalised comma ah sub limit m rightL and angbracket0 < a ∈ lessN , or equal angbracketleft h comma ah right angbracket period m right arrow infinity \ beginLet open{ a l i brace g n ∗} a sub alpha closing brace be a net of monotonically increasing positive elements of N with upper bound period It follows then \Phi { f , T } = f ( \cdot T) \ in N { ∗ } . that open brace a sub alpha closing braceL(hhm converges, ahmi) ≤ stronglylim suphh tom a, ah l periodmi ≤ h uh, period ahi. b period a in N open square(6.8) bracket 1 0 comma p period \end{ a l i g n ∗} 22 closing square bracket period .. From open parenthesis 6 period 8 closingm → parenthesis ∞ L open parenthesis angbracketleft h sub m comma open parenthesis a minus a sub alpha closing parenthesis h sub m right angbracket closing parenthesis less or equal \noindent Proof . \quad Without loss $ \{ h { m }\} ˆ{\ infty } { m = 1 }$ can b e rearranged so that the eigenvalues angbracketleftLet {aα} hbe comma a net open of monotonically parenthesis a minus increasing a sub alpha positive closing elementsparenthesis of h rightN with angbracket upper period bound Since . angbracketleft h comma $ \lambda { m }$ associated t o $ h { m }$ open parenthesisIt follows a minus that a{ subaα} alphaconverges closing strongly parenthesis to h aright l . angbracket u . b alpha.a ∈ Nright[10, arrowp . 0 22 comma ] . L open From parenthesis angbracketleft h form a decreasing sequence . \quad By hypothesis \quad $ \ langle h { m } , Ph { m }\rangle sub m comma(6.8)L(h ahhm sub, (a − ma rightα)hmi angbracket) ≤ closing parenthesis = supremum sub alpha L open parenthesis angbracketleft h sub m comma a sub \ leq \ langle h , Ph \rangle $ \quad for all proj ections $ P \ in N . $ alpha hhh, sub(a m− rightaα)hi angbracket. Since hh, closing(a − a parenthesisα)hiα→0,L period(hhm, ah Frommi) Corollary = supα L 6(h periodhm, aαh 1mi). From Corollary 6 . 1 Then $ \ langle h { m } , ah { m }\rangle \ leq \ langle h , ah \rangle , 0 < CapitalΦf,T ( Phia) = sub sup fα commaΦf,T (aα T) and open parenthesisΦf,T i s normal a closing on parenthesisN [10, § 3.6 =.1] supremum. sub alpha Capital Phi sub f comma T open parenthesis a suba alpha\ in closingN parenthesis ,We $ close as and with $ a Capital $ an example i Phi s a sub uniform fof comma(N , h limit) T− idominated s normal of finit on compactN open e linear square operators bracket positive . 1 0 comma spans S of 3 period 6 period 1 closing 2 squareproj bracket ections period [ 1 blacksquareLet 0 , pH . 23= ]L .(X,\quad µ) forFor a anyσ− finit generalised e measure limit space $(X, L µ $) andand set $ 0N as< a \ in N , $ Wemultiplication close with an example oper ofopen parenthesis N comma h closing parenthesis hyphen dominated compact operators period ∞ ∞ 2 \ beginLetators ..{ a H l i =g of n L∗}L to− thefunctions power of 2. open Then parenthesis0 < X T comma∈ L ( muL ( closingX, µ)) parenthesisb eing (N .., forh)− adominated sigma hyphen is finitthe e measure space .. open parenthesisL(same X\ aslangle comma muh closing{ m parenthesis} , ah.. and{ setm N}\ as multiplicationrangle ) oper\ hyphenleq \lim \sup \ langle h { m } , ah { m } \rangleators of L\ toleq the power\ langle of infinityh hyphen , functions ah \ periodrangle .... Then. \ 0tag less∗{ T$ in ( L to 6 the power . 8 of infinity ) $}\\ openm parenthesis\rightarrow L to the power\ infty of\end 2 open{Za l i parenthesis g n ∗} X comma mu closingZ parenthesis closing parenthesis .... b eing open parenthesis N comma h closing parenthesis hyphen 2 dµ(x) 2 measurable set sJ ⊂ is equivalent J | hm(x) | 2 are ≤ J | h(x) | dµby(x) forall| 2 1 ,whereX, which{ 1isanybasistotheof dominated is the same as | hm| some h| ∈L (X,µ)µ−a.e. hm}∞m= \noindentstatementLet $ \{ a {\alphadominated}\} $ be a net of monotonically increasing positive elements of $ N $ integral statement J bar h sub m open parenthesis x closing parenthesis bar to the power of 2 sub bar sub h√ sub m bar to the power of 2 to thewith powereigenvectors upper of d mu bound open of . parenthesisT. ItHence follows x any closing positive parenthesis compact sub are operator less or equal0 integral< T ∈ dominated L∞(L2(X, J µ bar)) i h s open(N , parenthesisg)− x closing parenthesis barthat to thedominated $ power\{ ofa 2 if d it{\ mu hasalpha by open a basis}\} parenthesis of$ eigenvectors xconverges closing parenthesis strongly whose sub modulus some to a for l allsquared . bar u . sub b hare $ bar . subdominated a 2 to the\ in power by N of measurable [ 1 sub0 in , L $ top the . 22 powersome ] . of\ 1quad openFrom$( parenthesis X comma 6 . mu sub 8 closing ) L parenthesis ( \ tolangle the powerh of set{ m sub} mu, hyphen ( a a to the− powera of{\ s subalpha period} e period) hpositive to{ them power}\ integrable ofrangle J sub comma function) to\ theleqg ∈ power$L1(X, of µ subset). For sub instance where X, the comma negative which open powers brace of sub the h Laplacian sub m closing brace sub infinity m = to the poweron of is sub 1 to the power of equivalent sub is any basis to the sub of \noindent $ \ langle h , ( a − a {\alpha }1 )n h \rangle . $ Since $ \ langle h eigenvectorsthe n− oftorus T period are dominated .... Hence any by positive the constant compact functionoperator 0 less1 ∈ L T in(T L). to the power of infinity open parenthesis L to the power of 2 open, parenthesis (7 a Summary− X commaa {\ mualpha closing} parenthesis) h closing\rangle parenthesis\alpha .... i{\ s ....rightarrow open parenthesis} N0 comma , square L ( root\ oflangle g closing parenthesish { m } hyphen, ahWe{ summarisem }\rangle the main) points = of\sup the review{\alpha and li} stL( the open\ questionslangle raisedh { m . } , a {\alpha } h { m } \rangledominated7 . 1) if it Summary .... . $ has From .... a basis Corollary .... of eigenvectors 6 . 1 .... whose modulus .... squared .... are .... dominated by .... some $positive\PhiThere integrable{ aref three , function main T } g non in( L - to a identical the ) power = classes of 1\sup open of Dixmier parenthesis{\alpha traces X}\ comma .Phi mu They{ closingf are , parenthesis , in T descending} ( period a For{\ instancealpha comma} ) $ the and $ \Phi { f , T }$ isnormalon $N [ 1 0 ,$ \S $ 3 . 6 . 1 ] . \ blacksquare $ negativeorder powers of of inclusion the Laplacian : on the ( original ) Dixmier traces (V1); the Connes – Dixmier traces (V2); theand n hyphen the Ces torusa` arero dominated invariant by Dixmier the constant traces function(V3) 1. in L to the power of 1 open parenthesis T to the power of n closing parenthesis period\ centerlineThe{ notionWeclose of measurable withanexample operator of with $( resp ect N t o , each h of these ) − classes$ dominated are non - identical compact operators . } 7 ... Summary The notion of Tauberian , ( 3 . 2 ) , is the strongest notion with ascending order of inclusion \ hspaceWe summarise∗{\ f i l l the} Let main\quad points of$ the H review = and L ˆ li{ st2 the} open(X, questions raised\mu period) $ \quad f o r a $ \sigma − $ finit e measure space \quad $ ( X , \mu ) $ \quad and set $ N $ as multiplication oper − 7 period 1 .. Summary T ( K1 ( K2 ( K3, There are three main non hyphen identical classes of Dixmier traces period .. They are comma in descending order of \noindentwherea t o r s o f $ L ˆ{\ infty } − $ functions . \ h f i l l Then $ 0 < T \ in L ˆ{\ infty } ( inclusion : .. the open parenthesisK = {T original∈ M closing| f (T parenthesis) = const Dixmier∀ f ∈ V traces}, i open= 1, 2 parenthesis, 3. V sub 1 closing parenthesis semicolon the LConnes ˆ{ 2 endash} (X, Dixmier traces\ openmui parenthesis) )1 $,∞ V\ subh f 2 i l closing l being$( parenthesisi semicolon N , and h the Ces ) a-grave− $ ro dominated is the same as invariant Dixmier traces open parenthesis V sub 3 closing parenthesis period \ [ The\ int notion{ ofstatement measurable} operatorJ \ withmid resph ect{ t om each} of( these x classes ) are\mid nonˆ hyphen{ 2 } identical{\mid period}ˆ{ d \mu ( x ) } { h { m } \midTheˆ{ notion2 }} of{ Tauberianare }\ commaleq open\ int parenthesis{ dominated 3 period} 2 closingJ \ parenthesismid h comma ( isx the ) strongest\mid notionˆ{ 2 with} d{\ ascendingmu } { orderby of} inclusion( x{ ) } { some } f o r a l l {\mid } { h \mid }ˆ{ measurable } { 2 } {\ in L ˆ{ 1 } (X, \mu }ˆ{ s e t } { ) }ˆ{ s } {\mu − Ta subsetneq}ˆ{ J K} sub{ . 1 subsetneq e . K}ˆ sub{\ 2subset subsetneq} K{ sub, } 3{ commawhere } X , which{\{} { h { m }\}}ˆ{ i s } {\ infty { m = }}whereˆ{ e q u i v a l e n t } { 1 } { i s any b a s i s } to the { o f }\ ] K sub i = open brace T in M sub 1 comma infinity bar f open parenthesis T closing parenthesis = const forall f in V sub i closing brace comma i = 1 comma 2 comma 3 period \noindent eigenvectors of $T . $ \ h f i l l Hence any positive compact operator $ 0 < T \ in L ˆ{\ infty } ( L ˆ{ 2 } (X, \mu ) ) $ \ h f i l l i s \ h f i l l $ ( N , \ sqrt { g } ) − $

\noindent dominated if it \ h f i l l has \ h f i l l a b a s i s \ h f i l l of eigenvectors \ h f i l l whose modulus \ h f i l l squared \ h f i l l are \ h f i l l dominated by \ h f i l l some

\noindent positive integrable function $ g \ in L ˆ{ 1 } (X, \mu ) . $ For instance , the negative powers of the Laplacian on

\noindent the $ n − $ torus are dominated by the constant function $ 1 \ in L ˆ{ 1 } ( T ˆ{ n } ) . $

\noindent 7 \quad Summary

\noindent We summarise the main points of the review and li st the open questions raised .

\noindent 7 . 1 \quad Summary

\noindent There are three main non − identical classes of Dixmier traces . \quad They are , in descending order of i n c l u s i o n : \quad the ( original ) Dixmier traces $ ( V { 1 } ) ; $ the Connes −− Dixmier traces $ ( V { 2 } ) ;$ andtheCes $ \grave{a} $ ro invariant Dixmier traces $ ( V { 3 } ) . $

The notion of measurable operator with resp ect t o each of these classes are non − i d e n t i c a l . The notion of Tauberian , ( 3 . 2 ) , is the strongest notion with ascending order of inclusion

\ [T \ subsetneq K { 1 }\ subsetneq K { 2 }\ subsetneq K { 3 } , \ ]

\noindent where

\ centerline { $ K { i } = \{ T \ in M { 1 , \ infty }\mid $ f $( T ) =$ const $ \ f o r a l l $ f $ \ in V { i }\} , i = 1 , 2 , 3 . $ } 28 .... S period Lord and F period Sukochev \noindenthline2828 \ h f i l l S . Lord and F . Sukochev S . Lord and F . Sukochev None of these sets i s an ideal of compact operators period .. The notion of measurable operator using \ [ Dixmier\ r u l e { traces3em}{ does0.4 pt not}\ achieve] T = K sub 1 comma however T to the power of plus = K sub 1 to the power of plus period .. This has the consequence that the None of these sets i s an ideal of compact operators . The notion of measurable operator operators T in K sub 1 do not decompose into unique measurable components+ + period .... In short comma K sub 1 does not \noindenthaveusing the propertyNone Dixmier of T in thesetraces K sub does sets 1 double not i s stroke achieve an idealrightT arrow= ofK1, barcompacthowever T bar in operatorsT K= subK1 . period .This\quad hasThe the consequence notion of measurable operator using DixmierConnesthat quoteright tracesthe doestrace theorem not achieve open parenthesis $ T Theorem = K ..{ 41 period} , 1 closing$ however parenthesis $Tˆ i s valid{ + for} any= rho K in ˆ V{ sub+ } sing{ to1 the} power. $ of\quad commaoperatorsThis to the has powerT the∈ ofK1 consequenceplusdo notopen decompose parenthesis that 4 intothe period unique 6 closing measurable parenthesis components period .. In particular . In short comma, K1 ifdoes M i s an nnot hyphen dimensional compact Riemannian manifold with Hodge Laplacian Capital Delta we have \noindenthave theoperators property $T T∈ K1 \⇒|inT |∈K K1.{ 1 }$ do not decompose into unique measurable components . \ h f i l l In short rho open parenthesis P closing parenthesis = c sub rho Res open parenthesis P closing+ parenthesis forall P in Op sub cl to the power of ρ ∈ V , (4.6). minus$ , n K openConnes{ parenthesis1 }$ ’ trace does M theorem closing not parenthesis ( Theorem 4 . 1 ) i s valid for any sing In particular , withif theM constanti s c sub rho greater 0 for the subset Capital Theta sub n of V sub sing to the power of plus such that rho open parenthesis open\noindent parenthesisan n−havedimensional 1 plus the Capital property compact Delta closing $ Riemannian T parenthesis\ in manifoldK to the{ 1 power}\ with ofRightarrow minusHodge n Laplacianslash 2 closing\mid∆ parenthesisweT have\mid greater\ in 0 periodK ..{ The1 set} Capital. $ Theta sub n \ hspace ∗{\ f i l l }Connes ’ trace theorem ( Theorem \quad−n4 . 1 ) i s valid for any $ \rho \ in V ˆ{ + } { s i n g ˆ{ , }} of singular traces is larger than the set ofρ( DixmierP ) = cρRes( tracesP ) open∀P parenthesis∈ Opcl (M) V sub 1 closing parenthesis period .. Further comma for rho in ( 4 . 6 ) . $ \quad In particular , if $M$ i s Capital Theta sub n we have + −n/2 with the constant cρ > 0 for the subset Θn of Vsing such that ρ((1 + ∆) ) > 0. The set Θn rho open parenthesis f open parenthesis 1 plus Capital Delta closing parenthesis to the power of minus n slash 2 closing parenthesis = rho of singular traces is larger than the set of Dixmier traces (V1). Further , for ρ ∈ Θn we have open\noindent parenthesisan open $ n parenthesis− $ dimensional 1 plus Capital Delta compact closing Riemannian parenthesis to manifold the power withof minus Hodge n slash Laplacian 2 closing parenthesis $ \Delta divided$ we by Vol have open parenthesis M closing parenthesis integral sub M fdx comma−n/2 forallZ f in L to the power of infinity open parenthesis M closing parenthesis −n/2 ρ((1 + ∆) ) ∞ comma\ [ \rho ( P ) =ρ(f(1 c +{\ ∆) rho) =} Res ( Pfdx, ) ∀\ff o∈ rL a l l(M),P \ in Op ˆ{ − n } { c l } (M Vol(M) M ) \where] dx i s the volume form on M open parenthesis Theorem 4 period 2 closing parenthesis period Forwhere any sp ectraldx i s triple the volume open parenthesis form on AM comma( Theorem H comma 4 . D 2 closing ) . parenthesis with angbracketleft D right angbracket to the power of −n + minus n inFor M sub any 1 sp comma ectral infinity triple define(A,H,D Capital) with ThetahD openi ∈ parenthesis M1,∞ define D closingΘ(D) parenthesist o b e the t o subset b e the of subsetVsing of V sub sing to the power of\noindent plussuch thatwithρ( thehDi− constantn) > 0. Define $ c the{\ noncommutativerho } > 0 $ integral for the subset $ \Theta { n }$ o f $ V ˆ{ + } { s i n g }$ suchsuch that that rho $ open\rho parenthesis( ( angbracketleft 1 + \ DDelta right angbracket) ˆ{ − to then power / of 2minus} n) closing> parenthesis0 . $ greater\quad 0 periodThe s .. e t Define $ \ theTheta { n }$ noncommutativeof singular integral traces is larger thanρ(a thehDi−n set), aof∈ ADixmier, ρ ∈ Θ( tracesD). $ ( V { 1 } ) . $ \quad Further , for $ \rhorho open\ parenthesisin \Theta a angbracketleft{ n }$ D we right have angbracket to the power of minus n closing parenthesis comma a in A comma rho in Capital This defines a family of continuous positive traces on A( in the implied C∗− norm ) ( Corollary Theta open parenthesis D closing parenthesis period 4 . 1 ) . Restricting the noncommutative integral t o the consideration of ρ ∈ V ( the set \ [ This\rho defines( a family f of ( continuous 1 + positive\Delta traces on) Â{ open − parenthesisn / in 2 the} implied) = C to\ f the r a c power{\1rho of * hyphen( norm( 1 closing + parenthesis\Delta of Dixmier traces or the set of Hardy – Littlewood – P o´ lya submajorisation ordered open) ˆ{ parenthesis − n Corollary / 2 } 4 period) }{ 1Vol closing ( parenthesis M ) period}\ int { M } fdx , \ f o r a l l f \ in L ˆ{\ infty } (M),continuous\ functionals] on m1,∞, Section 2 . 1 ) enables Lidksii typ e theorems , residue Restricting, and the heat noncommutative kernel theorems integral , t o see the Sectionconsideration 5 of and rho comments in V sub 1 open . parenthesis the set of Dixmier traces or the set .. of Hardy endash Littlewood endashThe functional P o-acute lya submajorisation ordered continuous functionals on m sub 1 comma infinity comma .. Section 2 period 1 closing parenthesis .. enables Lidksii typ e theorems comma .. residue comma .. and\noindent heat kernelwhere theorems $ commadx $ .. i see s the volume form on $M ( $ Theorem 4 . 2 ) . f(aT ), a ∈ A ⊂ L(H) Section 5 .. and comments period ForanyspectralThewhere functional f i s a trace triple on a two $( - sided A ideal , of H compact , D operators )$ withJ and $ 0\ 0 . $ \quad Define the noncommutative integral wherecharacterised f i s a trace on by a two linear hyphen functionals sided idealσ of: ` compact∞ → C applied operators to J sequencesand 0 less T of in J comma “ expectation i s the general values for hyphen ∞ mat” of noncommutative{hhm, ahmi}m=1 integration introduced by Connes period Such functionals can be characterised \ [ \rho ( a \ langle D \rangle∞ ˆ{ − n } ) , a \ in A, \rho \ in \Theta ( by linearfor some functionals orthonormal sigma : l basis to the power{hm}m of=1 infinityof eigenvectors right arrow of C appliedT. In particular to sequences of .. quotedblleft expectation values quotedblright D)... open brace angbracketleft\ ] h sub m comma ah sub m right angbracket closing brace sub m = 1 to the power of infinity for some orthonormal basis open brace h sub m closing brace sub m = 1 to the power of infinity of eigenvectors of T period In particular ∞ f open parenthesis aT closing parenthesisf( =aT sigma) = σ open({hhm parenthesis, ahmi} open) brace angbracketleft h sub m comma ah sub m right angbracket Row\noindent 1 infinityThis Row 2 defines m = 1 closing a family brace. for of all continuous a in L open parenthesis positivem = H 1 traces closing parenthesis on $ A period ( $ in the implied $ C ˆ{ ∗ } − $For norm flat t orii ) ( comma Corollary both commutative 4 . 1 ) and. noncommutativeforall commaa ∈ L( thisH). implies Restrictingf open parenthesis the ag noncommutative open parenthesis Capital integral Delta tclosing o the parenthesis consideration closing parenthesis of $ \ =rho f open parenthesis\ in V g{ open1 } parenthesis( $ the Capital set of Dixmier Deltatraces closing or parenthesis theFor set flat closing\quad t orii parenthesiso f , bothHardy commutative tau−− openLittlewood parenthesis and−− noncommutative a closingP $ parenthesis\acute{ ,o} this$ implies lya submajorisation ordered continuous functionals onfor any$ m trace{ 1 f on .. , a two\ infty hyphen} sided, ideal$ \quad J commaSection .. and g 2 .. a. positive 1 ) \quad boundedenables .. Borel Lidksii function .. typ such e that theorems , \quad r e s i d u e , \quad and heat kernel theorems , \quad see Secg open tion parenthesis 5 \quad and Capital comments Delta closing . parenthesisf(ag(∆)) in = J f(g period(∆))τ( ..a) Here Capital Delta i s the Laplacian open parenthesis resp period noncommutative Laplacian closing parenthesis .. on the torus and tau i s \ centerlinefor any{ traceThe functional f on a two} - sided ideal J , and g a positive bounded Borel function thesuch Lebesgue that integralg(∆) on∈ commutative J . Here t∆ oriii s open theparenthesis Laplacian resp ( resp period . noncommutative the unique faithful Laplacian normal finit ) e trace on on the typthe e II torussub 1 noncommutative and τ i s the t Lebesgue orii closing integral parenthesis on period commutative t orii ( resp . the unique faithful \ [If f N subset ( aT L open ) parenthesis , a H closing\ in parenthesisA \subset is a von NeumannL(H) algebra comma\ ] a positive compact operator T i s open parenthesis normal finit e trace on the typ e II1 noncommutative t orii ) . N comma hIf closingN ⊂ L parenthesis(H) is a von hyphen Neumann dominated algebra , a positive compact operator T i s (N , h)− dominated if comma .. for some orthonormal basis .. open brace∞ h sub m closing brace sub m = 1 to the power of infinity .. of eigenvectors of T comma if , for some orthonormal basis {hm}m=1 of eigenvectors of T, there exists h ∈ H ..\noindent there existswhere .. h in H f .. i such s a that trace on a two − sided ideal of compact operators $ J $ and $ 0 < T \ in such that k P hm k ≤ k P h k for all proj ections P ∈ N . If J is any two - sided ideal with Jbar , Ph $ sub i ms barthe less general or equal for bar Ph− bar .. for all proj ections P in N period .. If J is any two hyphen sided ideal with positive singular matpositive of noncommutative singular trace integration f , and 0 < introduced T ∈ J is ( byN , h Connes)− dominated . Such , functionalsthen f (·T ) i scan a positive be characterised tracenormal f comma linear and 0 functional less T in J is on openN parenthesis. N comma h closing parenthesis hyphen dominated comma then f open parenthesis times Tby closing linear parenthesis functionals i s a positive $ normal\sigma linear: functional\ e l l ˆ{\ infty }\rightarrow C $ applied to sequences of \quad ‘‘ expectation values ’’ \quad $ \{\on N periodlangle h { m } , ah { m }\rangle \} ˆ{\ infty } { m = 1 }$ \noindent for some orthonormal basis $ \{ h { m }\} ˆ{\ infty } { m = 1 }$ of eigenvectors of $T . $ In particular

\ begin { a l i g n ∗} f ( aT ) = \sigma ( \{\ langle h { m } , ah { m }\rangle \}\ begin { array }{ c}\ infty \\ m = 1 \end{ array })\\ f o r a l l a \ in L(H). \end{ a l i g n ∗}

\ centerline {For flat t orii , both commutative and noncommutative , this implies }

\ [ f ( ag ( \Delta ) ) = f ( g ( \Delta )) \tau ( a ) \ ]

\noindent for any trace f on \quad a two − sided ideal $J , $ \quad and $ g $ \quad a positive bounded \quad Borel function \quad such that $ g ( \Delta ) \ in J . $ \quad Here $ \Delta $ i s the Laplacian ( resp . noncommutative Laplacian ) \quad on the torus and $ \tau $ i s the Lebesgue integral on commutative t orii ( resp . the unique faithful normal finit e trace on the typ e $ I I { 1 }$ noncommutative t orii ) .

I f $ N \subset L ( H ) $ is a von Neumann algebra , a positive compact operator $T$ i s $ ( N , h ) − $ dominated i f , \quad for some orthonormal basis \quad $ \{ h { m }\} ˆ{\ infty } { m = 1 }$ \quad of eigenvectors of $ T , $ \quad there exists \quad $ h \ in H $ \quad such that $ \ parallel Ph { m }\ parallel \ leq \ parallel Ph \ parallel $ \quad for all proj ections $ P \ in N . $ \quad If $J$ is anytwo − sided ideal with positive singular trace f , and $ 0 < T \ in J $ i s $ ( N , h ) − $ dominated , then f $ ( \cdot T ) $ i s a positive normal linear functional on $ N . $ Measure Theory in Noncommutative Spaces .... 29 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 29 29 7 period 2 .. List of open questions \ [ Let\ r u .... l e { open3em}{ parenthesis0.4 pt }\ A] comma H comma D closing parenthesis .... b e a sp ectral triple .... open parenthesis as defined in the first paragraph of Section .... 1 closing parenthesis .... such that angbracketleft7 . 2 ListD right of angbracket open questions to the power of minus n in M sub 1 comma infinity open parenthesis as defined at open parenthesis 1 \noindentLet (A7,H,D . 2 )\bquad e a spList ectral of triple open ( questions as defined in the first paragraph of Section 1 ) such that period 5 closing−n parenthesis closing parenthesis period hDi ∈ M1,∞( as defined at ( 1 . 5 ) ) . Open .... Question .... 7 period 1 .... open parenthesis from .... Section .... 1 closing00 parenthesis period .... If A to the power of prime prime ....\noindent containsOpen noLet finite Question\ h rank f i l l operators$ ( 7 . is A 1 Capital ( , from Phi H sub , omega Section D open ) parenthesis $ 1 )\ h . f i l lIf theb eA acontains sp ectral no finitetriple rank\ h f i l l ( as defined in the first paragraph of Section \ h f i l l 1 ) \ h f i l l such that functionaloperators in open is parenthesisΦω ( the 1 period 1 closing parenthesis closing parenthesis .. normal ? \noindentOpenfunctional Question$ \ langle in7 period( 1 . 2 1 ....D ) ) open\ranglenormal parenthesisˆ{? − fromn ....}\ Sectionin ....M 1{ closing1 parenthesis , \ infty period} ....($ What as are defined necessary at and (1 sufficient . 5) ) . relationshipsOpen be Question hyphen 7 . 2 ( from Section 1 ) . What are necessary and sufficient \noindentrelationshipsOpen \ h be f i - l l Question \ h f i l l 7 . 1 \ h f i l l ( from \ h f i l l Sec tion \ h f i l l 1 ) . \ h f i l l I f $ A ˆ{\prime tween angbracketleft−n D right angbracket to the power of minus n .... and the * hyphen algebra A so that Capital Phi sub omega open parenthesis\primetween}$ the\ functionalhhD fi i l l contains inand open the parenthesis no∗− finitealgebra 1 period rankA 1so operatorsclosing that parenthesisΦω is( the $ closing functional\Phi parenthesis{\omega in ( .... 1} is. 1 independent( ) ) $ theis of omega open parenthesis for independent of ω ( for \noindentvariousfunctional sets of generalised in ( 1 limits . 1 ) ))?\quad normal $ ? $ various sets of generalised limits closing parenthesis ? ∞ OpenOpen Question Question 7 period 37 .... . 3 open ( parenthesis from Section from Section 3 ) . 3 closingIs parenthesisB ⊂ S∞(` period) satisfying .... Is B thesubset Ces S suba` infinityro open parenthesis l to the\noindent powerlimit of infinitypropertyOpen Question closing parenthesis 7 . 2 ....\ h satisfying f i l l ( from the Ces\ h a-grave f i l l Sec ro limit tion property\ h f i l l 1 ) . \ h f i l l What are necessary and sufficient relationships be − K+ open( from parenthesis[ 2 from 7 , open§5 ] ) squarea necessary bracket 2 and .... 7 sufficient comma S 5 condition closing square for bracketequality closing of a setparenthesisB ofpositive a necessary and sufficient condition for\noindent equalitymeasurable oftween a set K sub $ \ Blangle to the powerD of\ plusrangle ofpositiveˆ{ − measurablen }$ \ h f i l l and the $ ∗ − $ algebra $A$ so that $ \operatorsPhioperators{\ openomega parenthesis( e} . g( . $( e 3 period the . 3 ) functional g ) periodwith open the in parenthesis Tauberian ( 1 . 1 3 operators ) period ) \ h 3 f closingi l l? is parenthesis independent closing of parenthesis $ \omega .. with( the $ Tauberian f o r Question + operatorsOpen ? operators?7.4( from Section 3 ) . Is KPDL[1,∞)( defined in ( 3 . 3 ) ) equal to the \noindentOpenTauberian operatorsvarious sub ? sets to the of power generalised of Question limits 7 period 4 $ open ) parenthesis ? $ from Section 3 closing parenthesis period .... Is K sub PDL open squareOpen bracket Question 1 comma 7 infinity. 5 ( closing from parenthesis Section to3 .the 2 power ) . ofIs plus there open a parenthesismaximal c defined losed symmetricin open parenthesis 3 period 3 closing \noindent Open Question 7 . 3 \ h f i l l ( from Section 3 ) . \ h f i l l I s $ B \subset S {\ infty } ( parenthesissubideal closingU parenthesis2 equal to the Tauberian \ e l l ˆ{\ infty } ) $ \ h f i l l satisfying the1,w Ces $ \grave{a} $ ro limit property Openof QuestionM1,∞ 7such period that 5 ....T˜ open∩ U2 parenthesis= K2 ∩ U2? fromIs SectionL 3⊆ period U2? 2 closing parenthesis period .... Is there a maximal c losed symmetric 1 subidealOpen U sub 2 Question 7 . 6 . What is the relation between the s ets V 2 and Θ(D)( \noindentof Mdefined sub 1( comma above from infinity [) 2 \ suchh f i l that l 7 T-tilde , \S cap5 ] U ) sub a 2 necessary = K sub 2 cap and U sub sufficient 2 ? .. Is L to condition the power of for 1 commaof equality w subset of equal a set U sub $ 2 K ? ˆ{ + } { B }$ ofpositive measurable Opensingular .... Question traces .... 7? periodWhat 6 period are their .... What s ets is of the measurable relation between operators the s ets? V 1 divided by 2 .... and Capital Theta open parenthesis D closing parenthesis open parenthesis defined above closing parenthesis .... of \noindentAoperators Identifications ( e . g . ( in 3 . 3 ( ) 2 ) .\quad 3with 7 ) the Tauberian operators $ ? $ singularWe start traces with ? .. What preliminaries are their s . ets of Define measurable operators ? A .. Identifications .. in .. open parenthesis 2 period 3 .. 7 closing parenthesis \noindentWe start with$ Openpreliminaries{ o p period e r a t o r.. s Define}ˆ{ Question } { ? } 7 . 4 ($ fromSection3). \ h f i l l I s $ K ˆ{ + } { PDL [ 1 , \ infty ) } ( $ defined in ( 3Z .t 3 ) ) equal to the Tauberian Equation: open parenthesis A period 1 closing parenthesis1 .. alpha∗ sub g open parenthesis t closing parenthesis : = 1 divided by log open αg(t) := g (s)ds, t ≥ 1, (A.1) parenthesis 1 plus t closing parenthesis integral sublog(1 1 to + thet) power1 of t g to the power of * open parenthesis s closing parenthesis ds comma t greater\noindent equalOpen 1 comma Question 7 . 5 \ h f i l l ( from Section 3 . 2 ) . \ h f i l l Is there a maximal c losed symmetric subideal $ Uwhere{where2 }$ g to the power of * open parenthesisg∗(s) := inf s closing{t | µ( parenthesis| g |> t) < s :}, = inf µ i open s Lebesgue brace t bar measure mu open on parenthesis[1, ∞). bar( A g .bar 2 ) greater t closing parenthesis less\noindent s closingSet braceo f $ comma M { mu1 i s , Lebesgue\ infty measure}$ on such open that square $ bracket\ tilde 1 comma{T}\ infinitycap closingU { parenthesis2 } = period K { open2 }\ parenthesiscap AU { 2 } period? $ \ 2quad closingI parenthesiss $ L ˆ{ 1 , w }\subseteq U { 2 } ? $

Set ∞ \noindentEquation:Open open parenthesis\ h f i l l Question A periodm1 3,\∞ closingh:= f i{ lg l parenthesis∈7L .([1 6, .∞)) ..\ h| mα f ig sub l∈ l C 1Whatb([1 comma, ∞ is))} infinity. the relation : = open brace between g in L(A the to.3) the s power ets of $ infinityV \ f r open a c { 1 }{ 2 }$ parenthesis\ h f i l l and open $ square\Theta bracket( 1 comma D ) infinity ($ closing definedabove) parenthesis closing\ h parenthesis f i l l o f bar alpha sub g in C sub b open parenthesis open Define : p from `∞ t o bounded Borel functions on [0, ∞) by square bracket 1 comma infinity closing parenthesis closing parenthesis closing brace period \noindentDefine : psingular from l to the traces power of infinity$ ? $ t o\quad boundedWhat Borel are functions their on s open ets square of measurable bracket 0 comma operators infinity closing $ ? parenthesis $ by infinity Equation: open parenthesis A period 4 closing parenthesis .. p open parenthesis∞ open brace a sub k closing brace sub k = 0 to the power\noindent of infinityA \ closingquad Identifications parenthesis open parenthesis\quad tin closing\quad parenthesis( 2 . 3 :\ =quad sum a7 sub ) k chi sub open square bracket k comma k plus 1 ∞ X closing parenthesis open parenthesis t closingp({ak parenthesis}k=0)(t) := commaakχ[k,k t greater+1)(t), equal t ≥ 00; semicolon k = 0 (A.4) \noindent We start with preliminaries . \quad Define p c from l to the power of infinity t o continuous bounded functions on openk = square 0 bracket 0 comma infinity closing parenthesis by infinity Equation: open parenthesis A period 5 closing parenthesis .. p c open parenthesis open brace a sub k closing brace sub k = 0 to the \ begin { a l i g n ∗} power ofpc infinityfrom ` closing∞ t o continuousparenthesis open bounded parenthesis functions t closing on parenthesis[0, ∞) by : = sum open parenthesis a sub k plus open parenthesis t minus k closing\alpha parenthesis{ g } open( parenthesis t ) a sub: k = plus\ 1f r minus a c { 1 a sub}{\ klog closing parenthesis( 1 + closing t parenthesis ) }\ int chi subˆ{ opent } square{ 1 } bracketg ˆ{ k comma ∗ } k( plus s 1 closing ) parenthesis ds , open t parenthesis\geq 1 t closing , \ tag parenthesis∗{$ ( comma A .t greater 1 equal ) $} 0 semicolon k = 0 \endr from{ a l i boundedg n ∗} Borel or continuous functions on open square bracket 0 comma infinity closing∞ parenthesis t o l to the power of infinity by Equation: open parenthesis A period 6 closingX parenthesis .. r open parenthesis f closing parenthesis k : = f open parenthesis k closing pc({a }∞ )(t) := (a + (t − k)(a − a ))χ (t), t ≥ 0; (A.5) parenthesis\noindent commawhere k in open bracek 0k=0 comma 1 commak 2 commak period+1 periodk [k,k period+1) closing brace semicolon k = 0 \ hspace ∗{\ f i l l } $ g ˆ{ ∗ } ( s ) : = $ i n f $ \{ t \mid \mu ( \mid g \mid > t ) r from< boundeds \} Borel, or\mu continuous$ i s Lebesgue functions on measure[0, ∞) t on o ` $∞ [by 1 , \ infty ) . ( $ A . 2 )

\noindent Set r(f)k := f(k), k ∈ {0, 1, 2, ...}; (A.6) \ begin { a l i g n ∗} m { 1 , \ infty } : = \{ g \ in L ˆ{\ infty } ( [ 1 , \ infty )) \mid \alpha { g } \ in C { b } ( [ 1 , \ infty )) \} . \ tag ∗{$ ( A . 3 ) $} \end{ a l i g n ∗}

\noindent Define $ : p$ from $ \ e l l ˆ{\ infty }$ t o bounded Borel functions on $ [ 0 , \ infty ) $ by

\ begin { a l i g n ∗} \ infty \\ p ( \{ a { k }\} ˆ{\ infty } { k = 0 } ) ( t ) : = \sum a { k } \ chi { [ k , k + 1 ) } ( t ) , t \geq 0 ; \ tag ∗{$ ( A . 4 ) $}\\ k = 0 \end{ a l i g n ∗}

\noindent $ p c $ from $ \ e l l ˆ{\ infty }$ t o continuous bounded functions on $ [ 0 , \ infty ) $ by

\ begin { a l i g n ∗} \ infty \\ p c ( \{ a { k }\} ˆ{\ infty } { k = 0 } ) ( t ) : = \sum ( a { k } + ( t − k ) ( a { k + 1 } − a { k } )) \ chi { [ k , k + 1 ) } ( t ) , t \geq 0 ; \ tag ∗{$ ( A . 5 ) $}\\ k = 0 \end{ a l i g n ∗}

\noindent $ r $ from bounded Borel or continuous functions on $ [ 0 , \ infty ) $ t o $ \ e l l ˆ{\ infty }$ by

\ begin { a l i g n ∗} r(f)k:=f(k),k \ in \{ 0 , 1 , 2 , . . . \} ; \ tag ∗{$ ( A . 6 ) $} \end{ a l i g n ∗} 30 .... S period Lord and F period Sukochev \noindentLine30 1 hline30 Line\ h 2 f E i l : l LS to the . Lord power and of infinity F . open Sukochev parenthesis open square bracket S . 0 Lord comma and infinity F . Sukochev closing parenthesis closing parenthesis right arrow L to the power of infinity open parenthesis open square bracket 0 comma infinity closing parenthesis closing parenthesis by Line 3 E\ [ open\ begin parenthesis{ a l i g n e f d closing}\ r u parenthesis l e {3em}{0.4 open pt parenthesis}\\ t closing parenthesis : = integral sub t to the power of t plus 1 f open parenthesis s E : L ˆ{\ infty } ( [ 0 , \ infty )) \rightarrow L ˆ{\ infty } ( [ 0 , \ infty closing parenthesis ds comma t greater equal 0 semicolonE : L∞([0 open, ∞)) parenthesis→ L∞([0, A∞ period))by 7 closing parenthesis Line 4 L to the power of minus 1 : L to) the ) power by of\\ infinity open parenthesis open square bracket 1 comma infinity closing parenthesis closing parenthesis right arrow L to the power E ( f ) ( t ) : = \Zintt+1 ˆ{ t + 1 } { t } f ( s ) ds , t \geq 0 of infinity open parenthesis open square bracketE(f)(t) 0 := commaf infinity(s)ds, closing t ≥ 0; parenthesis (A.7) closing parenthesis by Line 5 L to the power of minus 1 open; parenthesis ( A . g closing 7 parenthesis ) \\ open parenthesist t closing parenthesis = g open parenthesis e to the power of t closing parenthesis comma t greaterL ˆ{ −equal1 0 semicolon} : L open ˆ{\ parenthesisinfty } A periodL(−1 : L [ 8∞ closing([1 1, ∞)) parenthesis ,→ L∞\ infty([0, ∞))by )) \rightarrow L ˆ{\ infty } ([ 0 , \ infty ) ) by \\ and finally the maps on l to the power of infinityL−1(g for)(t) j =ing N(e byt), t ≥ 0; (A.8) LEquation: ˆ{ − open1 } parenthesis(g)(t)=g(eˆ A period 9 closing parenthesis .. T sub j open{ t parenthesis} ) , open t brace\geq a sub k0 closing ; brace ( sub A k = . 0 to 8 the ∞ power) \end ofand{ infinitya l finallyi g n e closingd }\ the] maps parenthesis on ` : =for openj ∈ braceN by a sub k plus j closing brace sub k = 0 to the power of comma to the power of infinity Equation: open parenthesis A period 10 closing parenthesis .. C open parenthesis open brace a sub k Row 1 infinity Row 2 k = 0 closing brace.

: = open brace 1 divided by n plus 1 sum from k = 0 to n a sub∞ k closing brace∞ sub n = 0 to the power of infinity comma \noindentand the mapsand on finally L to the thepower maps of infinity on open $ \ e parenthesis lT lj({â{\k}k=0infty open) := { squarea}k$+j}k f=0 obracket r, $ 0 j comma\ in infinityN $ closing by (A parenthesis.9) closing parenthesis k=0 for a greater 0 by ∞ 1 X \ begin { a l i g n ∗} C({a } ) := { a }∞ , (A.10) Equation: open parenthesis A period 1 1 closingk k = parenthesis 0 n + .. 1 T subk an open=0 parenthesis f closing parenthesis open parenthesis t closing parenthesisT { j } : =( f open\{ parenthesisa { k t plus}\} a closingˆ{\ infty parenthesis} { commak =n Equation: 0 } open) parenthesis : = \{ A perioda 1{ 2k closing + parenthesis j }\} ..ˆ C{\ openinfty } { k =parenthesis 0 ˆ{ f, closing}}\ tag parenthesis∗{$ ( open A parenthesis . 9 t ) closing $}\\ parenthesisC( :\{ = 1 divideda { k by}\}\ t integralbegin sub{ 0array to the}{ powerc}\ infty of t f open\\ parenthesisk = s and the maps on L∞([0, ∞)) for a > 0 by closing0 \end parenthesis{ array }) ds :period = \{\ f r a c { 1 }{ n + 1 }\sum ˆ{ k = 0 } { n } a { k }\} ˆ{\ infty } { n =Lemma 0 } A, period\ tag ∗{ 1$ period ( .. A Let open. 10brace a ) sub $} n closing brace sub n = 0 to the power of infinity in l to the power of infinity .. and f in\end L to{ a the l i g power n ∗} of infinity open parenthesis open square bracket 0 comma infinity closing parenthesis closing parenthesis period .. With the Ta(f)(t) := f(t + a), (A.11) above definitions we have : \noindent and the maps on $ L ˆ{\ infty } (1 Z t [ 0 , \ infty ) ) $ f o r $ a > 0 $ by 1 period limint sub t right arrow infinity open parenthesisC(f)(t) := T subf j(s p)ds. sub open parenthesis c closing parenthesis(A. minus12) p sub open parenthesis c closing parenthesis T sub j closing parenthesis open parenthesist 0 open brace a sub n closing brace closing parenthesis open parenthesis t closing parenthesis\ begin { a l = i g 0n ∗} semicolon 2 period limint sub t right arrow infinity open parenthesis Cp sub open parenthesis c closing parenthesis minus p {a }∞ ∈ `∞ f ∈ L∞([0, ∞)). subT open{ Lemmaa parenthesis} (f)(t):=f(t+a), A . c 1 closing . parenthesisLet n n C=0 closing parenthesisand open parenthesisWith open the brace above\ atag sub∗{ definitions n$ closing ( A brace we . closing 1 parenthesis 1 ) $}\\ openC parenthesis( fhave ) t : closing ( parenthesis t ) := 0semicolon = \ f r a c 3{ period1 }{ opent }\ parenthesisint ˆ{ Tt sub} j{ rE0 minus} f rET ( sub sj closing ) parenthesis ds . \ opentag ∗{ parenthesis$ ( A f closing. 1 parenthesis 2 ) $in} c sub 0 semicolon 4 period open parenthesis CrE minus rEC closing parenthesis open parenthesis f closing parenthesis in c\end sub{ 0a comma l i g n ∗} for any j in N period 1. lim (Tjp(c) − p(c)Tj)({an})(t) = 0; Proof period .. Let open brace a sub k closingt→∞ brace sub k = 0 to the power of infinity in l to the power of infinity period .. Then \noindent Lemma A . 1 . \quad Let $ \{ a { n }\} ˆ{\ infty } { n = 0 }\ in \ e l l ˆ{\ infty }$ Line 1 infinity infinity Line 2 open parenthesis2. lim T(Cp sub(c) j− pp minus(c)C)({ pTan} sub)(t)j = closing 0; parenthesis open parenthesis open brace a sub k closing brace\quad closingand parenthesis $ f \ in open parenthesisL ˆ{\ infty t closingt}→∞ parenthesis( [ =0 sum , a sub\ infty k chi sub open) square ) . bracket $ \quad k commaWith k plus the 1 closing above parenthesis definitions we have : open parenthesis t plus j closing parenthesis minus sum3. a( subTjrE k− plusrET jj chi)(f sub) ∈ c open0; square bracket k comma k plus 1 closing parenthesis open \ begin { a l i g n ∗} parenthesis t closing parenthesis = 0 period Line 3 k =4. 0 k(CrE = 0 − rEC)(f) ∈ c0, 1 . \lim { t \rightarrow \ infty } (T { j } p { ( c ) } − p { ( c ) } T { j } Similarly comma foranyj ∈ N. )(open parenthesis\{ a T{ subn j}\} p c minus) p c to ( the power t ) of T = j to the 0 power ; \\ of closing2 . parenthesis\lim open{ t parenthesis\rightarrow open brace\ ainfty k closing} brace( Cp { ( c ) } − p∞ { (∞ c ) } C)( \{ a { n }\} ) ( t ) = 0 ; \\ 3 closingProof parenthesis . openLet parenthesis{ak}k=0 ∈ t closing` . Then parenthesis = 0 period .(TThis proves{ 1 periodj } ..rE We also− haverET { j } ) ( f ) \ in c { 0 } ; \\ 4 . ( CrE − rEC )Line ( 1open f )parenthesis\ in Cpc minus{ 0 } pC closing, \\ fparenthesis o r any open j parenthesis\ in N. open brace∞ a∞ sub k closing brace closing parenthesis open parenthesis\end{ a l i g tn ∗} closing parenthesis = 1 divided by tX integral sub 0 to theX power of t sum from k = 0 to infinity a sub k chi sub open square (Tjp − pTj)({ak})(t) = akχ[k,k+1)(t + j) − ak+jχ[k,k+1)(t) = 0. bracket k comma k plus 1 closing parenthesis open parenthesis s closing parenthesis ds minus sum from k = 0 to infinity 1 divided by k plus 1 sum\noindent from i =Proof 0 to k a . sub\quad i chi subLet open $ square\{ bracketa { k k}\} comma kˆ{\ plusinfty 1 closing} parenthesis{k =k 0 k == open 0 0 }\ parenthesisin \ te closing l l ˆ{\ parenthesisinfty } Line. 2 $ = \quad Then parenleftbiggSimilarly floorleft , t floor plus 1 divided by t minus 1 parenrightbigg open parenthesis C open parenthesis open brace a sub k closing brace closing parenthesis open parenthesis floorleft t floor closing parenthesis plus parenleftbigg 1 minus floorleft t floor divided by t parenrightbigg \ [ \ begin { a l i g n e d }\ infty \ infty \\ a sub floorleft t floor period T )({a (Tjpc − pc j k})(t) = 0. (THence { j } p − pT { j } )( \{ a { k }\} ) ( t ) = \sum a { k }\ chi { [ kbar ,This open k proves parenthesis + 1 1 . Cp ) We minus} also( pC have t closing + parenthesis j ) − open \ bracesum a suba { k closingk + brace j }\ bar subchi infinity{ [ less k or equal , k parenleftbigg + 1 vextendsingle-vextendsingle-vextendsingle-vextendsingle) } ( t ) = 0 . \\ ceilingleft t ceilingright divided by t minus 1 vextendsingle-vextendsingle-vextendsingle- k=0 k=0 i=0 vextendsinglek = 0 plus kvextendsingle-vextendsingle-vextendsingle-vextendsingle = 0 \end{ a l i g n e1 dZ}\t X] X floorleft1 X t floor divided by t minus 1 vextendsingle-vextendsingle- (Cp − pC)({a })(t) = a χ (s)ds − a χ (t) vextendsingle-vextendsingle to the powerk of parenrightbiggt k bar[k,k+1) a bar infinity rightk + 1 arrowi 0[k,k period+1) 0 ∞ ∞ k \noindent S i m i l a r l y , btc + 1 btc = ( − 1)(C({a })(btc) + (1 − )a . t k t btc \ [(THence{ j } p c − p c ˆ{ T } j ˆ{ )( \{ a } k \} ) ( t ) = 0 . \ ]

dte btc k (Cp − pC){a } k ≤ (| − 1| + | − 1|) k a k ∞ → 0. \noindent This proves 1 . \quad Wek a∞ l s o havet t

\ [ \ begin { a l i g n e d } ( Cp − pC ) ( \{ a { k }\} ) ( t ) = \ f r a c { 1 }{ t }\ int ˆ{ t } { 0 } \sum ˆ{ k = 0 } {\ infty } a { k }\ chi { [ k , k + 1 ) } ( s ) ds − \sum ˆ{ k = 0 } {\ infty }\ f r a c { 1 }{ k + 1 }\sum ˆ{ i = 0 } { k } a { i }\ chi { [ k , k + 1 ) } ( t ) \\ = ( \ f r a c {\ l f l o o r t \ rfloor + 1 }{ t } − 1 ) ( C ( \{ a { k }\} )( \ l f l o o r t \ rfloor ) + ( 1 − \ f r a c {\ l f l o o r t \ rfloor }{ t } ) a {\ l f l o o r t \ rfloor } . \end{ a l i g n e d }\ ]

\noindent Hence

\ [ \ parallel ( Cp − pC ) \{ a { k }\}\ parallel {\ infty }\ leq ( \arrowvert \ f r a c {\ l c e i l t \ r c e i l }{ t } − 1 \arrowvert + \arrowvert \ f r a c {\ l f l o o r t \ rfloor }{ t } − 1 \arrowvert ˆ{ ) } \ parallel a \ parallel \ infty \rightarrow 0 . \ ] Measure Theory in Noncommutative Spaces .... 3 1 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 3 1 3 1 Similarly comma \ [ Line\ r u 1l e open{3em parenthesis}{0.4 pt }\ Cp] sub c minus p c to the power of C closing parenthesis open parenthesis open brace a sub k closing brace closing parenthesis open parenthesis t closing parenthesis = 1 divided by t integral sub 0 to the power of t sum from k = 0 to infinity open parenthesis a sub kSimilarly plus open parenthesis , s minus k closing parenthesis open parenthesis a sub k plus 1 minus a sub k closing parenthesis closing parenthesis chi\noindent sub open squareS i m i l abracket r l y , k comma k plus 1 closing parenthesis open parenthesis s closing parenthesis ds Line 2 infinity Line 3 minus sum Z t k=0 C open parenthesis open brace a subC k closing brace1 closingX parenthesis open parenthesis n closing parenthesis plus open parenthesis t minus \ [ \ begin { a l i g n e d } ((Cp Cpc − pc{ )(c{a}k})( −t) = p c ˆ(a{k C+ (}s − )(k)(ak+1 − a\{k))χ[k,ka +1){(s)kds}\} ) ( t ) = \ f r a c { 1 }{ t } n closing parenthesis open parenthesis C open parenthesist 0 open brace a sub k closing brace closing parenthesis open parenthesis n plus 1 \ int ˆ{ t } { 0 }\sum ˆ{ k = 0 } {\∞infty } ( a { k } + ( s − k ) ( a { k + closing parenthesis minus C open parenthesis open brace a sub k closing brace closing parenthesis∞ open parenthesis n closing parenthesis closing parenthesis1 } − chia sub{ k open} square)) bracket\ chi n comma{ [ n plus k 1 , closing k parenthesis + 1 open ) } parenthesis( s t closing ) ds parenthesis\\ Line 4 n = 0 Line 5 = 1 \ infty \\ X divided by t sum from k = 0− to floorleftC({ak t} floor)(n) + 1 ( dividedt − n)(C by({a 2k open})(n + parenthesis 1) − C({a ak} sub)(n)) kχ plus[n,n+1) a sub(t) k plus 1 closing parenthesis minus 1 divided − \sum C( \{ a { k }\} ) ( n ) + ( t − n ) ( C ( \{ a { k } by t integral sub t to the power of ceilingleft t ceilingright a sub floorleft t floor plus open parenthesisn = 0 s minus floorleft t floor closing parenthesis open\} parenthesis) ( a n sub + floorleft 1 t floor ) − plus 1C( minus a sub\{ floorlefta { t floork }\} closing parenthesis) ( dsn Line ) 6 minus ) \ Cchi open{ parenthesis[ n open , brace n k=0 +a sub 1 k closing ) } brace( closing t ) parenthesis\\ 1 X open1 parenthesis1 floorleftZ dte t floor closing parenthesis minus open parenthesis t minus floorleft t floor = (a + a ) − a + (s − btc)(a − a )ds closingn parenthesis = 0 \\ open parenthesis Ct open2 parenthesisk k+1 opent bracebtc a sub k closingbt bracec+1 closingbtc parenthesis open parenthesis floorleft t floor btc t plus= 1 closing\ f r a c { parenthesis1 }{ t minus}\sum C openˆ{ parenthesisk = open 0 } brace{\ l fa l sub o o r k closingt brace\ rfloor closing}\ parenthesisf r a c { 1 open}{ parenthesis2 } ( floorleft a { k t floor} + closing a { k +parenthesis 1 } closing) − parenthesis \ f r a c { period1 }{−tC(}\{ak})(intbtc) −ˆ{\(t −l bct ec)( i lC({akt})(bt\cr+ c 1) e i− l C}({{ak}t)(}btc))a. {\ l f l o o r t \ rfloor } + ( s − \ l f l o o r t \ rfloor ) ( a {\ l f l o o r t \ rfloor + 1 } − a {\ l f l o o r t WeWe recall recall that that \ rfloorC open parenthesis} ) ds open\\ brace a sub k closing brace closing parenthesis open parenthesis n plus 1 closing parenthesis minus C open parenthesis − C( \{ a { k }\} )( \ l f l o o r t \ rfloor ) − ( t − \ l f l o o r t \ rfloor open brace a sub k closing brace closing parenthesis open parenthesisn + 1 n closing parenthesisan+1 = parenleftbigg n plus 1 divided by n plus 2 minus )(C( \{ aC({{akk})(}\}n + 1) − C)(({ak})(n) =\ l ( f l o o r− 1)Ct ({ak\})(rfloorn) − + 1 ) − C( \{ a { k } 1 parenrightbigg C open parenthesis open brace a sub k closingn brace+ 2 closing parenthesisn open+ 2 parenthesis n closing parenthesis minus a sub n \} )( \ l f l o o r t \ rfloor )). \end{ a l i g n e d }\ ] plus 1 dividedso that by n plus 2 so that limint n right arrow infinity bar C open parenthesis open brace a sub k closing brace closing parenthesis open parenthesis n plus 1 closing \noindent We recall that lim | C({ak})(n + 1) − C({ak})(n) |= 0. parenthesis minus C open parenthesis openn→∞ brace a sub k closing brace closing parenthesis open parenthesis n closing parenthesis bar = 0 period AlsoAlso \ [C(C parenleftbigg\{ 1 divideda { k by}\} 2 open parenthesis) (open n + brace 1 a sub ) k plus− 1C( plus a sub\{ k closinga brace{ k closing}\} parenthesis) ( parenrightbigg n ) = ( \ f r a c { n + 1 }{ n + 2 } − 1 ) C ( \{ a { k }\} ) ( n ) − \ f r a c { a { n open parenthesis n closing parenthesis1 = 1 divided by 2 parenleftbigg1 an+1 a 0 sub n plus 1 divided by n plus 1 minus a sub 0 divided by n plus 1 + 1 }}{ n + 2 }\ ] C( ({ak+1 + ak}))(n) = ( − ) + C({ak})(n) parenrightbigg plus C open parenthesis2 open brace a sub k2 closingn + 1 bracen + closing 1 parenthesis open parenthesis n closing parenthesis so thatso that limint n right arrow infinity vextendsingle-vextendsingle-vextendsingle-vextendsingle C parenleftbigg 1 divided by 2 open parenthesis open brace\noindent a sub kso plus that 1 plus a sub k closing brace1 closing parenthesis parenrightbigg open parenthesis n closing parenthesis minus C open lim |C( ({ak+1 + ak}))(n) − C({ak})(n)| = 0. parenthesis open brace a sub k closing bracen→∞ closing2 parenthesis open parenthesis n closing parenthesis vextendsingle-vextendsingle-vextendsingle- \ [ \lim { n \rightarrow \ infty }\mid C( \{ a { k }\} ) ( n + 1 ) − C vextendsingleNow = 0 period ( Now\{ a { k }\} ) ( n ) \mid = 0 . \ ] Line 1 bar open parenthesis Cp sub c minus p c to the power of C closing parenthesis open brace a sub k closing brace bar infinity less C dte 1 or equal vextendsingle-vextendsingle-vextendsingle-vextendsinglek (Cpc − pc ){ak} k ∞ ≤ | − 1| k a k ∞ + ceilingleft|C{ (ak+1 t+ ceilingrightak)}(btc) − dividedC({ak})( bybtc t)| minus 1 vextendsingle-vextendsingle- \noindent Also t 2 vextendsingle-vextendsingle bar a bar infinity plus vextendsingle-vextendsingle-vextendsingle-vextendsinglek a k ∞ C braceleftbigg 1 divided by 2 open + | C({a })(btc + 1) − C({a })(btc) | +3 → 0. parenthesis a sub k plus 1 plus a sub k closing parenthesisk bracerightbigg openk parenthesis floorleftt t floor closing parenthesis minus C open parenthesis\ [C( open\ f r brace a c { a1 sub}{ k2 closing} ( brace\{ closinga parenthesis{ k + open 1 parenthesis} + a floorleft{ k t}\} floor closing) parenthesis ) ( vextendsingle-vextendsingle- n ) = \ f r a c { 1 }{ 2 } vextendsingle-vextendsingle( \ f rThese a c { a results{ n demonstrate + Line 1 2}}{ plus 2n bar . C + open 1 parenthesis} − \ f open r a c { bracea a{ sub0 }}{ k closingn brace + closing 1 } parenthesis) + C open ( parenthesis\{ a floorleft{ k t} floor\} plus) 1 closing ( n parenthesis ) \ ] minus C open parenthesis open brace a sub k closing brace closing parenthesis open parenthesis floorleft t floor ∞ closing parenthesis bar plus 3 bar a bar infinity dividedSetE0 = byrE t: rightL∞([0 arrow, ∞)) 0→ period` Consider . These results demonstrate 2 period \noindent so that Z n+1 Z n+j+1 Line 1 Set E to the power of prime0 = rE0 : L to the power of infinity open parenthesis open square bracket 0 comma infinity closing parenthesis (TjE − E Tj)f(n) = f(t + j)dt − f(t)dt = 0. closing parenthesis right arrow Row 1 infinity Row 2 periodn . Line 2 openn parenthesis+j T sub j E to the power of prime minus E to the power of \ [ \lim { n \rightarrow \ infty }\arrowvert C( \ f r a c { 1 }{ 2 } ( \{ a { k + 1 } + prime TWe sub have j closing parenthesis f open parenthesis n closing parenthesis = integral sub n to the power of n plus 1 f open parenthesis t plus j closinga { k parenthesis}\} dt) minus ) integral ( n sub n ) plus− j to theC( power of\{ n plusa j plus{ k 1 f}\} open parenthesis) ( t closing n ) parenthesis\arrowvert dt = 0 period= 0 . \ ] We have i=0 Z n+1 1 Z t 1 X Z i+1 Line 1 bar open parenthesis| (E0C E− toCE the0)f power(n) |= of| prime C minusf(s)dsdt CE− to the power of primef(s)ds closing| parenthesis f open parenthesis n closing t n + 1 parenthesis bar = vextendsingle-vextendsingle-vextendsingle-vextendsingle-vextendsinglen 0 n i integral sub n to the power of n plus 1 1 divided by t \noindent Now integral sub 0 to the power of t f open parenthesis s closingZ n+1 1 parenthesisZ t dsdt minus1 Z 1n+1 divided by n plus 1 sum from i = 0 to n integral sub i = | f(s)dsdt − f(s)ds| to the power of i plus 1 f open parenthesis s closing parenthesist ds vextendsingle-vextendsingle-vextendsingle-vextendsingle-vextendsinglen + 1 Line 2\ [ =\ begin vextendsingle-vextendsingle-vextendsingle-vextendsingle{ a l i g n e d }\ parallel ( Cp { cn } integral − 0 p sub c n ˆto{ theC } power0 ) of n\{ plus 1a 1 divided{ k }\}\ by t integralparallel sub 0 to the\ powerinfty \ leq \arrowvert \ f r a c {\ l c e i l t \ r c e i l 1}{Z tt } − 1 Z \narrowvert+1 \ parallel a \ parallel \ infty of t f open parenthesis s closing parenthesis dsdt≤ t minus∈ [ sup 11) divided| f by(s) nds plus− 1 integralf sub(s)ds 0| to the power of n plus 1 f open parenthesis s + \arrowvert C \{\ f r a c { 1 }{ 2 } ( a { k + 1 } + a { k } ) \} ( \ l f l o o r t closing parenthesis ds vextendsingle-vextendsingle-vextendsingle-vextendsinglen,n+ t 0 n Line+ 1 30 less or equal t in open square bracket supremum n comma \ rfloor ) − C( \{ a { k }\} )( \ l f l o o r t \ rfloor ) \arrowvert \\ n plus 1 closing parenthesis vextendsingle-vextendsingle-vextendsingle-vextendsingle n+1 − 1 + 1 1 divided by t integral sub 0 to the power of t f open + \mid C( \{ a { k }\} ≤ )( \ l f l o o r t k f\krfloor ∞ → 0. + 1 ) − C( \{ a { k } parenthesis s closing parenthesis ds minus 1 divided by n plus 1 integraln sub 0 n to the power of n plus 1 f open parenthesis s closing parenthesis ds vextendsingle-vextendsingle-vextendsingle-vextendsingle\} )( \ l f l o o r t \ rfloor ) \ Linemid 4 less+ or equal 3 \ Rowf r a c 1{\ n underbarparallel to thea power\ parallel of plus 1 minus\ 1infty plus 1}{ underbart } Row\rightarrow 2These n n . bar equations f bar0 infinity . demonstrate\end right{ a arrow l i g n e 30d period}\ and] 4. These equations demonstrate 3 and 4 period blacksquare \noindent These results demonstrate 2 .

\ [ \ begin { a l i g n e d }\ l e f t . Set E ˆ{\prime } = rE : L ˆ{\ infty } ( [ 0 , \ infty )) \rightarrow \ e l l \ begin { array }{ c}\ infty \\ . \end{ array } Consider \ right . \\ (T { j } E ˆ{\prime } − E ˆ{\prime } T { j } ) f ( n ) = \ int ˆ{ n + 1 } { n } f ( t + j ) dt − \ int ˆ{ n + j + 1 } { n + j } f ( t ) dt = 0 . \end{ a l i g n e d }\ ]

\noindent We have

\ [ \ begin { a l i g n e d }\mid ( E ˆ{\prime } C − CE ˆ{\prime } ) f ( n ) \mid = \arrowvert \ int ˆ{ n + 1 } { n }\ f r a c { 1 }{ t }\ int ˆ{ t } { 0 } f ( s ) dsdt − \ f r a c { 1 }{ n + 1 }\sum ˆ{ i = 0 } { n }\ int ˆ{ i + 1 } { i } f ( s ) ds \arrowvert \\ = \arrowvert \ int ˆ{ n + 1 } { n }\ f r a c { 1 }{ t }\ int ˆ{ t } { 0 } f ( s ) dsdt − \ f r a c { 1 }{ n + 1 }\ int ˆ{ n + 1 } { 0 } f ( s ) ds \arrowvert \\ \ leq t \ in [ \sup { n , n + } 1 ) \arrowvert \ f r a c { 1 }{ t }\ int ˆ{ t } { 0 } f ( s ) ds − \ f r a c { 1 }{ n + 1 }\ int ˆ{ n + 1 } { 0 } f ( s ) ds \arrowvert \\ \ leq \ l e f t (\ begin { array }{ cc } n {\underline {\}}ˆ{ + 1 } − 1 + 1 {\underline {\}}\\ n & n \end{ array }\ right ) \ parallel f \ parallel \ infty \rightarrow 0 . \end{ a l i g n e d }\ ]

\noindent These equations demonstrate 3 and $ 4 . \ blacksquare $ 32 .... S period Lord and F period Sukochev \noindenthline3232 \ h f i l l S . Lord and F . Sukochev S . Lord and F . Sukochev Lemma A period 2 period .. If g in m sub 1 comma infinity comma then : \ [ Line\ r u 1l e 1{3em period}{0.4 xi open pt }\ parenthesis] 1 minus p sub open parenthesis c closing parenthesis r closing parenthesis L to the power of minus 1 open parenthesis alpha sub g closing parenthesis = 0 for all xi in BL sub open parenthesis c closing parenthesis open square bracket 0 comma infinityLemma closing parenthesis A . 2 . semicolonIf g Line∈ m 21, 2∞ period, then xi : to the power of prime r open parenthesis 1 minus E closing parenthesis L to the power \noindent Lemma A . 2 . \quad I f $ g \ in m { 1 , \ infty } , $ then : of minus 1 open parenthesis alpha sub1. g ξclosing(1 − p parenthesisr)L−1(α ) = = 0 0forallξ for all xi∈ toBL the[0 power, ∞); of prime in BL period Proof period .. The following arguments were(c) first publishedg in a more general(c) form in open square bracket 2 .. 7 comma S 2 period 2 \ [ \ begin { a l i g n e d } 1 . \ xi ( 10 − p−1 { ( c ) 0 } r ) L ˆ{ − 1 } ( \alpha { g } ) closing square bracket period .. We 2. ξ r(1 − E)L (αg) = 0forallξ ∈ BL. =include 0f them o r for a completenessl l \ xi period\ in ..BL Let f{ =( L to c the power ) } of[ minus 0 1 open, \ parenthesisinfty ); alpha sub\\ g closing parenthesis in C sub b 2Proof . \ xi . ˆ{\Theprime following} r arguments ( 1 were− firstE published ) L ˆ{ in − a more1 } general( \alpha form in{ [ 2g } 7) , = 0 f o r a l l open parenthesis open square bracket 0 comma infinity closing parenthesis closing−1 parenthesis and c open parenthesis s closing parenthesis = \ xi ˆ{\§2 .prime 2 ] .}\ Wein includeBL them . \end for{ completenessa l i g n e d }\ ] . Let f = L (αg) ∈ Cb([0, ∞)) and c(s) = log log open(1 parenthesis + es). Set 1 plus e to the power of s closing parenthesis period .. Set k open parenthesis s closing parenthesis = 1 divided by c open parenthesis s closing parenthesis integral sub e to the power of s to the power s+1 of e to the power of s plus 1 g to the power of * open1 parenthesisZ e u closing parenthesis du comma s greater equal 0 period \noindentLet theta beProof a state . \ ofquad C subThe b open following parenthesisk(s) = arguments open squareg∗(u were) bracketdu, first s ≥ 0 comma0. published infinity closing in a parenthesis more general closing form parenthesis in [ such2 \quad that 7 , \S 2 . 2 ] . \quad We c(s) s thetainclude = theta them T sub for j for completeness j in N period .. For . \ examplequad Let commae $ thetaf = = xi L bar ˆ{ C − sub b1 open} parenthesis( \alpha open{ squareg } bracket) \ 0in commaC infinity{ b } ( [ 0 , \ infty ) )$and$c ( s ) =$log$( 1 + eˆ{ s } ) . $ \quad Set closingLet parenthesisθ be a closing state parenthesis of Cb([0, ∞ where)) such that θ = θTj for j ∈ N. For example , θ = ξ | Cb([0, ∞)) 0 0 xi inwhere BL openξ ∈ squareBL[0, ∞ bracket), θ = 0ξ commawhere infinityξ ∈ BL closingc[0, ∞), parenthesisor ξ r for commaξ ∈ BL. thetaThen = xi where xi in BL sub c open square bracket 0 comma infinity\ [ k closing ( sparenthesis ) = comma\ f r a or c { xi1 to}{ thec power ( of prime s r ) for}\ xi toint theˆ power{ e ˆ of{ primes + in BL 1 period}} { ..e Then ˆ{ s }} g ˆ{ ∗ } ( s+1 s+1 s uLine ) 1 theta du open , parenthesis s \geq1 kZ closinge0 parenthesis . \ ] = theta1 Z opene parenthesis 1 divided1 Z bye c open parenthesis s closing parenthesis integral θ(k) = θ( g∗(u)du) = θ( g∗(u)du) − θ( g∗(u)du) sub e to the power of s to the powerc(s) ofes e to the power of s plusc(s) 1 g1 to the power of * openc(s) parenthesis1 u closing parenthesis du closing parenthesis = theta open parenthesis 1 divided by c open parenthesis ss+1 closing parenthesis integral subs+1 1 to the power of e to the power of s plus 1 g to \noindent Let $ \theta $ bea state of1 Z $Ce { b } ( [1 0Z e , \ infty ) )$ suchthat $ \theta the power of * open parenthesis u closing parenthesis= θ( du closingg parenthesis∗(u)du) − minusθ( theta parenleftbiggg∗(u)du) 1 divided by c open parenthesis s closing =parenthesis\theta integralT sub{ 1j to}$ the power f o r of $ e j to the\ in powerc(s)N of1 s g . to $ the\ powerquad ofFor *c( opens example+ 1) parenthesis1 $ , u closing\theta parenthesis= du\ xi parenrightbigg\mid LineC { b } 2( = theta [ open 0 parenthesis , \ infty 1 divided) by ) c $ open where parenthesisc(s + 1) s closing parenthesis( integralc(s + sub 1) 1 to the power of e to the power of s plus 1 g to $ \ xi \ in BL [ 0 , = θ\((1infty− ),)f(s + 1))\theta = lim 1=− \ xi $)θ(f where) = 0. $ \ xi \ in BL { c } [ the power of * open parenthesis u closing parenthesisc du(s) closing parenthesiss→∞ minus thetac(s) open parenthesis 1 divided by c open parenthesis s plus0 1 , closing\ infty parenthesis) integral , $ sub or 1to $ the\ xi powerˆ{\ ofprime e to the} powerr $ of f so rplus $ 1\ gxi to theˆ{\ powerprime of *}\ openin parenthesisBL u . closing $ \quad parenthesisThen du closingNow parenthesis set Line 3 = theta parenleftbigg open parenthesis 1 minus c open parenthesis s plus 1 closing parenthesis divided by c open \ [ \ begin { a l i g n e d }\theta ( k ) = \theta ( \ f r a c { 1 }{ c ( s ) }\ int ˆ{ e ˆ{ s + parenthesis s closing parenthesis closing parenthesisK(s) = f sup open parenthesis| f(t) − f(s) s| plus, s 1≥ closing0. parenthesis parenrightbigg = limint s right arrow infinity to1 the}} { powere ˆ of{ opens }} parenthesisg ˆ{ ∗ 1 } minus( c u open ) parenthesis du ) s plus = 1 closing\theta parenthesis( \ f divided r a c { 1 by}{ c openc parenthesis ( s s ) closing}\ int parenthesisˆ{ e ˆ{ s +closing 1 parenthesis}} { 1 } thetag ˆ open{ ∗ parenthesis } ( u f closing ) parenthesis du ) =− 0 periodt \∈theta[s, s + 1) ( \ f r a c { 1 }{ c ( s ) }\ int ˆ{ e ˆ{ s }} { 1 } g ˆ{ ∗ } ( u ) du ) \\ NowThen set =Line\ 1theta K open parenthesis( \ f r a cs{ closing1 }{ parenthesisc ( = s supremum ) }\ barint f openˆ{ e parenthesis ˆ{ s +t closing 1 }} parenthesis{ 1 } minusg ˆ{ f ∗open } parenthesis( u s ) closing du parenthesis) − \ bartheta comma( s greater\ f r a c equal{ 1 0}{ periodc Line ( 2 s t in open +Z et square 1 ) bracket}\ intZ s commaes ˆ{ e s plus ˆ{ s 1 closing + parenthesis 1 }} { 1 } g ˆ{ ∗ } ( 1 ∗ 1 ∗ uThen ) du ) \\ K(s) = t ∈ sup [s, s + 1)| g (u)du − g (u)du| c(t) 1 c(s) 1 =Line\ 1theta K open parenthesis( ( s 1 closing− parenthesis \ f r a c { c = t in ( supremum s + open 1 square ) }{ bracketc (s comma s s ) plus} 1) closing f parenthesis ( s vextendsingle- + 1 ) ) = \lim { s \rightarrow \ inftyZ}eˆs+1{ ( } 1 − \ fZ r ae cs { c ( s + 1 ) }{ c ( s ) } vextendsingle-vextendsingle-vextendsingle-vextendsingle1 1 divided∗ by c open1 parenthesis∗ t closing parenthesis integral sub 1 to the power of e ) \theta ( f ) = 0 . ≤\end{ a l i g ng e d(u}\)du] − g (u)du to the power of t g to the power of * open parenthesisc(s) 1 u closing parenthesisc(s + 1) du1 minus 1 divided by c open parenthesis s closing parenthesis s+1 s integral sub 1 to the power of e to the power1 Z ofe s g to the power ofc *(s open) parenthesis1 Z e u closing parenthesis du vextendsingle-vextendsingle- vextendsingle-vextendsingle-vextendsingle≤ Line 2 lessg∗( oru) equaldu + (1 1 divided− by) c open parenthesisg∗(u)du s closing parenthesis integral sub 1 to the power c(s) s c(s + 1) c(s) of\noindent e to the powerNow of s e s t plus 1 g to the powere of * open parenthesis u closing parenthesis1 du minus 1 divided by c open parenthesis s plus 1 c(s) closing parenthesis integral sub 1 to the power of e to the power≤ ofk s( gs) to + the|1 − power of| *f( opens). parenthesis u closing parenthesis du Line 3 less or\ [ \ equalbegin 1 divided{ a l i g n eby d } c openK parenthesis ( s ) s closing = parenthesis\sup \ integralmid subf e ( toc the(s t+ power 1) ) of− s to thef power ( of s e to ) the power\mid of s, plus s 1 g to\geq 0 . \\ the powerHence of * open parenthesis u closing parenthesis du plus parenleftbigg 1 minus c open parenthesis s closing parenthesis divided by c open parenthesist \ in s plus[ 1 closing sparenthesis , s + parenrightbigg 1 ) \end 1 divided{ a l i g n by e d c}\ open] parenthesis s closing parenthesis integral sub 1 to the power of e to the power of s g to the power of * open parenthesis u closingθ(K) parenthesis = 0. du Line 4 less or equal k open parenthesis s closing parenthesis plus vextendsingle-vextendsingle-vextendsingle-vextendsingle 1 minus c open parenthesis s closing parenthesis divided by c open parenthesis s plus 1\noindent closingWe parenthesis haveThen vextendsingle-vextendsingle-vextendsingle-vextendsingle f open parenthesis s closing parenthesis period Hence | θ((1 − pr)(f)) |=| θ(f(t) − f(btc)) |=| θ(f(t − 1) − f(btc) | \ [ \thetabegin open{ a l parenthesisi g n e d } K K closing ( s parenthesis ) = = 0 t period\ in \sup{ [ } s , s + 1 ) \arrowvert \ f r a c { 1 }{ c (We t have ) }\ int ˆ{ e ˆ{ t }}≤ θ{(| 1f(t}− 1)g− ˆf{(b ∗tc) }|) ≤(θ(K u(t − 1)) ) = θ du(K) =− 0. \ f r a c { 1 }{ c ( s ) }\ int ˆ{ e ˆ{ s }} { 1 } g ˆ{ ∗ } ( u ) du \arrowvert \\ LineAlso 1 bar theta open parenthesis open parenthesis 1 minus pr closing parenthesis open parenthesis f closing parenthesis closing parenthesis bar\ =leq bar theta\ f r a c open{ 1 parenthesis}{ c ( f open s parenthesis ) }\ int t closingˆ{ e parenthesis ˆ{ s + minus 1 f}} open{ parenthesis1 } g ˆ{ floorleft ∗ } t( floor u closing ) parenthesis du − closing \ f r a c { 1 }{ c ( s + 1 ) }\ int ˆ{ e ˆ{ s }} { 1 } g ˆ{ ∗ } ( u ) du \\ parenthesis bar = bar| thetaθ((1 − openpcr)( parenthesisf)) |=| θ((t f− open btc)( parenthesisf(t) − f(dt te)) minus + (1 − 1 closing(t − btc parenthesis))(f(t) − f(b minustc))) | f open parenthesis floorleft t floor closing \ leq \ f r a c { 1 }{ c ( s ) }\ int ˆ{ e ˆ{ s + 1 }} { e ˆ{ s }} g ˆ{ ∗ } ( u ) du + parenthesis bar Line 2 less or equal theta open parenthesis≤ θ bar(| f f( opent) − f parenthesis(dte) |) + θ( t| minusf(t − 1) 1 closing− f(btc parenthesis) |) minus f open parenthesis floorleft t( floor 1 closing− parenthesis \ f r a c { c bar closing( s parenthesis ) }{ c less ( or equal s theta + open 1 parenthesis ) } ) K\ f open r a c { parenthesis1 }{ c t minus ( s 1 closing ) }\ parenthesisint ˆ{ closinge ˆ{ s }} { 1 } parenthesisg ˆ{ ∗ } =( theta u open ) parenthesis du \\ K closing parenthesis = 0≤ periodθ(K(t)) + θ(K(t − 1)) = 2θ(K) = 0. \ leq k ( s ) + \arrowvert 1 − \ f r a c { c ( s ) }{ c ( s + 1 ) }\arrowvert AlsoThis demonstrates 1 . fLine ( 1 bar s theta ) open . \end parenthesis{ a l i g n open e d }\ parenthesis] 1 minus p c to the power of r closing parenthesis open parenthesis f closing parenthesis closing parenthesis bar = bar theta open parenthesis open parenthesis t minus floorleft t floor closing parenthesis open parenthesis f open parenthesis t closing parenthesis minus f open parenthesis ceilingleft t ceilingright closing parenthesis closing parenthesis plus open parenthesis 1 minus\noindent open parenthesisHence t minus floorleft t floor closing parenthesis closing parenthesis open parenthesis f open parenthesis t closing parenthesis minus f open parenthesis floorleft t floor closing parenthesis closing parenthesis closing parenthesis bar Line 2 less or equal theta open parenthesis bar\ [ f\theta open parenthesis( K t closing ) =parenthesis 0 . minus\ ] f open parenthesis ceilingleft t ceilingright closing parenthesis bar closing parenthesis plus theta open parenthesis bar f open parenthesis t minus 1 closing parenthesis minus f open parenthesis floorleft t floor closing parenthesis bar closing parenthesis Line 3 less or equal theta open parenthesis K open parenthesis t closing parenthesis closing parenthesis plus theta open parenthesis\noindent KWe open have parenthesis t minus 1 closing parenthesis closing parenthesis = 2 theta open parenthesis K closing parenthesis = 0 period This demonstrates 1 period \ [ \ begin { a l i g n e d }\mid \theta ( ( 1 − pr ) ( f ) ) \mid = \mid \theta ( f ( t ) − f ( \ l f l o o r t \ rfloor )) \mid = \mid \theta ( f ( t − 1 ) − f ( \ l f l o o r t \ rfloor ) \mid \\ \ leq \theta ( \mid f ( t − 1 ) − f ( \ l f l o o r t \ rfloor ) \mid ) \ leq \theta ( K ( t − 1 ) ) = \theta ( K ) = 0 . \end{ a l i g n e d }\ ]

\noindent Also

\ [ \ begin { a l i g n e d }\mid \theta ( ( 1 − p c ˆ{ r } ) ( f ) ) \mid = \mid \theta ( ( t − \ l f l o o r t \ rfloor ) ( f ( t ) − f ( \ l c e i l t \ r c e i l )) + ( 1 − ( t − \ l f l o o r t \ rfloor ) ) ( f ( t ) − f ( \ l f l o o r t \ rfloor ))) \mid \\ \ leq \theta ( \mid f ( t ) − f ( \ l c e i l t \ r c e i l ) \mid ) + \theta ( \mid f ( t − 1 ) − f ( \ l f l o o r t \ rfloor ) \mid ) \\ \ leq \theta ( K ( t ) ) + \theta ( K ( t − 1 ) ) = 2 \theta ( K ) = 0 . \end{ a l i g n e d }\ ]

\noindent This demonstrates 1 . Measure Theory in Noncommutative Spaces .... 33 \noindenthlineMeasureMeasure Theory Theory in Noncommutative in Noncommutative Spaces Spaces \ h f i l l 33 33 Similarly \ [ bar\ r u open l e {3em parenthesis}{0.4 pt 1}\ minus] E closing parenthesis open parenthesis f closing parenthesis open parenthesis s closing parenthesis bar = vextendsingle-vextendsingle-vextendsingle-vextendsingleSimilarly f open parenthesis s closing parenthesis minus integral sub s to the power of s plus 1 f open parenthesis t closing parenthesis dt vextendsingle-vextendsingle-vextendsingle-vextendsingle less or equal t in supremum open square bracket\ centerline s comma{ S s i mplus i l a 1 r closingl y } parenthesis barZ fs+1 open parenthesis t closing parenthesis minus f open parenthesis s closing parenthesis bar = K open parenthesis s closing| (1 − E parenthesis)(f)(s) |= |f period(s) − f(t)dt| ≤ t ∈ sup [s, s + 1) | f(t) − f(s) |= K(s). \ [ Hence\mid ( 1 − E ) ( fs ) ( s ) \mid = \arrowvert f ( s ) − \ int ˆ{ s +bar 1Hence theta} { opens } parenthesisf ( open t parenthesis ) dt 1\ minusarrowvert E closing parenthesis\ leq t open\ parenthesisin \sup{ f closing[ } parenthesiss , s closing + parenthesis 1 ) bar\mid less orf equal ( theta t open ) parenthesis− f bar ( open s parenthesis ) \mid 1 minus= E closing K ( parenthesis s ) open . parenthesis\ ] f closing parenthesis bar closing parenthesis less or equal theta open parenthesis K| closingθ((1 − E parenthesis)(f)) |≤ θ( =| (1 0− periodE)(f) |) ≤ θ(K) = 0. ThisThis demonstrates demonstrates 2 period2. blacksquare Lemma A . 3 . For every ξ ∈ BL(c)[0, ∞)( resp .CBL(c)[0, ∞),B(C)(c)[0, ∞)) \noindentLemmathere A existsHence period 3ξ period0 ∈ .. For every xi in BL sub open parenthesis c closing parenthesis open square bracket 0 comma infinity closing parenthesis open parenthesis resp period CBL sub open parenthesis c closing parenthesis open square bracket 0 comma infinity closing parenthesis \ [ \mid \theta ( ( 1 − E ) ( f ) ) \mid \ leq \theta ( \mid ( 1 − comma B open(resp.CBL parenthesis, (C C))(and closing parenthesisξ0∈ sub open parenthesis(resp. cCBL, closingB(C)) parenthesis openξ square∈ bracket 0 comma infinity closing 0 E )BL( (.CBL f )[0,B∞\mid )(c))for[0\, everyleq \thetaBLthat( K−ξ r )−1 =there 0=0∀ .exists\ ] g∈m ,∞.BL(c)[0, ∞) parenthesisresp closing parenthesis(c) ), B ..(C there exists xi to∞ the)))such power of primeξ in )L (αg )(t) 1 0 0 ∞ BLProof open parenthesis . Let sub respξ ∈ subBL period(c)[0, ∞ to). theSet powerξ of= openξp(c). parenthesisIt i s easily resp sub verified CBL to theξ i power s a state of period on sub` open parenthesis c closing parenthesiswhich to i the s translationpower of CBL invariant sub open bysquare Lemma bracket 0 A comma . 1 . 1 to . the Moreoverpower of comma , B infinity sub closing parenthesis comma to the power\noindent of openThis parenthesis demonstrates C sub B open $ parenthesis 2 . C\ blacksquare to the power of closing$ parenthesis closing parenthesis sub closing parenthesis sub open Lemma A . 3 . \quad For every $ 0 \ xi−1 \ in BL { (−1 c ) } [ 0 , \ infty ) ( $ resp $ . parenthesis c closing parenthesis to the power(ξ − ξ ofr) openL (α parenthesisg) = ξ(1 − andp(c)r for)L open(αg square) = 0 bracket 0 comma every infinity closing parenthesis closing CBLparenthesis{ ( closing c parenthesis ) } [ such 0 to , the power\ infty of xi to),B(C) the power of prime in BL that{ open( parenthesis c ) } to the[ power 0 of , open\ infty parenthesis) sub due t o Lemma A . 2 . Now let ξ0 ∈ BL. Set ξ = ξ0rE. It i s easily verified ξ i s a state on xi) to$ the\quad powerthere of resp exists sub minus $ xi\ toxi theˆ{\ powerprime of prime}\ rin to the$ power of period sub closing parenthesis L to the power of minus 1 open L∞([0, ∞)) ( resp .C ([0, ∞))) which is translation invariant for a ∈ by Lemma A . 1 . 3 . parenthesis to the power of CBLb comma sub alpha sub g closing parenthesis open parenthesisN t closing parenthesis to the power of B open Let a = j + k where parenthesis\ [BL{ ( } C{ closingresp parenthesis}ˆ{ ( closing resp } parenthesis{ . }ˆ{ there. } ={ 0CBL forall} existsˆ{ CBL g in} m{ sub( 1 sub c comma ) }ˆ{ infinity, } { to[ the power 0of , } xi subB period{\ infty to }ˆ{ ( Cthe} power{ ), of in BL}ˆ{ sub)) open parenthesis} { B(C c closing} parenthesisˆ{ ( and open} square{ ) bracket{ ( 0 c comma ) }} infinityf o r { closing[ } parenthesis0 , every {\ infty ) ) ) such }ˆ{\ xi ˆ{\prime }\ in } BL{ that } ( ˆ{ ( }ˆ{ resp } {\ xi }ˆ{ . } { − \ xi ˆ{\prime } Proof period .. Let xi in BL sub open parenthesis c closing parenthesisThen open square bracket 0j comma∈ Nand infinityk ∈ (0, 1) closing. parenthesis period .. r }ˆ{ CBL , } { ) L ˆ{ − 1 } ( }ˆ{ B(C)) } {\alpha { g } ) ( t ) } the re { = Set xi to the power of primeZ =n+1+ xi pk sub open parenthesisZ n+1 c closingZ n+1+ parenthesisk period .. It i s easily verified xi to the power of prime i s a state on0 l to\ f the o r powera l l } of infinitye x i s t s which{ g i sf(s)\dsin= m {f(s1)ds}}+ˆ{\ xi }f(s{)ds, \ infty }ˆ{\ in } { . } BL { ( c ) } [ 0translation , \ infty invariant) by\ Lemma] n+k .. A periodn 1+ periodk 1 periodn+1 .. Moreover comma openand parenthesis xi minus xi to the power of prime r closing parenthesis L to the power of minus 1 open parenthesis alpha sub g closing parenthesis = xi open parenthesis 1 minus p sub open parenthesis c closing parenthesis r closing parenthesis L to the power of minus 1 open parenthesis\noindent alphaProof sub .g closing\quad parenthesisLet $ \ =xi 0 \ in BL Z{ n+1( c ) } Z n[+1+k 0 , \ infty ) . $ \quad Set $ \ xi ˆ{\prime } ξ0rE(T f) = ξ0( f(s)ds) + ξ0( f(s)ds) = due\ xi t o Lemmap { A( .. period c 2 ) period} . Now $ let\quad xi to theItk power i s easilyof prime in verified BL period .. $ Set\ xi xi =ˆ{\ xi toprime the power}$ of i prime sastateon rE period It i s $ easily\ e l l ˆ{\ infty }$ which i s n+k n+1 verified xi i s a state on L to theZ powern+1 of infinity openZ n+k parenthesis openZ squaren+1 bracket 0 comma infinity closing parenthesis closing parenthesis translationopen parenthesis invariant resp= periodξ0( by C sub Lemmaf(s b)ds open)\ +quadξ parenthesis0( A .f( 1s open)ds .) 1 square =. ξ\0quad( bracketMoreoverf( 0s) commads) = infinity ,ξ0rE(f) closing. parenthesis closing parenthesis closing parenthesis which is translationn invariant+k for a in Nn by Lemma A .. periodn 1 period 3 period Let a = j plus k where \ [(Equation:Hence\ xi j inξ0rE− N andi s \ translationxi k inˆ open{\prime parenthesis invariant} r 0 comma for ) all 1 La closing > ˆ{0. − parenthesisMoreover1 } ( period , \alpha .. Then integral{ g } sub) n plus = k\ toxi the power( of 1 n plus− 1 plusp { ( kc f open ) } parenthesisr ) s closing L ˆ{ − parenthesis1 } ds( = integral\alpha sub{ ng plus} k) to the = power 0 \ of] n plus 1 f open parenthesis s closing parenthesis ds plus 0 −1 0 −1 integral sub n plus 1 to the power of n plus(ξ − 1ξ plusr)L k f(α openg) = parenthesisξ r(1 − E)L s closing(αg) = parenthesis 0 ds and \noindentdue tdue o Lemma t o Lemma A . A 2 .\quad . 2 . Now l e t $ \ xi ˆ{\prime }\ in BL . $ \quad Set $ \ xi = Line 1 xi to the powerξ = ofγC prime∈ CBL rE open[0, parenthesis∞) T sub k f closing parenthesisγ L∞([0 =, xi∞ to)) the ( power.C of([0 prime, ∞))) parenleftbigg. integral sub n \ xi ˆ{\primeNow} letrE . $ It(c i) s easilyfor any verified singular state $ \ xi on$ i sastateonresp $Lˆb {\ infty } ( [ 0 plus k toThen the power of n plus 1 f open parenthesis s closing parenthesis ds parenrightbigg plus xi to the power of prime parenleftbigg integral sub, n\ plusinfty 1 to the) power ) $ of n plus 1 plus k f open parenthesis s closing parenthesis ds parenrightbigg Line 2 = xi to the power of prime parenleftbigg( resp $ integral . C sub{ nb plus} k( to the [ power 00 of , n plus\ infty 1 f open parenthesis) ) )s closing $ which parenthesis is translation ds parenrightbigg invariant plus xi to for the power $ a of ξ = ξp(c) = γCp(c) = γp(c)C prime\ in parenleftbiggN $ by Lemma integral A sub\quad n to the.1.3.Let power of n plus k f $a open parenthesis = j s + closing k$ parenthesis where ds parenrightbigg = xi to the power of prime by Lemma A . 1 . 2 where γp is a singular state on `∞. Hence ξ0 ∈ CBL. Similarly , if parenleftbigg integral sub n to the power of(c) n plus 1 f open parenthesis s closing parenthesis ds parenrightbigg = xi to the power of prime rE ξ ∈ open\ begin parenthesis{ a l i g n ∗} f closing parenthesis period \ tagHence∗{$ xi j to the\ in powerN of prime and rE ki s translation\ in ( invariant 0 for , all 1 a greater ) 0 . period $} Then .. Moreover\\\ int commaˆ{ n + 1 + k } { n +open k } parenthesisf ( xi s minus ) xi to ds the power = \ ofint primeˆ{ rn closing +B parenthesis(C 1) }[0{, ∞n) L, to + the power k } off minus ( 1 open sparenthesis ) ds alpha+ \ subint g closingˆ{ n + 1 + k } { n + 1 } f ( s ) ds (c) parenthesis = xi to the power of prime r open0 parenthesis 1 minus E closing parenthesis0 L to the power of minus 1 open parenthesis alpha sub ξ C = ξp(c)C = ξCp(c) = ξp(c) = ξ g\end closing{ a l parenthesis i g n ∗} = 0 0 0 0 0 dueby t o Lemma Lemma A A .. period . 1 . 2 2 period again . Hence ξ ∈ B(C). Conversely , if ξ = γ C ∈ CBL with γ a ∞ \noindentNowsingular let xiand = gamma state on C in` CBLthen sub open parenthesis c closing parenthesis open square bracket 0 comma infinity closing parenthesis for any singular state gamma on L to the power of infinity open parenthesis open square bracket 0 comma infinity closing parenthesis closing parenthesis 0 0 0 open\ [ \ begin parenthesis{ a l i g n resp e d }\ periodxi Cˆ{\ subprime b open} parenthesisξrE= ξ rE ( open= γ TCrE square{ =k γ bracket}rECf 0 ) comma = infinity\ xi ˆ closing{\prime parenthesis} ( closing\ int parenthesisˆ{ n + closing 1 } { n + k } f ( s ) ds ) + 0 \ xi ˆ{\prime } ( \ int∞ ˆ{ n + 1 + k } { n + 1 } f parenthesisby Lemma period A . 1 . 4 where γ rE i s a singular state on L ([0, ∞)) ( resp .Cb([0, ∞))). (Then sSimilarly ) ds , if ) \\ =xi to\ thexi powerˆ{\ ofprime prime} = xi( p sub\ int openˆ parenthesis{ n + c closing 1 } { parenthesisn + = k gamma} f Cp (sub open s parenthesis ) ds c closing ) + parenthesis\ xi ˆ{\ = gammaprime } ( \ int ˆ{ n + k } { n } f ( s ) ds ) = \ xi ˆ{\prime } ( \ int ˆ{ n + 1 } { n } p sub open parenthesis c closing parenthesis C 0 fby ( Lemma s .... ) A period ds 1 ) period = 2 where\ xi ˆ gamma{\prime p sub} openrE parenthesisξ (∈ B(C f), c closing ) . parenthesis\end{ a l i g.... n e is d }\ a singular] state on l to the power of 0 0 0 infinity period .... Hence xi to the power of primeξC = inξ rEC CBL= periodξ CrE ....= Similarlyξ rE = ξ comma .... if xi in B open parenthesis C closing parenthesis sub open parenthesis c closing parenthesis open square bracket 0 comma infinity closing parenthesis by Lemma A . 1 . 4 again . Hence ξ ∈ B(C)(c)[0, ∞). comma\noindent xi to theHence power $ of\ primexi ˆ C{\ =prime xi p sub} openrE parenthesis $ i s translationc closing parenthesis invariant C = xi Cp for sub all open parenthesis $ a > c closing0 . parenthesis $ \quad =Moreover xi , p sub open parenthesis c closing parenthesis = xi to the power of prime \ [(by Lemma\ xi A− .. period \ xi 1ˆ period{\prime 2 again} periodr ) .. Hence L ˆ{ xi − to the1 power} ( of prime\alpha in B{ openg } parenthesis) = C\ closingxi ˆ{\ parenthesisprime } periodr .. Conversely( 1 − commaE if xi ) to the L ˆ power{ − of1 prime} =( gamma\alpha to the{ powerg } of) prime = C in 0 CBL\ ] with gamma to the power of prime a singular state on l to the power of infinity then xi = xi to the power of prime rE = gamma to the power of prime CrE = gamma to the power of prime rEC \noindentby Lemmadue A .... t period o Lemma 1 period A \ 4quad where. gamma 2 . to the power of prime rE i s a singular state on L to the power of infinity open parenthesis open square bracket 0 comma infinity closing parenthesis closing parenthesis open parenthesis resp period C sub b open parenthesis open square bracket\ hspace 0∗{\ commaf i linfinity l }Now closing l e t $parenthesis\ xi = closing\gamma parenthesisC closing\ in parenthesisCBL { period( c .... Similarly ) } [ comma 0 if , \ infty ) $ for any singular state $ \xigamma to the$ power on of prime $ L ˆ in{\ B openinfty parenthesis} ( C [ closing 0 parenthesis , \ infty comma xi) C = ) xi to ($ the power resp of prime $. rEC C ={ xib to} the power( [ of prime 0 CrE, =\ infty xi to the power) of) prime ) rE . = $ xi by Lemma .... A .... period 1 period 4 again period .... Hence xi in B open parenthesis C closing parenthesis sub open parenthesis c closing parenthesis\noindent openThen square bracket 0 comma infinity closing parenthesis period blacksquare \ [ \ xi ˆ{\prime } = \ xi p { ( c ) } = \gamma Cp { ( c ) } = \gamma p { ( c ) } C \ ]

\noindent by Lemma \ h f i l l A. 1 . 2where $ \gamma p { ( c ) }$ \ h f i l l is a singular state on $ \ e l l ˆ{\ infty } . $ \ h f i l l Hence $ \ xi ˆ{\prime }\ in CBL . $ \ h f i l l S i m i l a r l y , \ h f i l l i f $ \ xi \ in $

\ begin { a l i g n ∗} B(C) { ( c ) } [ 0 , \ infty ), \\\ xi ˆ{\prime } C = \ xi p { ( c ) } C = \ xi Cp { ( c ) } = \ xi p { ( c ) } = \ xi ˆ{\prime } \end{ a l i g n ∗}

\noindent by Lemma A \quad . 1 . 2 again . \quad Hence $ \ xi ˆ{\prime }\ in B ( C ) . $ \quad Conversely , if $ \ xi ˆ{\prime } = \gamma ˆ{\prime } C \ in CBL $ with $ \gamma ˆ{\prime }$ a s i n g u l a r s t a t e on $ \ e l l ˆ{\ infty }$ then

\ [ \ xi = \ xi ˆ{\prime } rE = \gamma ˆ{\prime } CrE = \gamma ˆ{\prime } rEC \ ]

\noindent by Lemma A \ h f i l l . 1 . 4 where $ \gamma ˆ{\prime } rE $ i s a singular state on $Lˆ{\ infty } ( [ 0 , \ infty ) ) ($ resp $. C { b } ( [ 0 , \ infty ) ) ) . $ \ h f i l l Similarly , if

\ begin { a l i g n ∗} \ xi ˆ{\prime }\ in B(C), \\\ xi C = \ xi ˆ{\prime } rEC = \ xi ˆ{\prime } CrE = \ xi ˆ{\prime } rE = \ xi \end{ a l i g n ∗}

\noindent by Lemma \ h f i l l A \ h f i l l . 1 . 4 again . \ h f i l l Hence $ \ xi \ in B(C) { ( c ) } [ 0 , \ infty ). \ blacksquare $ 34 S . Lord and F . Sukochev

Corollary A . 1 . We have the equalities as in ( 2 . 3 9 ) , ( 2 . 4 0 ) and ( 2 P∞ . 4 1 ) . Proof . We note g = n=1 µn(T )χ[n,n+1) ∈ m1,∞ for all T ∈ M1,∞. The result for ( 2 . 3 9 ) follows from the diagram

VDL[1,∞) VDLc[1,∞) L q L q (A.13)

WBL[0,∞) A=.3 WBL A=.3 WBLc[0,∞)

where L denotes the i somorphism b etween WBL[1,∞) ( resp . WBLc[1,∞)) and VDL[1,∞) ( resp . −1 VDLc[1,∞)) induced by the map L ( A . 8 ) , and A=.3 denotes the statement of Lemma A . 3 . The result for ( 2 . 4 0 ) and ( 2 . 4 1 ) follows from an identical diagram . Acknowledgements The authors thank N . Kalton for many discussions concerning Dixmier traces and symmetric functionals . References [ 1 ] Chamseddine A . H . , Connes A . , The spectral action principle , C o m m . M a t h . P h y s . 186 ( 1 997 ) , 731 – 750 , hep-th/9 66001. [ 2 ] Connes A . , Noncommutative geometry , Academic Press , Inc . , San Diego , CA , 1 994 . [ 3 ] Connes A . , Gravity coupled with matter and the foundation of non - commutative geometry , C o m m . M a th . P h y s . 182 ( 1 996 ) , 1 5 5 – 1 76 , h ep - t h / 9 6 0 3 0 5 3 . [ 4 ] Segal I . E . , A non - commutative extension of abstract integration , A n n . of M at h . ( 2 ) 57 ( 1 953 ) , 40 1 – 457 . [ 5 ] Kunze R . A ., Lp Fourier transforms on locally compact unimodular groups , T ra n s . A m er . M at h . S o c . 89 ( 1 958 ) , 5 1 9 – 540 . [ 6 ] Stinespring W . F . , Integration theorems for gages and duality for unimodular groups , T r a n s . A m e r . M a t h . S o c . 90 ( 1 959 ) , 1 5 – 56 . [ 7 ] Nelson E . , Notes on non - commutative integration , J . Fun ct . A n al . 15 ( 1 974 ) , 1 3 – 1 1 6 . [ 8 ] Fack T . , Kosaki H . , Generalised s− numbers of τ− measurable operators , Pacific J . Math . 123 ( 1 986 ) , 269 – 300 . [ 9 ] Pisier G . , Xu Q . , Non - commutative Lp− spaces , in H a n db oo k of t h e g eo met r y of B an a ch s pa c e s , Vol . 2 , North - Holland , Amsterdam , 2003 , 1 459 – 1 5 1 7 . [ 1 0 ] Pederson G . K ., C∗− algebras and their automorphism groups , London Mathematical Society Monographs , Vol . 1 4 , Academic Press , London – New York , 1 979 . [ 1 1 ] S imon B . , Trace ideals and their applications , 2 nd ed . , Mathematical Surveys and Monographs , Vol . 1 20 , American Mathematical Society , Providence , RI , 2005 . [ 1 2 ] von Neumann J . , Some matrix inequalities and metrization of metric - space , Rev . Tomsk . Univ . 1 ( 1 937 ) , 2 86 – 300 . [ 1 3 ] Bratteli O . , Robinson D . W . , Operator algebras and quantum statistical mechanics .1.C∗− and W ∗− algebras , symmetry groups , decomposition of states , 2 nd ed . , Texts and Monographs in Physics , Springer - Verlag , New York , 1 987 . [ 14 ] Cipriani F . , Guido D . , Scarlatti S . , A remark on trace properties of K− cycles , J . Operator . Theory 35 ( 1 996 ) , 1 79 – 1 89 , f u n ct - a n / 9 5 0 6 0 3 . [ 1 5 ] Carey A . L . , Rennie A . , Sedaev A . , Sukochev F . , The Dixmier trace and asymptotics of zeta functions , J . F nu c t . A n a l . 249 ( 2 7 ) , 2 53 – 283 , mat h . O A / 6 1 1 62 9 . 34 .... S period Lord and F period Sukochev \noindent 34 \ h f i l l S . Lord and F . Sukochev hline [ 1 6 ] Benameuar M . - T . , Fack T . , Type II non - commutative geometry . I . Dixmier trace in von CorollaryNeumann A period algebras 1 , period .. We have the equalities as in open parenthesis 2 period 3 .. 9 closing parenthesis comma .. open parenthesis \ [ \ r u l e {3em}{0.4 pt }\ ] 2 period 4 0 closing parenthesis .. andA open d v .parenthesis M a t h2 . period199 4( 20061 closing ) , 29 parenthesis – 87 . period Proof period[ 1 7 ] .. Connes We note A .g , = The sum action sub functional n = 1 to in the noncommutative power of infinity geometry mu n , Co open m parenthesis m . M T closinga t h . P parenthesis h ys . chi sub open square bracket1 n 1 comma 7 ( 1 988 n plus ) , 673 1 closing – 683 . parenthesis [ 1 8 ] Dunford in m sub N 1 . comma , Schwartz infinity J . T .. . , for Linear all T operators in M sub . Part 1 comma I . General infinity theory period , .. The result for .. open \noindent Corollary A . 1 . \quad We have the equalities as in ( 2 . 3 \quad 9 ) , \quad ( 2 . 4 0 ) \quad and ( 2 . 4 1 ) . parenthesisJohn 2 Wiley period& 3Sons .. 9 , closing Inc . , New parenthesis York , Proof . \quad We note $ g = \sum ˆ{\ infty } { n = 1 }\mu n ( T ) \ chi { [ n follows from the diagram 1 988 . ,V n sub DL + open 1 square ) }\ bracketin 1 commam { infinity1 , closing\ infty parenthesis}$ \ Vquad sub DLf o sub r a c l open l $ square T \ bracketin M 1 comma{ 1 infinity , closing\ infty parenthesis} . $ Equation:\quad The open result parenthesis for A\quad period( 13 2 closing . 3 \ parenthesisquad 9 ) .. L shortparallel L shortparallel W sub BL open square bracket 0 comma infinity closing parenthesis A = period 3 W sub BL A = period 3 W sub BL sub c open square bracket 0 comma infinity closing parenthesis \noindentwhere .... Lfollows .... denotes from .... the the .... diagram i somorphism b etween .... W sub BL open square bracket 1 comma infinity closing parenthesis open parenthesis resp period W sub BL sub c open square bracket 1 comma infinity closing parenthesis closing parenthesis .... and .... V sub DL open\ begin square{ a l i bracket g n ∗} 1 comma infinity closing parenthesis open parenthesis resp period V V{ subDL DL sub [ c open 1 square , \ bracketinfty 1 comma) } V infinity{ DL closing{ c parenthesis} [closing 1 , parenthesis\ infty induced) }\\ by theL map\ shortparallel L to the power of minus L 1\ shortparallel open parenthesis A\ ....tag period∗{$ ( 8 closing A parenthesis . 13 comma ) $}\\ andW A ={ periodBL 3 [ denotes 0 the , statement\ infty of Lemma) } AA ....{ period= } 3. period 3 .... W The{ BL } A result{ = } for open. parenthesis 3 W { 2BL period{ c 4 0} closing[ parenthesis 0 , \ andinfty open parenthesis) } 2 period 4 1 closing parenthesis follows from an identical diagram\end{ a lperiod i g n ∗} blacksquare Acknowledgements \noindentThe authorswhere thank\ Nh periodf i l l Kalton$ L $ for\ manyh f i l l discussionsdenotes concerning\ h f i l l Dixmierthe \ h tracesf i l l i and somorphism symmetric b etween \ h f i l l $ W { BL [functionals 1 , period\ infty ) } ( $ resp $ . W { BL { c } [ 1 , \ infty ) } ) $ \ h f i l l and \ h f i l l $ VReferences{ DL [ 1 , \ infty ) } ( $ resp . open square bracket 1 closing square bracket .. Chamseddine A period H period comma Connes A period comma The spectral action principle\noindent comma$ .. V C{ o mDL .. m{ periodc } ..[ M .. 1 a t h , period\ infty .. P h y s) period} ) 186 $ open induced parenthesis by the 1 997 map closing $ parenthesis L ˆ{ − comma1 } 731( $ endash A \ h f i l l . 8 ) , and 750$ A comma{ = } . 3 $ denotes the statement of LemmaA \ h f i l l . 3 . \ h f i l l The h e p hyphen t h slash 9 .. 6 6 0 0 1 period \noindentopen squareresult bracket for 2 closing ( 2 square . 4 0 bracket ) and .. ( Connes 2 . 4 A 1 period ) follows comma from Noncommutative an identical geometry diagram comma Academic $ . \ blacksquare Press comma Inc $ period comma San Diego comma CA comma 1 994 period \noindentopen squareAcknowledgements bracket 3 closing square bracket .. Connes A period comma Gravity coupled with matter and the foundation of non hyphen commutative geometry comma .. C o .. m m .. period .. M a th period \noindentP .. h y sThe period authors 182 open thank parenthesis N . 1 Kalton 996 closing for parenthesis many discussions comma 1 5 5 endash concerning 1 76 comma Dixmier .. h ep traces hyphen andt h slash symmetric 9 .. 6 0 3 0 5 3 period \noindentopen squarefunctionals bracket 4 closing . square bracket .. Segal I period E period comma A non hyphen commutative extension of abstract integration comma A n n .. period of .. M at h period open parenthesis 2 closing parenthesis 57 open parenthesis 1 953 closing parenthesis comma 40 1 endash\noindent 457 periodReferences open square bracket 5 closing square bracket .. Kunze R period A period comma L sub p Fourier transforms on locally compact unimodular groups\ hspace comma∗{\ f .. i l T l } ra[ n 1 s ]period\quad .. AChamseddine .. m er period M A .. . at H .. . h , period Connes S o c A period . , The89 spectral action principle , \quad C o m \quad m . \quad M \quad a t h . \quad Phy s . 186 ( 1997 ) , 731 −− 750 , open parenthesis 1 958 closing parenthesis comma 5 1 9 endash 540 period \ centerlineopen square{h bracket e p − 6 closingt h / square 9 \quad bracket6 6 .. 0 Stinespring 0 1 . } W period F period comma Integration theorems for gages and duality for unimodular groups comma .. T r a n s period .. A .. m e r period .. M a t h period \ centerlineS o c period{ 90[ open2 ] \ parenthesisquad Connes 1 959 A closing . , parenthesis Noncommutative comma 1 geometry 5 endash 56 , period Academic Press , Inc . , San Diego , CA , 1 994 . } open square bracket 7 closing square bracket .. Nelson E period comma Notes on non hyphen commutative integration comma J period Fun ..\ hspace ct period∗{\ Af n i al l l period} [ 3 15 ] \ openquad parenthesisConnes A 1 974 ., closing Gravity parenthesis coupled comma with 1 3 matterendash 1 and1 6 period the foundation of non − commutative geometry , \quad C o \quad m m \quad . \quad M a th . open square bracket 8 closing square bracket .. Fack T period comma Kosaki H period comma Generalised s hyphen numbers of tau hyphen measurable\ centerline operators{P \quad commahys Pacific . J 182(1996) period Math period ,155 123 open−− parenthesis1 76 , 1\ 986quad closingh ep parenthesis− t h / comma 9 \quad 269 endash6 0 3 300 0 5period 3 . } open square bracket 9 closing square bracket .. Pisier G period comma Xu Q period comma Non hyphen commutative L to the power of p hyphen\ centerline spaces comma{ [ 4 ] in\ ..quad H a nSegal db oo k I of . t E h .. . e ,Anon .. g eo .. met− commutative r y .. of B an a extension ch .. s pa c e of s comma abstract Vol period integration 2 comma , A n n \quad . o f \quad Math. (2)57(1953) ,401 −− 457 . } North hyphen Holland comma Amsterdam comma 2003 comma 1 459 endash 1 5 1 7 period \ hspaceopen square∗{\ f ibracket l l } [ 5 1 ] 0\ closingquad squareKunzeR.A bracket .. $. Pederson , G periodL { p K} period$ Fourier comma C transforms to the power on of * locally hyphen algebras compact and unimodular their groups , \quad T ra n s . \quad A \quad m er . M \quad at \quad h . S o c . 89 automorphism groups comma London Mathematical Society Monographs comma \ centerlineVol period 1{ 4( comma 1 958 Academic ) , 5 1 Press 9 −− comma540 London . } endash New York comma 1 979 period open square bracket 1 1 closing square bracket .. S imon B period comma Trace ideals and their applications comma 2 nd ed period comma Mathematical\ hspace ∗{\ f Surveys i l l } [ and 6 ] Monographs\quad Stinespring comma Vol period W . F 1 20 . comma , Integration theorems for gages and duality for unimodular groups , \quad T r a n s . \quad A \quad m e r . \quad M a t h . American Mathematical Society comma Providence comma RI comma 2005 period \ centerlineopen square{ bracketSoc.90(1959) 1 2 closing square bracket ,15 .. von−− Neumann56 . } J period comma Some matrix inequalities and metrization of metric hyphen space comma Rev period .. Tomsk period .. Univ period 1 open parenthesis 1 937 closing parenthesis comma \ centerline2 86 endash{ 300[ 7 period ] \quad Nelson E . , Notes on non − commutative integration , J . Fun \quad ct .Anal . 15(1974) ,13 −− 1 1 6 . } open square bracket 1 3 closing square bracket .. Bratteli O period comma Robinson D period W period comma Operator algebras and quantum[ 8 ] \quad statisticalFack mechanics T . , period Kosaki 1 period H . C, toGeneralised the power of * $hyphen s − and$ W numbers to the power of of $* hyphen\tau algebras− $ comma measurable operators , Pacific J . Math . 123 ( 1 986 ) , 269 −− 300 . [symmetry 9 ] \quad groupsPisier comma G decomposition . , XuQ . of , states Non − commacommutative 2 nd ed period $ comma L ˆ{ p Texts} −and$ Monographs spaces, in in Physics\quad commaHandbook Springer hyphen of th \quad e \quad g eo \quad met r y \quad o f B an a ch \quad spaces ,Vol.2 , Verlag comma New \ centerlineYork comma{North 1 987 period− Holland , Amsterdam , 2003 , 1 459 −− 1 5 1 7 . } open square bracket 14 closing square bracket .. Cipriani F period comma Guido D period comma Scarlatti S period comma A remark on trace\ hspace properties∗{\ f i of l l K} [ hyphen 1 0 ] cycles\quad commaPedersonG.K J period Operator $. period Theory , Cˆ 35{ open ∗ } parenthesis − $ algebras 1 996 closing and parenthesis their automorphism comma groups , London Mathematical Society Monographs , 1 79 endash 1 89 comma f u n ct hyphen a n slash 9 5 0 6 0 3 period \ centerlineopen square{ bracketVol . 1 1 5 closing 4 , Academic square bracket Press .. Carey , London A period−− L periodNew York comma , Rennie 1 979 A . period} comma Sedaev A period comma Sukochev F period comma The Dixmier trace and asymptotics of zeta functions comma \ hspaceJ period∗{\ Ff nu i l ..l } c[ t 1period 1 ] A\quad .. n a lS period imon 249 B open . , parenthesis Trace ideals 2 7 closing and parenthesis their applications comma 2 53 endash , 2 nd283 edcomma . , .. Mathematical mat h period O Surveys and Monographs , Vol . 1 20 , A slash 6 1 1 62 9 period \ centerlineopen square{ bracketAmerican 1 6 closing Mathematical square bracket Society .. Benameuar , Providence M period hyphen , RI T, period2005 comma. } Fack T period comma Type II non hyphen commutative geometry period I period Dixmier trace in von Neumann algebras comma \ hspaceA d v∗{\ periodf i l .. l M} [ a 1 t h 2 period ] \quad 199 openvon parenthesis Neumann J2006 . closing , Some parenthesis matrix commainequalities 29 endash and 87 period metrization of metric − space , Rev . \quad Tomsk . \quad Univ . 1 ( 1 937 ) , open square bracket 1 7 closing square bracket .. Connes A period comma The action functional in noncommutative geometry comma Co m\ centerline .. m period M{2 .. 86 a t−− h period300 P . h} ys period 1 1 7 open parenthesis 1 988 closing parenthesis comma 673 endash 683 period open square bracket 1 8 closing square bracket .. Dunford N period comma Schwartz J period T period comma Linear operators period Part I\ periodhspace General∗{\ f i l theory l } [ 1 comma 3 ] \ Johnquad WileyBratteli ampersand O . Sons , Robinson comma Inc D period . W comma . , Operator New York commaalgebras and quantum statistical mechanics $ .1 988 1 period . C ˆ{ ∗ } − $ and $ W ˆ{ ∗ } − $ a l g e b r a s , \ hspace ∗{\ f i l l }symmetry groups , decomposition of states , 2 nd ed . , Texts and Monographs in Physics , Springer − Verlag , New

\ centerline {York , 1 987 . }

\ hspace ∗{\ f i l l } [ 14 ] \quad Cipriani F . , Guido D . , Scarlatti S . , A remark on trace properties of $ K − $ cycles , J . Operator . Theory 35 ( 1 996 ) ,

\ centerline {1 79 −− 1 89 , f u n ct − a n / 9 5 0 6 0 3 . }

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\ centerline {J . F nu \quad c t . A \quad nal .249(27) ,253 −− 283 , \quad math .OA/611629 . }

\ hspace ∗{\ f i l l } [ 1 6 ] \quad Benameuar M . − T . , FackT . , Type II non − commutative geometry . I . Dixmier trace in von Neumann algebras ,

\ centerline {A d v . \quad Ma t h . 199 ( 2006 ) , 29 −− 87 . }

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Carey A period L period comma Sukochev F period A period comma Dixmier traces and \ [ \ r u l e {3em}{0.4 pt }\ ] some applicationsGrundlehren in noncommutative der Mathematischen geometry Wissenscha comma ..f R− t usen s ia, Vol n . 274 , Springer - Verlag , Berlin , 1 994 . M a t h period[ 47 ] S Connes .. u r v A e . y , s Moscovici 61 open Hparenthesis . , The local 2006 index closing formula parenthesis in noncommutative comma 1 geometry 39 endash , 1 9G 9 eo comma m m . F a th period OA .. slash 0 6 8 3 7 5u period n c t . A nal . 5 \ centerline { [ 1 9 ] \quad Carey A . L . , Sukochev F . A . , Dixmier traces and some applications in noncommutative geometry , \quad R us s i a n } open square bracket 20 closing square bracket ..( 1 Guido 995 ) ,D 1 period 74 – 243 comma . Isola T period comma Dimensions and singular traces for spectral triples comma with applications to fractals comma J period F u nc t period \noindentA n al periodM a 203 t open h . parenthesis S \quad urveys61( 2003 closing parenthesis 2006 comma ) 362 ,139 endash−− 400199 comma ,math m a t h period .OA O\ ..quad A slash/ 0 0 2 6 0 2 8 1 3 0 87 period 5 . [open 20 ] square\quad bracketGuido 2 1 D closing . , Isolasquare bracket T . , .. Dimensions Pietsch A period and comma singular About traces the Banach for envelope spectral of l triples sub 1 comma , with infinity applications comma to fractals , J . F u nc t . Rev period Mat period Complut period 22 open parenthesis 2 9 closing parenthesis comma 209 endash 2 26 period \noindentopen squareAn bracket al . 22 203 closing ( square2003 )bracket , 362 .. Dixmier−− 400 J ,period m a comma t h . Existence O \quad deA traces / 0 non 2 0 normales 2 1 0 comma 8 . C period R period Acad period[ 2 1Sci ] period\quad ParisPietsch S acute-e A r . period , About A hyphen the B Banach 262 open envelope parenthesis of 1 966 $ closing l { parenthesis1 , \ commainfty A} 1 1, 7 $ endash Rev A . 1 Mat1 8 period . Complut . 22 ( 2 9 ) , 209 −− 2 26 . [open 22 ] square\quad bracketDixmier 23 closing J . square , Existence bracket .. de Lorentz traces G period non normales G period comma , C . A R contribution . Acad . to Sci the theory. Paris of divergent S $ \acute sequences{e} $ commar . A A− cB262 ta period ( M1966 .. ath ) period ,A117 80 open−− parenthesisA 1 1 1 8 948 . closing parenthesis comma 1 67 endash 1 90 period [open 23 ] square\quad bracketLorentz 24 closing G . square G . , bracket A contribution .. Sucheston L to period the comma theory Banach of divergent limits comma sequences A .. m e r period , A M c ..ta a t . h M period\quad M ..ath . 80 ( 1 948 ) , 1 67 −− 1 90 . o[ nth 24 ly ]74\ openquad parenthesisSucheston 1 967 L closing . , Banach parenthesis limits comma , 308 A \ endashquad m 31 e 1 period r . M \quad a t h . M \quad o nth ly 74 ( 1 967 ) , 308 −− 31 1 . [open 25 ] square\quad bracketCarey 25 Aclosing . , square Phillips bracket J .. . Carey , Sukochev A period F comma . , Phillips Spectral J period flow comma and Dixmier Sukochev F traces period comma , \quad SpectralA \quad flow d v . \quad M \quad a t h . \quad 1 73 ( 2003 ) , 68 −− 1 1 3 . and Dixmier traces comma .. A .. d v period .. M .. a t h period .. 1 73 open parenthesis 2003 closing parenthesis comma 68 endash 1 1 3 period\ centerline {m \quad ath .OA/0205076 . } m .. at h period O A slash 0 205 0 7 6 period \ hspaceopen square∗{\ f i bracket l l } [ 26 26 closing ] \quad squareKrein bracket S . .. G Krein . , S Petunin period G period Yu . comma I . , Petunin Semenov Yu E period . M I . period , Interpolation comma Semenov of E period linear M operators , \quad Translations of Mathematical period comma Interpolation of linear operators comma .. Translations of Mathematical \ centerlineMonographs{ commaMonographs Vol period , Vol 54 comma . 54 American , American Mathematical Mathematical Society comma Society Providence , Providence comma R period , R .I period I . ,comma 1 982 1 982 . } period open square bracket 27 closing square bracket .. 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A comma . , Symmetric Semenov E period functional M period and singular traces , Posi − comma Sukochev F period A period comma Symmetric functional and singular traces comma Posi hyphen \noindenttiv i t y 2 opentiv parenthesis i ty2(1998 1 998 closing ) ,47 parenthesis−− 75 comma . 47 endash 75 period [open 29 ] square\quad bracketDodds 29 P closing . G square . , de bracket Pagter .. Dodds B . P , period Sedaev G Aperiod . A comma . , Semenov de Pagter E B . period M . comma , Sukochev Sedaev F A period. A . A , period Singular symmetric functionals , comma Semenov E period M period comma Sukochev F period A period comma Singular symmetric functionals comma \ hspaceZap period∗{\ f Nauchn i l l }Zap period . Nauchn Sem period . Sem S period . S hyphen . − Peterburg period . Otdel Otdel period . Mat Mat . period Inst Inst . Steklov period Steklov . ( POMIperiod open ) 290 parenthesis ( 2002 ) , 42 −− 71 ( English transl . : \quad J. 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