Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras

TWISTED CYCLIC THEOR Y AND AN INDEX THEOR Y F OR THE GA UGE INV ARIANT KMS ST A TE ON CUNTZ ALGEBR AS A.L. Carey ∗ , J. Phillips ♯ , A. Rennie ∗ , † ∗ Mathematica l Sciences Institute, Australian National Univ ersity , Canb erra, A CT, A UST RALIA ♯ Departmen t of Mathemat ics and S tatistics, Univ ersity of Vic toria, Victoria, BC, CANAD A † Departmen t of Mathematics, Univ ersity of Copen hagen, Cop enhagen, Denmark Abstract This pap er presents, b y example, an ind ex theory ap p ropriate to algebras without trace. Whilst we w ork exclusiv ely with the Cuntz alg eb ras the exp osition is designed to ind icate ho w to develo p a general theory . Our main resu lt is an index theorem (formulate d in terms of sp ectral ﬂ ow) using a twisted cyclic co cycle wh er e the twisting comes from the mo dular automorphism group for the canonical gauge action on th e C unt z algebra. W e introdu ce a mo d iﬁed K 1 -group of the Cuntz algebra so as to p air with this t wisted co cycle. As a corollary we obtain a noncomm utativ e geometry interpretatio n for Araki’s not ion of relativ e entrop y in this example. W e also n ote the conn ection of this example to the theory of n oncomm utativ e man if olds . Contents 1. In tro duction 1 2. Some bac kground 3 2.1. Semiﬁnite noncomm u tativ e geometry 3 2.2. The Cuntz algebras an d th e canonical Kasparo v mo du le 4 2.3. The mapping cone algebra and APS b ound ary cond itions 6 3. The mo dular sp ectral triple of the Cuntz algebras 7 4. Mo dular K 1 16 5. An L (1 , ∞ ) lo cal in dex form u la 20 5.1. The sp ectral ﬂo w f orm ula: correction terms 20 5.2. A lo cal index form u la for th e Cu n tz algebras 21 6. Concluding Remarks 25 6.1. Relativ e en tropy 25 6.2. Manifold structures 25 6.3. Outlo ok 26 References 26 1. Introduction In this pap er we initiate an extension of index theory to algebras withou t trace. W e tak e the Cu n tz algebras O n [Cu] as basic examples. In the abs en ce of a non -trivial trace on the C u nt z algebras, ou r approac h is to use a KMS state, [BR2], to d eﬁne an in dex pairing using sp ectral ﬂ o w. The state we use is th e uniqu e KMS state f or the canonical T 1 gauge action on O n . As O n is a graph algebra, we can imp ort many of the tec h niques of [PR] wh ere the semiﬁnite v ers ion of the lo cal index f ormula was used to calculate sp ectral ﬂow inv ariant s of a class of Cuntz -K r ieger algebras. T h e Cu n tz algebras give us an excellen t testing ground for the ideas required to deal with index theory in a t yp e I I I setting. 1 2 The appr oac h is motiv ated by [CPRS 2] where a semiﬁnite lo cal index formula in noncomm utativ e geometry is prov ed. This semiﬁnite theory is review ed in Section 2 together with notation for the Cuntz algebras. In [PR] the semiﬁnite theory was applied to certain graph C ∗ -algebras. The new idea explained there w as the constru ction of a Kasparo v A, F - m o dule for the graph algebra A of a lo cally ﬁnite graph with no sources where F = A T 1 is th e ﬁ xed p oint algebra for the natural T 1 gauge action. This construction ap p lies to the Cu n tz algebra b ecause it is a graph algebra of this type. A K -theoretic r eﬁnement for th e index theorem in [PR] w as dev elop ed in [CPR] where the o dd Kasparo v mo dule of [PR ] is ‘dou b led up ’ on a half inﬁn ite cylinder to an ev en Kasparov M ( F , A ) , F -modu le, where M ( F , A ) is the mapping cone algebra for the inclus ion of th e ﬁxed p oin t algebra. Our idea is to mo dify this tracial case so as to extend, as far as is p ossible, these results to the Cuntz algebra. W e easily observe that there is a Kasparo v mo dule for the Cuntz algebra and hence that we ha ve a K 0 ( F )-v alued pairing with M ( F , A ). Ho w ever, in the absence o f a trace w e need a new idea. The primary result of this pap er in tro duces a mo diﬁed sp ectral triple ( referred to as a ‘mo dular sp ectral triple’) with whic h we can compute an index pairing. Our metho d, of emplo ying a KMS functional instead of a trace, leads to v arious sub tleties. Restricting the KMS state to the ﬁ xed p oint algebra F giv es a trace on F , and so a homomorphism on K 0 ( F ). Ho w ever, when w e p ass to the Morita equiv alen t algebra of compact endomorph ism s on our K asp aro v mo dule, we ﬁnd that the fu nctional we are forced to emplo y on this new algebra do es not r esp ect all Mur ra y-vo n Neumann equ iv alences. It is this fact that leads to the consideration of ﬁner in v arian ts than those obtained from ordinary K -theory in the KMS or ‘t wisted setting’. W e sho w that mo dular sp ectral t riples lead t o ‘twisted residue co cycles’ using a v ariation on the semiﬁnite residue co cycle of [CPRS2]. I t is well known that suc h t wisted co cycles cannot pair with ordinary K 1 , r ather we in tro duce, in Section 4, a sub stitute which we term ‘m o dular K 1 ’. It is a s emigroup and, as is explained in our main theorem (Theorem 5.5), there is a general sp ectral ﬂo w form u la whic h deﬁnes the p airing of m o dular K 1 with our ‘t w isted r esid ue co cycle’. Th er e is an analogy with the lo cal ind ex formula of noncommutat ive geometry in the L 1 , ∞ -summable case, ho wev er, there are imp ortan t d iﬀeren ces: the u sual residue co cycle is replaced by a t wisted residue co cycle and the Dixmier trace arising in the s tand ard situation is replaced by a K MS-Dixmier functional. The common ground with [CPRS2] stems from the use of the general sp ectral ﬂo w form u la of [CP2] to d eriv e the t wisted resid u e co cycle and this h as th e corollary that w e ha ve a homotop y inv arian t. F or th e Cuntz algebras the m ain result is T heorem 5.6 and its Corollary where we compu te, for particular m o dular unitaries in matrix algebras o ve r the Cuntz algebras, the p recise numerical v alues arising from the general f ormalism. W e use [CPR] to see that these numerical v alues pro v id e strong evidence that the mapp ing cone KK-theory of Section 2 is pla ying a (y et to b e fully und ersto o d) role. In the ﬁ n al Section we note that there is a p hysical interpretation of the sp ectral ﬂow inv arian t w e are calculating in terms of Araki’s notion of relativ e entrop y of t wo KMS states. W e also show that our mo du lar sp ectral triples for the Cu n tz algebras satisfy t wisted v ers ions of Con n es’ axioms for noncomm utative manifolds. W e plan to return to th is matter and to the appropriate cohomolog ical setting for our index theorem elsewhere. Already , in work in p rogress [CR T], w e ha ve unco vered furth er examples w hic h indicate there is a complex an d interesting theory to b e u ndersto o d . The organisation is sum marised in the C on tents list. Section 2 is review material w hic h p laces this article in con text. T he Cu n tz algebra example b egins on S ection 3 and the main n ew material is in Sections 4 and 5. Ac kno wle dgements W e would lik e to thank Ian Putnam, Nigel Higson, Ryszard Nest, Sergey Nesh vey ev and K ester T ong for advice and comments. The ﬁrst named author wa s supp orted by 3 the Au stralian Research Council, th e Cla y Mathematic s Institute, and the Erwin S c hro dinger In sti- tute (where some of th is pap er was written). Th e s econd named author ac knowledges the su pp ort of NSER C (Canada) while the third named author thanks Statens Naturvid ensk ab elige F orsknin gsr ˚ ad, Denmark. All authors are grateful for the su pp ort of the Banﬀ Inte r national Researc h Station where some of this researc h was und ertak en. 2. Some back ground 2.1. Semiﬁnite noncomm utative geometry. W e b egin with some semiﬁn ite v ersions of standard deﬁnitions and r esults f ollo wing [CPRS 2]. Let φ b e a ﬁxed faithful, normal, semiﬁnite trace on a v on Neumann algebra N . Let K N b e the φ -compact op erators in N (that is the norm closed ideal generated b y the pro j ections E ∈ N with φ ( E ) < ∞ ). Deﬁnition 2.1. A semiﬁnite sp e ctra l t riple ( A , H , D ) is giv e n by a Hilb ert sp ac e H , a ∗ -algebr a A ⊂ N wher e N is a semiﬁnite von Neumann algebr a acting on H , and a densely deﬁne d unb ounde d self-adjoint op er ator D aﬃliate d to N su c h that [ D , a ] is densely deﬁne d and e xtends to a b ounde d op er ator in N for al l a ∈ A and ( λ − D ) − 1 ∈ K N for al l λ 6∈ R . The triple is said to b e even if ther e is Γ ∈ N such that Γ ∗ = Γ , Γ 2 = 1 , a Γ = Γ a for al l a ∈ A and D Γ + Γ D = 0 . Otherwise it is o dd . Note that if T ∈ N and [ D , T ] is b oun ded, then [ D , T ] ∈ N . W e recall from [FK] that if S ∈ N , th e t-th generalized singular v alue of S f or eac h r eal t > 0 is giv en by µ t ( S ) = in f {|| S E || : E is a pro jection in N with φ (1 − E ) ≤ t } . The ideal L 1 ( N , φ ) consists of those op er ators T ∈ N suc h that k T k 1 := φ ( | T | ) < ∞ wh ere | T | = √ T ∗ T . In the Type I setting this is the usual trace class ideal. W e will denote the norm on L 1 ( N , φ ) b y k · k 1 . An alternativ e deﬁnition in terms of singular v alues is that T ∈ L 1 ( N , φ ) if k T k 1 := R ∞ 0 µ t ( T ) dt < ∞ . When N 6 = B ( H ), L 1 ( N , φ ) n eed n ot b e complete in this norm but it is complete in the norm || . || 1 + || . || ∞ . (where || . || ∞ is the uniform n orm). W e use the notation L (1 , ∞ ) ( N , φ ) =  T ∈ N : k T k L (1 , ∞ ) := su p t> 0 1 log(1 + t ) Z t 0 µ s ( T ) ds < ∞  . The reader should note that L (1 , ∞ ) ( N , φ ) is often tak en to mean an ideal in the algebra e N of φ - measurable op erators aﬃliated to N . Our notation is h ow ev er consisten t with that of [C] in th e sp ecial case N = B ( H ). With this con ve ntion the ideal of φ -compact op erators, K ( N ), consists of those T ∈ N (as op p osed to e N ) suc h that µ ∞ ( T ) := lim t →∞ µ t ( T ) = 0 . Deﬁnition 2.2. A semiﬁnite sp e ctr al triple ( A , H , D ) r elative to ( N , φ ) with A uni tal is (1 , ∞ ) - summable if ( D − λ ) − 1 ∈ L (1 , ∞ ) ( N , φ ) for al l λ ∈ C \ R . It f ollo ws that if ( A , H , D ) is (1 , ∞ )-summable then it is n -sum mable (with resp ect to the trace φ ) for all n > 1. W e next need to brieﬂy d iscuss Dixmier traces. F or more inform ation on semiﬁnite Dixmier traces, see [CPS2]. F or T ∈ L (1 , ∞ ) ( N , φ ), T ≥ 0, the fu nction F T : t → 1 log(1 + t ) Z t 0 µ s ( T ) ds is b ounded. There are certain ω ∈ L ∞ ( R + ∗ ) ∗ , [CPS2, C], wh ich deﬁne (Dixmier) traces on L (1 , ∞ ) ( N , φ ) b y setting φ ω ( T ) = ω ( F T ) , T ≥ 0 4 and extending to all of L (1 , ∞ ) ( N , φ ) by linearit y . F or eac h su c h ω we write φ ω for the asso ciated Dixmier trace. Eac h Dixmier trace φ ω v anish es on the ideal of trace class op erators. Whenev er the function F T has a limit at inﬁn it y , all Dixmier traces r eturn that limit as their v alue. Th is leads to the notion of a measurable op erator [C, LSS], that is, one on which all Dixmier traces tak e the same v alue. W e no w introdu ce (a sp ecial case of ) the analytic sp ectral ﬂo w formula of [CP1, CP2]. This formula starts w ith a s emiﬁ nite sp ectral triple ( A , H , D ) and computes the φ sp ectral ﬂ o w from D to u D u ∗ , where u ∈ A is un itary with [ D , u ] b ound ed, in the case wh ere ( A , H , D ) is n -summable for n > 1 (Theorem 9.3 of [CP2 ]): (1) sf φ ( D , u D u ∗ ) = 1 C n/ 2 Z 1 0 φ ( u [ D , u ∗ ](1 + ( D + tu [ D , u ∗ ]) 2 ) − n/ 2 ) dt, with C n/ 2 = R ∞ −∞ (1 + x 2 ) − n/ 2 dx . Th is real num b er sf φ ( D , u D u ∗ ) is a pairing of th e K -homology class [ D ] of A with the K 1 ( A ) class [ u ] [C P RS2]. There is a geometric w ay to view th is form u la. It is sho wn in [CP2] that the f u nctional X 7→ φ ( X (1 + ( D + Y ) 2 ) − n/ 2 ) determines an exact one-form for X in the tangen t sp ace, N sa , of an aﬃne sp ace D + N sa mo delled on N sa . Thus (1) rep resen ts the int egral of this one-form along the path {D t = (1 − t ) D + tu D u ∗ } p ro vided one app reciates that ˙ D t = u [ D , u ∗ ] is a tangen t vec tor to this path. In [CP R S 2], the local ind ex formula in noncommutati ve geometry of [CM] wa s extended to semiﬁ nite sp ectral triples. In the simplest terms, the lo cal index form u la is a pairing of a ﬁnitely su mmable sp ectral triple ( A , H , D ) with the K -theory of the C ∗ -algebra A . Our approac h in this pap er is insp ired by the follo win g theorem (see also [CPRS2, CM, H]). Theorem 2.3 ([CPS2]) . L et ( A , H , D ) b e an o dd (1 , ∞ ) -summable semiﬁnite sp e ctr al triple, r elative to ( N , φ ) . Then for u ∈ A unitary the p airing of [ u ] ∈ K 1 ( A ) with ( A , H , D ) is given by h [ u ] , ( A , H , D ) i = sf φ ( D , u D u ∗ ) = lim s → 0 + s φ ( u [ D , u ∗ ](1 + D 2 ) − 1 / 2 − s ) . In p articular, the limit on the right exists. 2.2. The Cun tz algebras and the canonical Kasparo v mo dule. F or n ≥ 2, the Cu n tz algebra [Cu] on n generators, O n , is the (universal) C ∗ -algebra generated b y n isometries S i , i = 1 , ..., n , sub ject only to the relation P n i =1 S i S ∗ i = 1 . The pro j ections S i S ∗ i will b e denoted by P i and more generally we will wr ite P µ = S µ S ∗ µ . F or µ ∈ { 1 , 2 , ..., n } k = n k w e w rite S µ = S µ 1 S µ 2 · · · S µ k and S ∗ µ = S ∗ µ k S ∗ µ k − 1 · · · S ∗ µ 1 . Usin g the fact that S ∗ i S j = δ ij , one can s ho w that ev ery w ord in the S i , S ∗ j can b e w r itten in the form S µ S ∗ ν , w here µ ∈ n k and ν ∈ n l are multi- in dices. W e will write | µ | = k and | ν | = l f or the length of such multi-i n dices. As the family of monomials { S µ S ∗ ν } is closed un der m ultiplication and inv olution, we hav e (2) O n = span { S µ S ∗ ν : µ ∈ n k , ν ∈ n m , k , m ≥ 0 } . If z ∈ T 1 , then th e family { z S j } is another C untz -Kr ieger f amily w hic h generates O n , and the un iv ersal prop erty of O n giv es a homomorphism σ z : O n → O n suc h that σ z ( S e ) = z S e . Th e homomorph ism σ z is an in verse for σ z , so σ z ∈ Au t O n , and a routine argumen t sho w s that σ is a strongly cont inuous action of T 1 on O n . It is called the gauge action . Av er aging o v er σ with resp ect to n ormalised Haar measure giv es a p ositiv e, faithful exp ectatio n Φ of O n on to the ﬁxed-p oin t algebra F := O σ n : Φ( a ) := 1 2 π Z T 1 σ z ( a ) dθ for a ∈ O n , z = e iθ . T o simplify n otation, w e let A = O n b e the C untz algebra and F = A σ , the ﬁxed p oint algebra for the T 1 gauge action. The algebras A c , F c are deﬁned as the ﬁ nite linear sp an of th e generators. Righ t 5 m ultiplication mak es A into a righ t F -mo d ule, and similarly A c is a righ t m o dule o v er F c . W e deﬁne an F -v alued inner pro d uct ( ·|· ) R on b oth these mo dules by ( a | b ) R := Φ( a ∗ b ) . Deﬁnition 2.4. L et X b e the right F C ∗ -mo dule obtaine d by c ompleting A (or A c ) in the norm k x k 2 X := k ( x | x ) R k F = k Φ( x ∗ x ) k F . The algebra A acting by left multiplicatio n on X provides a rep r esen tation of A as adjoin table op erators on X . Let X c b e the cop y of A c ⊂ X . The T 1 action on X c is unitary and extend s to X , [PR]. F or all k ∈ Z , the pr o jection on to the k -th sp ectral su bspace of the T 1 action is th e op erator Φ k on X : Φ k ( x ) = 1 2 π Z T 1 z − k σ z ( x ) dθ , z = e iθ , x ∈ X . Observe th at Φ 0 restricts to Φ on A and on generators of O n w e hav e (3) Φ k ( S α S ∗ β ) =  S α S ∗ β | α | − | β | = k 0 | α | − | β | 6 = k . W e qu ote the follo wing result from [PR]. Lemma 2.5. The op er ators Φ k ar e adjointa b le endomorphisms of the F -mo dule X such that Φ ∗ k = Φ k = Φ 2 k and Φ k Φ l = δ k ,l Φ k . If K ⊂ Z then the sum P k ∈ K Φ k c onver ges strictly to a pr oje ction in the endomorp hism algebr a. The sum P k ∈ Z Φ k c onver ges to the identity op er ator on X . F or al l x ∈ X , the su m x = P k ∈ Z Φ k x = P k ∈ Z x k c onver ges in X . The u nb ound ed op erator of the next pr op osition is of course the generator of th e T 1 action on X . W e refer to L an ce’s b o ok, [L, Chapters 9,10], for in formation on un b ound ed op erators on C ∗ -mo dules. Prop osition 2.6. [PR] L et X b e the right C ∗ - F -mo dule of D eﬁnition 2.4. Deﬁne D : X D ⊂ X to b e the line ar sp ac e X D = { x = X k ∈ Z x k ∈ X : k X k ∈ Z k 2 ( x k | x k ) R k < ∞} . F or x ∈ X D deﬁne D ( x ) = P k ∈ Z k x k . Then D : X D → X is a is self-adjoint, r e gular op er ator on X . Remark . On generators in O n (regarded as elemen ts of X c ⊂ X ) w e ha ve D ( S α S ∗ β ) = ( | α | − | β | ) S α S ∗ β . W e will need the follo wing tec hnical r esult fr om [PR] later: Lemma 2.7. F or al l a ∈ A and k ∈ Z , a Φ k ∈ E nd 0 F ( X ) , the c omp act F line ar endomorphisms of the right F mo dule X . If a ∈ A c then a Φ k is ﬁnite r ank. In tro duce the r ank one op erator Θ R x,y b y Θ R x,y z = x ( y | z ) R . Then b y [PR, Lemma 4.7], for k ≥ 0, Φ k = P | µ | = k Θ R S µ ,S µ where for the Cuntz algebras the su m is ﬁnite. F or th e negativ e subspaces the form u la in [PR ] giv es, in the Cuntz algebras Φ − k = 1 n k P | µ | = k Θ R S ∗ µ ,S ∗ µ . Theorem 2.8. [PR] L et X b e the right F mo dule of Deﬁnition 2.4. L et V = D (1 + D 2 ) − 1 / 2 . Then ( X, V ) is an o dd Kasp ar ov mo dule for A - F and so deﬁnes an element of K K 1 ( A, F ) . Giv en the hyp otheses of the Th eorem, we may write D as D = P k ∈ Z k Φ k . Remarks . The constructions in [PR] imply immediately that we obtain a class in K K 1 ( O n , F ). Theorem 2.8 is part of an ind ex theorem p ro ved in [PR]. Th e p airing of ( X , V ) w ith unitaries u in K 1 ( A ) giv es a K 0 ( F ) v alued ind ex, and wr iting P = X [0 , ∞ ) ( D ), it is giv en by (4) h [ u ] , [( X , V )] i = [ker( P uP )] − [cok er ( P uP )] 6 where the square b rac ke ts denote the K 0 class of the r elev ant k ern el p ro jections. Ho wev er, the main result of [PR] (wh ic h f ails for the Cu n tz algebras) r equ ires a faithfu l semiﬁn ite gauge inv ariant lo we r semi-con tin u ous trace φ on A . 2.3. The mapping cone algebra and APS b oundary conditions. In [CPR] w e reﬁn ed [PR] by sho win g that K 0 ( F )-v alued ind ices could also b e obtained fr om an ev en index pairing in K K -theory using APS b oundary conditions, sim ilar to [APS3]. W e brieﬂy review this result, as it p ro vides an in terpr etation of the mo d ular index p airings in Section 5. W e use the n otation M k ( B ) to d enote the algebra of k × k matrices o v er an algebra B . If F ⊂ A is a sub - C ∗ -algebra of the C ∗ -algebra A , then the mapping cone algebra for th e inclusion is M ( F, A ) = { f : R + = [0 , ∞ ) → A : f is con tin u ous and v anishes at inﬁnit y , f (0) ∈ F } . When F is an id eal in A it is kno w n that K 0 ( M ( F, A ) ) ∼ = K 0 ( A/F ), [Pu]. In general, K 0 ( M ( F, A ) ) is the set of homotopy classes of partial isometries v ∈ M k ( A ) with range and source p ro jections v v ∗ , v ∗ v in M k ( F ), with op eration the direct sum and in verse − [ v ] = [ v ∗ ]. All this is pr o v ed in [Pu]. F ollo wing [C PR] w e now explain our n oncomm utativ e analogue of the A tiya h -P ato di-Singer index theorem [APS1 ]. Note that w hen we are working with matrix algebras o ver A or M ( F, A ) w e inﬂ ate D to D ⊗ I k and so on. Deﬁnition 2.9. L et ( X , D ) b e an unb ounde d Kasp ar ov mo dule and form the algebr aic tensor pr o duct of L 2 ( R + ) and X . We c omplete the line ar sp an of the e lementary tensors in the algebr aic tensor pr o duct (these ar e functions fr om R + to X ) in the norm arising fr om the inner pr o duct h ξ , η i := Z ∞ 0 ( ξ ( t ) | η ( t )) X dt and write the c ompletion as L 2 ( R + ) ⊗ X and denote this sp ac e by E . An extended L 2 -function f : R + → X is a function of the form f = g + x 0 such that g is i n L 2 ( R + ) ⊗ X and x 0 is a c onstant function with D ( x 0 ) = 0 , that is x 0 ∈ ker( D ) = Φ 0 ( X ) . We denote the sp ac e of extende d L 2 -functions by ˆ E and deﬁne the F -value d i nner pr o duct on ˆ E by h g + x | h + y i ˆ E := h g | h i E + h x | y i X . No w, certain K asp aro v A, F -mo du les extend to Kasparo v M ( F , A ) , F -mo d ules: Prop osition 2.10 ([CPR]) . L et ( X , D ) b e an u ng r ade d unb ounde d Kasp ar ov mo dule for C ∗ -algebr as A, F with F ⊂ A a sub algebr a such that AF = A . Supp ose that D also c ommutes with the left action of F ⊂ A , and that D has discr ete sp e ctrum. Then the p air ( ˆ X , ˆ D ) =  E ˆ E  ,  0 − ∂ t + D ∂ t + D 0  with AP S b oundary c onditions is a gr ade d unb ounde d Kasp ar ov mo dule for the mapping c one algebr a M ( F, A ) . By APS b ound ary conditions we mean let P = X R + ( D ) and take th e d omain of ˆ D to initially b e dom ˆ D = { ξ ∈ span of elementa r y tensors in ˆ X : P ξ 1 (0) = 0 , (1 − P ) ξ 2 (0) = 0 , ˆ D ξ ∈ ˆ X } . In [CPR] w e sh o w that APS b oun dary conditions mak e sens e for the self adjoint closure of ˆ D and no tec hnical obstru ctions exist to w orking with th is closure on its natur al d omain. Strictly sp eaking w e should also men tion that th e u nb ound ed Kasparo v mo du le is deﬁn ed for a certain smo oth algebra A ⊂ A , and we will supp ose that this is the case, and th at F ⊂ A . T o explain the app earance in the second comp onent of ˆ X of the r igh t F , C ∗ -mo dule ˆ E , we h a ve to recall that the diﬀerent treatmen t of k er( D ) (whic h has the restriction of the inn er p ro du ct on X ) is to accoun t for ‘extended L 2 solutions’ corresp ondin g to the zero eigen v alue of D , just as in [APS1, pp 58-60] . 7 If v is a partial isometry in M k ( A ) ∼ (the minimal unitization) setting e v ( t ) = 1 − vv ∗ 1+ t 2 − iv t 1+ t 2 iv ∗ t 1+ t 2 v ∗ v 1+ t 2 ! , deﬁnes e v as a pr o jection in M k ( F , A ) (the k × k matrices o ver the mapp ing cone). Wh en v is a unitary , denoted u sa y , then u is trivially a partial isometry with range and source in (the unitization of ) M k ( F ), so we obtain a class in K 0 ( M ( F, A ) ) whic h we denote by [ e u ] −  1 0 0 0  . In the statemen t of the next result (whic h is a sp ecial case of the main theorem of [CPR]) we su ppress th e subscript k . Prop osition 2.11 ([CPR]) . L et ( X , D ) b e an ungr ade d unb ounde d Kasp ar ov mo dule for (pr e-) C ∗ - algebr as A , F with F ⊂ A a sub algebr a such that AF = F . Supp ose that D also c ommutes with the left action of F ⊂ A , and that D has discr ete sp e ctrum. L et ( ˆ X , ˆ D ) b e the u nb ounde d Kasp ar ov M ( F, A ) , F mo dule of Pr op osition 2.10. Then for any u ni tary u ∈ A such that u ∗ [ D , u ] is b ounded and comm utes w ith D we have the fol lowing e qu ality of index p airings with values in K 0 ( F ) : h [ u ∗ ] , [( X, D )] i := Index( P u ∗ P ) = h [ e u ] −  1 0 0 0  , [( ˆ X , ˆ D )] i ∈ K 0 ( F ) . Mor e over, if v ∈ A is a p artial i sometry, with v v ∗ , v ∗ v ∈ F and v ∗ [ D , v ] b ounde d and c ommuting with D we have h [ e v ] −  1 0 0 0  , [( ˆ X , ˆ D )] i = − Index( P v P : v ∗ v P ( X ) → v v ∗ P ( X )) ∈ K 0 ( F ) = Ind ex( P v ∗ P : v v ∗ P ( X ) → v ∗ v P ( X )) ∈ K 0 ( F ) . (5) W e r emark that the hypothesis that D and v ∗ [ D , v ] comm u te can b e considerably relaxed (with considerable eﬀort). W e will see later how this theorem assists us when ¯ A = O n and O n is equipp ed with its natural KMS state. Before turn ing to this w e compute K 0 ( M ( F, A ) ) for th e examples we ha ve in mind . Let A = O n and let A b e an y dense smo oth su balgebra such that th e ﬁxed p oin t algebra for the mo d ular automorphism, F , is conta in ed in A . Using K 1 ( A ) = K 1 ( F ) = 0, the six term sequence in K -theory b ecomes 0 → K 0 ( M ( F, A )) → K 0 ( F ) → K 0 ( A ) → K 1 ( M ( F, A )) → 0 . No w K 0 ( F ) = Z [1 /n ] while K 0 ( A ) = Z n − 1 , [Da v]. A careful analysis of the map K 0 ( F ) → K 0 ( A ) sho ws that it is indu ced b y inclusion, [CPR]. Sin ce K 0 ( A ) = { 0 , I d, 2 I d, ..., ( n − 2) I d } for the Cu n tz algebra, this map is onto . Hence K 1 ( M ( F, A )) = 0 and K 0 ( M ( F, A )) = ( n − 1) Z [1 /n ]. 3. The modu lar spe ctral triple o f th e Cuntz algebras The C untz algebras do not p ossess a faithful gauge inv arian t trace. Th ere is how ev er a uniqu e state whic h is KMS for the gauge action, namely ψ := τ ◦ Φ : O n → C , w here Φ : O n → F is th e exp ectati on and τ : F → C the uniqu e faithfu l normalised trace. As the C unt z algebras satisfy the hypotheses of [PR] (they are graph algebras of a locally ﬁnite graph with no sou r ces), the generator of th e gauge action D acting on the r igh t C ∗ - F -mo dule X giv es us a K asp aro v mo dule ( X, D ). As with tracial graph algebras, we take th is class as our starting p oin t. Ho w ever we immediately encounte r a diﬃculty that there are no un itaries to pair with, sin ce K 1 ( O n ) = 0. Nev ertheless, there are man y partial isometries with range and sour ce in the ﬁxed p oint algebra ( O n is generated by s uc h elemen ts), so the APS pairing of the previous section is a v ailable. W e would like to compute a numerical pairing using a sp ectral triple and w e u s e the K asparo v mo du le for this p u rp ose. 8 Let H = L 2 ( O n ) b e the GNS Hilb ert space giv en by th e faithful state ψ = τ ◦ Φ . That is, the inn er pro du ct on O n is d eﬁned b y h a, b i = ψ ( a ∗ b ) = ( τ ◦ Φ)( a ∗ b ) . Then D extends to a self-adjoin t unb ou n ded op erator on H , [PR], and w e d en ote th is closur e by D from n o w on. The representati on π of O n on H b y left m u ltiplicatio n is b ounded and nondegenerate, and the dense subalgebra sp an { π ( S µ S ∗ ν ) } is in the smo oth domain of the deriv ation δ = ad( |D | ). W e d enote the left action of an elemen t a ∈ O n b y π ( a ) so that π ( a ) b = ab for all b ∈ O n . Th is distinction b et we en elemen ts of O n as ve ctors in L 2 ( O n ) and op erators on L 2 ( O n ) is sometimes crucial. T hus we see th at the cent r al algebraic structures of the gauge s p ectral triple on a tracial graph algebra are m ir rored in this construction. What diﬀers fr om the tracial s itu ation is the analytic information. W e b egin by obtaining some information ab out th e trace on F , the corresp onding state on O n and the asso ciated mo d ular theory . Lemma 3.1. The tr ac e τ : F → C satisﬁes τ ( S µ S ∗ ν ) = δ µ,ν 1 n | µ | . Pr o of. First of all, w e must h a v e | µ | = | ν | in ord er that S µ S ∗ ν ∈ F , and then τ ( S µ S ∗ ν ) = τ ( S µ S ∗ µ S µ S ∗ ν ) = τ ( S µ S ∗ ν S ν S ∗ ν ) = τ ( S µ S ∗ ν S µ S ∗ µ ) = τ ( S ν S ∗ ν S µ S ∗ ν ) = δ µ,ν τ ( S µ S ∗ µ ) = δ µ,ν τ ( S ν S ∗ ν ) . Th u s when ever | µ | = | ν | we h a v e τ ( S µ S ∗ µ ) = τ ( S ν S ∗ ν ). S ince there are exactly n k distinct S µ all with orthogonal ranges s o th at P | µ | = k S µ S ∗ µ = 1, the r esult follo w s.  Let S ﬁrst d enote the op erator a 7→ a ∗ deﬁned on O nc as a subsp ace of L 2 ( O n ). The conjugate-linear adjoin t of S exists, is denoted F and will b e explicitly calc u lated on the subspace O nc in the next lemma. It satisﬁes F ( S µ S ∗ ν ) = n ( | µ |−| ν | ) S ν S ∗ µ . In particular, F is den s ely deﬁn ed so that S is closable. So w e use the same symb ol S to denote the closure and also F w ill denote th e closure of F r estricted to O nc . Then S has a p olar d ecomp osition as S = J ∆ 1 / 2 = ∆ − 1 / 2 J, F = J ∆ − 1 / 2 = ∆ 1 / 2 J, where ∆ = F S , where J is an an tilinear map, J 2 = 1. Th e T omita-T ak esaki m o dular theory , [KR], shows that ∆ − it π ( O n ) ′′ ∆ it = π ( O n ) ′′ , J π ( O n ) ′′ J ∗ = ( π ( O n ) ′′ ) ′ , where π ( O n ) ′′ is the we ak closure of th e left action of O n on L 2 ( O n ). How ev er, all of these op erators can b e exp licitly calculated on the subspace O nc whic h is in fact a T omita algebra. Lemma 3.2. The algebr a O nc = span { S µ S ∗ ν } with the inner pr o duct, h a | b i = ψ ( a ∗ b ) = τ ◦ Φ ( a ∗ b ) arising fr om the state ψ = τ ◦ Φ i s a T omita algebr a (exc ept for the trivial diﬀer enc e that our i nner pr o duct is line ar in the se c ond c o or dinate). Pr o of. Since th e inner pro duct on O nc comes from th e GNS constr u ction given by the faithfu l state ψ = τ ◦ Φ th e left action of O nc on itself is in volutiv e, faithful (and hence isometric), and nondegenerate. This take s care of T ak esaki’s Axioms (I), (I I), (II I) for a T omita algebra [T a]. Next, as m en tioned ab o ve, the op erator S on O nc is just the mapp ing on O n , a 7→ S ( a ) = a ∗ , whic h, on generators, is S ( S µ S ∗ ν ) = S ν S ∗ µ . W e deﬁn e the conjugate linear map F on generators (and extend by linearity) via: F ( S µ S ∗ ν ) = n ( | µ |−| ν | ) S ν S ∗ µ . T o see that F is th e adjoin t of S it suﬃces to c heck the deﬁning equation: hS ( a ) | b i = hF ( b ) | a i on generators a = S α S ∗ β and b = S µ S ∗ ν . Th en, hS ( a ) | b i = τ ◦ Φ( S α S ∗ β S µ S ∗ ν ) while hF ( b ) | a i = τ ◦ Φ( S µ S ∗ ν S α S ∗ β ) n ( | µ |−| ν | ) . 9 No w, if | µ | + | α | − | ν | − | β | 6 = 0 then b oth terms are 0 hen ce equal. While if | µ | + | α | − | ν | − | β | = 0 , then hS ( a ) | b i = τ ( S α S ∗ β S µ S ∗ ν ) while hF ( b ) | a i = τ ( S µ S ∗ ν S α S ∗ β ) n ( | µ |−| ν | ) . In the second case where | β | − | µ | = | α | − | ν | , w e assume th at | β | − | µ | = | α | − | ν | ≥ 0 as the case ≤ 0 is very s im ilar. No w, S α S ∗ β S µ S ∗ ν = 0 u nless β = µλ , wh ence S α S ∗ β S µ S ∗ ν = S α S ∗ λ S ∗ ν and | λ | = | β | − | µ | . Th en since S α S ∗ λ S ∗ ν 6 = 0 we must hav e α = ν λ, and hence w e ha ve S α S ∗ β S µ S ∗ ν = S α S ∗ α where β = µλ and α = ν λ and | λ | = | β | − | µ | . Similary , S µ S ∗ ν S α S ∗ β = 0 unless S µ S ∗ ν S α S ∗ β = S β S ∗ β where α = ν γ , β = µγ and | γ | = | α | − | ν | . W e note that sin ce | λ | = | β | − | µ | = | α | − | ν | = | γ | , we ha ve that the t wo expressions S α S ∗ β S µ S ∗ ν and S µ S ∗ ν S α S ∗ β are b oth n on zero at the same time with the same cond ition: β = µλ and α = ν λ. Finally , τ ( S α S ∗ β S µ S ∗ ν ) = τ ( S α S ∗ α ) = n −| α | while τ ( S µ S ∗ ν S α S ∗ β ) n ( | µ |−| ν | ) = τ ( S β S ∗ β ) n ( | µ |−| ν | ) = n −| β | n ( | µ |−| ν | ) = · · · = n −| α | . Th u s, F is the adjoin t of S and so b oth are closable. This tak es care of T ak esaki’s Axiom (IX). W e immed iately dedu ce that ∆( S µ S ∗ ν ) = F S ( S µ S ∗ ν ) = n ( | ν |−| µ | ) S µ S ∗ ν so that ∆ 1 / 2 ( S µ S ∗ ν ) = n (1 / 2)( | ν |−| µ | ) S µ S ∗ ν and J ( S µ S ∗ ν ) = n (1 / 2)( | µ |−| ν | ) S ν S ∗ µ so that S = J ∆ 1 / 2 , F = ∆ 1 / 2 J, as required. Moreo v er, for all z ∈ C w e hav e: ∆ z ( S µ S ∗ ν ) = n z ( | ν |−| µ | ) S µ S ∗ ν where for w ∈ C we tak e n w := e w log ( n ) . W e remark that eac h S µ S ∗ ν is an eigenv ector of ∆ for the nonzero eigen v alue n ( | ν |−| µ | ) , and so eac h eigen v alue h as in ﬁnite m ultiplicit y . W e quickly review T ake saki’s remaining axioms for a T omita algebra. First, there is the u n-num b ered axiom that eac h ∆ z : O nc → O nc is an algebra homomorphism. C learly , eac h ∆ z is a linear isomor- phism, and it suﬃces to chec k multiplicativi ty on the generators. This is a calculatio n based on the follo wing fact: ( S µ S ∗ ν )( S α S ∗ β ) = 0 u nless either | ν | ≥ | α | and ν = αλ where ( S µ S ∗ ν )( S α S ∗ β ) = S µ S ∗ β λ or | ν | ≤ | α | an d α = ν γ where ( S µ S ∗ ν )( S α S ∗ β ) = S µγ S ∗ β . W e remind the reader that th is axiom says that as op erators on O n : π (∆ z ( a )) = ∆ z π ( a )∆ − z in particular , π (∆ it ( a )) = ∆ it π ( a )∆ − it . Axiom (IV): S (∆ z ( a )) = ∆ − z ( S ( a )) for all a ∈ O nc and all z ∈ C . This is a s tr aigh tforwa r d calculatio n . Axiom (V): h ∆ z ( a ) | b i = h a | ∆ z ( b ) i f or all a, b ∈ O nc , and z ∈ C . Another easy calculation. Axiom (VI): h ∆( S ( a )) |S ( b ) i = h b | a i for all a, b ∈ O nc . This is equiv alent to hF ( a ) |S ( b ) i = h b | a i . Axiom (VI I): The function z 7→ h a | ∆ z ( b ) i is analytic on C for eac h a, b ∈ O nc . Again an easy calculation since our inner p ro du cts are linear in the second v ariable. Finally w e hav e: Axiom (VI I I): F or eac h t ∈ R the sub space (1 + ∆ t )( O nc ) is dense in O nc . In fact, eac h generator S µ S ∗ ν is an eigen v ector of (1 + ∆ t ) with p ositiv e eigen v alue: 1 + n t ( | ν | − | µ | ) , and hence (1 + ∆ t )( O nc ) = O nc .  Lemma 3.3. The gr oup of mo dular automorphisms of the von Neu mann algebr a O ′′ n gener ate d by the left action of O n on L 2 ( O n ) (which is the same as the v on Neumann algebr a gener ate d by the left action of O nc on L 2 ( O nc ) = L 2 ( O n ) ) i s given on the gener ators by (6) σ t ( π ( S µ S ∗ ν )) := ∆ it π ( S µ S ∗ ν )∆ − it = π (∆ it ( S µ S ∗ ν )) = n it ( | ν | − | µ | ) π ( S µ S ∗ ν ) . 10 Pr o of. This is a str aightforw ard calculation obtained b y ev aluating these op erators on a generator S α S ∗ β and using the un-num b er ed Axiom th at ∆ it is an algebra h omomorphism on O nc .  Remarks. A sp ecial case of the KMS condition on the mo dular automorph ism group of the state ψ , [T a], (for t = i ) is the follo w ing: ψ ( xy ) = ψ ( σ i ( y ) x ) f or x, y ∈ π ( O n ) . The pro of is elemen tary: τ ◦ Φ( xy ) = h x ∗ | y i = hS ( x ) | y i = hF ( y ) | x i = τ ◦ Φ( S F ( y ) x ) = τ ◦ Φ(∆ − 1 ( y ) x ) = τ ◦ Φ( σ ( y ) x ) . F rom now on we r efer to this as the K MS condition for th e s tate ψ . Corollary 3.4. With O n acting on H := L 2 ( O n ) we let D b e the gener ator of the natur al u nitary implementation of the g auge action of T 1 on O n . Then we have ∆ = n −D or e it D = ∆ − it/log n . T o con tinue, we r ecall the u nderlying righ t C ∗ - F -mo dule, X , whic h is the completion of O n for the norm k x k 2 X = k Φ( x ∗ x ) k F . Lemma 3.5. Any F - line ar endomorp hism T of the mo dule X which pr eserves the c opy of O n inside X , extends u niquely to a b ounde d op er ator on the Hilb ert sp ac e H = L 2 ( O n ) . Pr o of. F or any x ∈ X we ha ve, by [L, Prop osition 1.2], ( T x | T x ) R ≤ k T k 2 E nd ( x | x ) R in F + . Letting k T k ∞ denote the op erator norm on H w e estimate u sing x ∈ O n : k T k 2 ∞ = su p k x k H ≤ 1 h T x | T x i H = sup k x k H ≤ 1 τ (( T x | T x ) R ) ≤ su p k x k H ≤ 1 k T k 2 E nd τ (( x | x ) R ) = k T k 2 E nd .  In particular, the ﬁn ite rank endomorp hisms of the p re- C ∗ mo dule O nc (acting on the left) satisfy this condition, and w e den ote the algebra of all these endomorphisms b y E n d 00 F ( O nc ). Prop osition 3.6. L et N b e the v on Neu mann algebr a N = ( E nd 00 F ( O nc )) ′′ , wher e we take the c ommu- tant inside B ( H ) . Then N is semiﬁnite, and ther e e xists a faithful, semiﬁnite, normal tr ac e ˜ τ : N → C such that for al l r ank one e ndomorphisms Θ R x,y of O nc , ˜ τ (Θ R x,y ) = ( τ ◦ Φ)( y ∗ x ) , x, y ∈ O nc . In addition, D is aﬃliate d to N and O n , acting on the left on X , is a sub algebr a of N . Pr o of. W e deﬁn e ˜ τ as a supremum of an increasing sequence of v ector states, as in [PR], which ensures that ˜ τ is normal. First for | µ | 6 = 0 we deﬁne for T ∈ N ω µ ( T ) := h S µ , T S µ i + 1 n | µ | h S ∗ µ , T S ∗ µ i . T ogether w ith ω 1 ( T ) := h 1 , T 1 i , this giv es a collection of p ositiv e ve ctor states on N . W e deﬁne ˜ τ ( T ) = ω 1 ( T ) + lim L ր X µ ∈ L ω µ ( T ) , where L ranges o ve r the ﬁnite s ubsets of the ﬁn ite path space E ∗ of the graph und erlying O n . With this deﬁnition, the pro of in [PR, Lemma 5.11] can b e applied almost ve r batim to this case. T he on ly real c hange in the pro of o ccurs on page 121 of [PR ]: the line b efore th e phr ase “the last inequalit y follo wing” shou ld b e replaced by: = k T k X s ( µ )= v, | µ | = k τ ( p r ( µ ) ) = k T k n k τ ( p v ) < ∞ . 11 Rather than rep eat the pro of here, w e simply obser ve for the reader’s b eneﬁt that to c hec k the trace prop erty (on endomorphisms ) only requires that τ is a trace on F , not all of O n . Here is the formal calculatio n for rank one op erators: ˜ τ (Θ R w ,z Θ R x,y ) = ˜ τ (Θ R w ( z | x ) ,y ) = τ ◦ Φ( y ∗ w ( z | x )) = τ (( y | w ( z | x ))) = τ (( y | w )( z | x )) = τ (( z | x )( y | w )) = ˜ τ (Θ R x ( y | w ) ,z ) = ˜ τ (Θ R x,y Θ R w ,z ) . Next we must sho w that D is aﬃliated to N . Ho w ever, w e h a ve already noted that the sp ectral pro jections of D are ﬁ nite s u ms of r ank one endomorp hisms of X c , in the paragraph immediately preceding Th eorem 2.8. This pro ves the claim. Th at A c em b eds in N follo w s from Lemm a 2.7 and the fact that the Φ k sum to the identit y . Since A is the unique C ∗ -completion of A c w e see that π em b eds A in N .  Unfortunately , in con trast to the situation in [PR], th is trace is n ot what we need for d eﬁning summa- bilit y . This can b e seen from the follo wing calculations. F or k ≥ 0 ˜ τ (Φ k ) = ˜ τ ( X | ρ | = k Θ R S ρ ,S ρ ) = τ ( X | ρ | = k ( S ρ | S ρ )) = τ ( X | ρ | = k S ∗ ρ S ρ ) = X | ρ | = k 1 = n k . Similarly , for k < 0 we hav e ˜ τ (Φ k ) = n k . Hence with resp ect to this trace we cannot exp ect D to satisfy an y su mmabilit y criterion. Deﬁnition 3.7. W e deﬁne a new weight on N + : let T ∈ N + then τ ∆ ( T ) := sup N ˜ τ (∆ N T ) wher e ∆ N = ∆( P | k |≤ N Φ k ) . Remarks . Since ∆ N is ˜ τ -trace-class, we see that T 7→ ˜ τ (∆ N T ) is a norm al p ositiv e linear fun ctional on N and hence τ ∆ is a n ormal weigh t on N + whic h is easily seen to b e faithful and semiﬁnite. W e now give another wa y to deﬁne τ ∆ whic h is n ot only conceptually usefu l b ut also make s a num b er of imp ortant prop erties straigh tforward to v erify . Notation . Let M b e the relativ e commutan t in N of the op erator ∆. Equiv alen tly , M is the relativ e comm utant of the set of sp ectral pr o j ections { Φ k | k ∈ Z } Clearly , M = P k ∈ Z Φ k N Φ k . Deﬁnition 3.8. As ˜ τ r estricte d to e ach Φ k N Φ k is a faithful ﬁnite tr ac e with ˜ τ (Φ k ) = n k we deﬁne b τ k on Φ k N Φ k to b e n − k times the r estriction of ˜ τ . Then, b τ := P k b τ k on M = P k ∈ Z Φ k N Φ k is a faithful normal semiﬁnite tr ac e b τ with b τ (Φ k ) = 1 for al l k . W e use b τ to giv e an alternativ e expression for τ ∆ b elo w T his alternativ e might b e a v oidable bu t at the exp ense of a d etailed use of [PT]. Ho w ever, (see the b ottom of page 61 of [PT]), t he semiﬁniteness of τ ∆ restricted to M dep ends on the existence of a n ormal τ ∆ -in v arian t pro jection (suc h as Ψ deﬁned b elo w) f rom N on to M . Lemma 3.9. An element m ∈ N i s in M if and only if it is in the ﬁxe d p oint algebr a of the action, σ τ ∆ t on N deﬁne d for T ∈ N by σ τ ∆ t ( T ) = ∆ it T ∆ − it . Both π ( F ) and the pr oje c tions Φ k b e long to M . The map Ψ : N → M deﬁne d b y Ψ ( T ) = P k Φ k T Φ k is a c onditional exp e ctation onto M and τ ∆ ( T ) = b τ (Ψ( T )) for al l T ∈ N + . That i s, τ ∆ = b τ ◦ Ψ so that b τ ( T ) = τ ∆ ( T ) for al l T ∈ M + . Final ly, if one of A, B ∈ M is b τ -tr ac e-class and T ∈ N then τ ∆ ( AT B ) = τ ∆ ( A Ψ( T ) B ) = b τ ( A Ψ( T ) B ) . Pr o of. The ﬁrst t wo statemen ts are imm ed iate. Also, the fact that Ψ is a unital norm one pro jection of N on to M (and hen ce a normal conditional exp ectatio n by T omiya ma’s theorem [T]) is clear. Only 12 the last assertions of the L emma n eed pr o of. T o this end let T ∈ N + , then τ ∆ ( T ) = su p N ˜ τ (∆ N T ) = sup N ˜ τ (∆( X | k |≤ N Φ k ) T ) = su p N ˜ τ ( X | k |≤ N ∆Φ k T ) = sup N ˜ τ ( X | k |≤ N n − k Φ k T ) = sup N X | k |≤ N n − k ˜ τ (Φ k T Φ k ) = X k ∈ Z n − k ˜ τ (Φ k T Φ k ) = b τ (Ψ( T )) . Hence if T ∈ M then b τ ( T ) = b τ (Ψ( T )) = τ ∆ ( T ) . Finally the last statemen t follo ws from the fact that Ψ( AT B ) = A Ψ( T ) B by T omiy ama’s Theorem [T].  Lemma 3.10. The mo dular automorphism gr oup σ τ ∆ t of τ ∆ is inner and given by σ τ ∆ t ( T ) = ∆ it T ∆ − it . The weight τ ∆ is a KMS weight for the gr oup σ τ ∆ t , and σ τ ∆ t | O n = σ τ ◦ Φ t . Pr o of. This follo ws fr om: [KR, Th m 9.2.38], wh ich giv es us the KMS prop erties of τ ∆ : the mo dular group is inner since ∆ is aﬃliate d to N . The ﬁn al statemen t ab out the restriction of the mo d ular group to O n is clear.  The rew ard for having sacriﬁced a trace on N for a trace on M is the follo wing. Lemma 3.11. Supp ose g is a function on R such that g ( D ) is τ ∆ tr ac e-class in M , then for al l f ∈ F we have τ ∆ ( π ( f ) g ( D )) = τ ∆ ( g ( D )) τ ( f ) = τ ( f ) X k ∈ Z g ( k ) . Pr o of. First note that τ ∆ ( g ( D )) = b τ ( P k ∈ Z g ( k )Φ k ) = P k ∈ Z g ( k ) b τ (Φ k ) = P k ∈ Z g ( k ) . Now, τ ∆ ( π ( f ) g ( D )) = b τ ( π ( f ) X k ∈ Z g ( k )Φ k ) = X k ∈ Z g ( k ) b τ ( π ( f )Φ k ) = X k ∈ Z g ( k ) b τ k ( π ( f )Φ k ) = X k ∈ Z g ( k ) n − k ˜ τ ( π ( f )Φ k ) . So it su ﬃces to see for eac h k ∈ Z , w e h a ve ˜ τ ( π ( f )Φ k ) = n k τ ( f ) . F or all f ∈ F , f is a norm limit of ﬁn ite su ms of terms like S α S ∗ β , | α | = | β | = r . So we compute for f = S α S ∗ β . Recall that we hav e the form u lae Φ k = X | µ | = k Θ R S µ ,S µ , k > 0 , Φ k = n − k X | µ | = | k | Θ R S ∗ µ ,S ∗ µ , k < 0 . and Φ 0 = Θ R 1 , 1 where, with µ the p ath of length zero, we are using the notation 1 = S µ . First for k ≥ 0 ˜ τ ( π ( f )Φ k ) = ˜ τ ( π ( f ) X | µ | = k Θ R S µ ,S µ ) = ˜ τ ( X | µ | = k Θ R f S µ ,S µ ) = X | µ | = k τ ◦ Φ( S ∗ µ f S µ ) = X | µ | = k τ ( S ∗ µ S α S ∗ β S µ ) = n k − r δ α,β = n k 1 n | α | δ α,β = n k τ ( S α S ∗ β ) = n k τ ( f ) . A similar calculation holds for k < 0 using the other formula for Φ k in this case. Since all f ∈ F c are linear com binations S α S ∗ β , | α | = | β | , we get for all f ∈ F c , the f orm ula τ ∆ ( π ( f ) g ( D )) = τ ∆ ( g ( D )) τ ( f ) = X k ∈ Z g ( k ) τ ( f ) . 13 No w, the r ight hand side is a n orm-con tinuous function of f . T o s ee th at the left sid e is norm-cont inuous w e do it in more generalit y . Let T ∈ N , then sin ce b τ is a trace on M we get: | τ ∆ ( T g ( D )) | = | b τ (Ψ( T g ( D )) | = | b τ (Ψ( T ) g ( D )) | ≤ k Ψ( T ) k b τ ( | g ( D ) | ) ≤ k T k b τ (( | g ( D ) | ) = k T k τ ∆ ( | g ( D ) | ) . That is the left h and side is norm -con tin u ous in T and so we h a ve the f ormula: τ ∆ ( π ( f ) g ( D )) = τ ∆ ( g ( D )) τ ( f ) = X k ∈ Z g ( k ) τ ( f ) for all f ∈ F .  Remarks. T he inequalit y ab o ve clearly holds in more generalit y . That is, if T ∈ N and B ∈ L 1 ( M , τ ∆ ) then: | τ ∆ ( T B ) | ≤ k T k ∞ τ ∆ ( | B | ) = k T k ∞ k B k 1 . (7) Prop osition 3.12. W e have (1 + D 2 ) − 1 / 2 ∈ L (1 , ∞ ) ( M , τ ∆ ) . That is, τ ∆ ((1 + D 2 ) − s/ 2 ) < ∞ f or al l s > 1 . Mor e over, for al l f ∈ F lim s → 1 + ( s − 1) τ ∆ ( π ( f )(1 + D 2 ) − s/ 2 ) = 2 τ ( f ) so that π ( f )(1 + D 2 ) − 1 / 2 is a me asur able op e r ator in the sense of [C] . Pr o of. Let s > 1. T hen τ ∆ ((1 + D 2 ) − s/ 2 ) = b τ ( P k ∈ Z (1 + k 2 ) − s/ 2 Φ k ) = P k ∈ Z (1 + k 2 ) − s/ 2 . Hence, (1 + D 2 ) − s/ 2 is τ ∆ -trace-cl ass in M for all R e ( s ) > 1 and lim s → 1 + ( s − 1) τ ∆ ((1 + D 2 ) − s/ 2 ) = 2 . By Lemma 3.11 we hav e the equalit y: τ ∆ ( π ( f )(1 + D 2 ) − s/ 2 ) = X k ∈ Z (1 + k 2 ) − s/ 2 τ ( f ) for all f ∈ F . Hence, lim s → 1 + ( s − 1) τ ∆ ( π ( f )(1 + D 2 ) − s/ 2 ) = 2 τ ( f ) and π ( f )(1 + D 2 ) − 1 / 2 is measurable, for all f ∈ F .  W e wish to extend our conclusions ab out τ ∆ and lim s → 1 + ( s − 1) τ ∆ ( T (1 + D 2 ) − s/ 2 ) to the w hole vo n Neumann algebra N . Unfortun ately , these limits do n ot exist f or general T ∈ N and we are forced to consider generalised limits as in the Dixmier trace theory . Deﬁnition 3.13. L et ˜ ω b e a state on L ∞ ( R + ) which satisﬁes the c ondition that if g ∈ L ∞ ( R + ) is r e al-value d then lim inf t →∞ g ( t ) ≤ ˜ ω ( g ) ≤ lim sup t →∞ g ( t ) . Cle arly any such state is identic al ly 0 on C 0 ( R + ) and also on any function which is essential ly c om- p actly supp orte d. Mor e over, if g has a limit at ∞ then ˜ ω ( g ) = lim t →∞ g ( t ) . We deﬁne ˜ ω − lim t →∞ g ( t ) := ˜ ω ( g ) . The existence of such states (with eve n more pr op erties) can b e found in [CPS 2, Corollary 1.6]. In order to ev aluate su c h states ˜ ω on fu nctions g of the form s 7→ ( s − 1) τ ∆ ( T (1 + D 2 ) − s/ 2 ) for s > 1 w e need to do a translation: let s = 1 + 1 /r then letting s → 1 + is the same as letting r → ∞ . And w e consider ( s − 1) τ ∆ ( T (1 + D 2 ) − s/ 2 ) = 1 r τ ∆  T  (1 + D 2 ) − 1 / 2  1+1 /r  . 14 Of cour se, the limit of the left h and side of this equation exists as s → 1 + if and only if the limit of the righ t h and sid e exists as r → ∞ and in this case they are equal. Abuse of nota t ion: ˜ ω − lim r →∞ 1 r τ ∆  T  (1 + D 2 ) − 1 / 2  1+1 /r  b ecomes ˜ ω − lim s → 1 + ( s − 1) τ ∆ ( T (1 + D 2 ) − s/ 2 ) . Remarks. Since τ ∆ ( T (1 + D 2 ) − s/ 2 ) = b τ (Ψ( T )(1 + D 2 ) − s/ 2 ) these generalised traces are taking place completely inside M with resp ect to the trace b τ . T hat is, we are in the no w well-understo o d semiﬁn ite situation. Prop osition 3.14. L et ˜ ω b e a state on L ∞ ( R + ) which satisﬁes the c ondition ab ove. The f u nctional b τ ω on N deﬁne d by b τ ω ( T ) = 1 2 ˜ ω − lim s → 1 + ( s − 1) τ ∆ ( T (1 + D 2 ) − s/ 2 ) is a state. F or T = π ( a ) ∈ π ( O n ) ⊂ N the fol lowing (or dinary) limit exists and b τ ω ( π ( a )) = 1 2 lim s → 1 + ( s − 1) τ ∆ ( π ( a )(1 + D 2 ) − s/ 2 ) = τ ◦ Φ( a ) , the original KMS state ψ = τ ◦ Φ on O n . Pr o of. First we observ e that τ ∆ ( T (1 + D 2 ) − s/ 2 ) is ﬁnite for s > 1 for all T ∈ N , since w e sh ow ed in the pro of of the p revious Prop osition that: | τ ∆ ( T (1 + D 2 ) − s/ 2 ) | ≤ k T k τ ∆ ((1 + D 2 ) − s/ 2 ) . Therefore, ( s − 1) τ ∆ ( T (1 + D 2 ) − s/ 2 ) is u niformly b ounded and so the generalised limit exists as s → 1 + . It is easy to see that this fu nctional is p ositiv e on N + and by the pr evious prop osition b τ ω (1) = 1, so that b τ ω is a state on N . No w, one easily c h ecks b y calculating on generators that f or π ( a ) ∈ π ( O nc ), Ψ( π ( a )) = π (Φ( a )) ∈ π ( F c ) and since Ψ is norm con tinuous w e h a ve that Ψ( π ( a )) = π (Φ( a )) ∈ π ( F ) for all a ∈ O n . Th u s b y Prop osition 3.12, for a ∈ O n (letting f = Φ( a )) w e hav e τ ◦ Φ( a ) = 1 2 lim s → 1 + ( s − 1) τ ∆ ( π (Φ( a ))(1 + D 2 ) − s/ 2 ) = 1 2 lim s → 1 + ( s − 1) τ ∆ (Ψ( π ( a ))(1 + D 2 ) − s/ 2 ) = b τ ω ( π ( a )) . Of course b τ ω is a tru e Dixmier-trace since for T ∈ N with T ≥ 0, we ha ve Ψ( T ) ∈ M , Ψ( T ) ≥ 0, and (1 + D 2 ) − 1 / 2 ∈ L (1 , ∞ ) ( M , b τ ). Th u s ˜ ω − lim r →∞ 1 r τ ∆ ( T ((1 + D 2 ) − 1 / 2 ) 1+1 /r ) = ˜ ω − lim r →∞ 1 r b τ (Ψ( T )((1 + D 2 ) − 1 / 2 ) 1+1 /r ) and the righ t h and sid e is a tru e Dixmier-trace on the semiﬁnite algebra M provi d ed w e choose ˜ ω as in [CPS2, T heorem 3.1].  W e su mmarise our construction to date. 0) W e ha ve a ∗ -subalgebra A = O nc of the Cu ntz algebra faithfully represent ed in N with the latter acting on the Hilb ert space H = L 2 ( O n , ψ ), 1) there is a faithful n ormal semiﬁnite weigh t τ ∆ on N such that the m o dular automorph ism group of τ ∆ is an inn er automorp h ism group ˜ σ of N w ith ˜ σ | A = σ , 2) τ ∆ restricts to a faithful semiﬁnite trace b τ on M = N σ , w ith a faithful normal pro jection Ψ : N → M satisfying τ ∆ = b τ ◦ Ψ on N . 15 3) With D th e generator of the one p arameter group implementing σ on H w e h a ve : [ D , π ( a )] extends to a b ound ed op erator (in N ) for all a ∈ A and for λ in the resolv ent set of D , ( λ − D ) − 1 ∈ K ( M , τ ∆ ), where K ( M , τ ∆ ) is the id eal of compact op erators in M relativ e to τ ∆ . In particular, D is aﬃliated to M . T erminology/Deﬁnition . The triple ( O nc , H , D ) along with N , τ ∆ constructed in this section satisfying prop erties (0) to (3) ab o ve we w ill refer to as a (un ital) mo dular sp ectral triple . F or matrix algebras A = O nc ⊗ M k o v er O nc , ( O nc ⊗ M k , H ⊗ M k , D ⊗ I d k ) is also a mo du lar sp ectral triple in the obvio u s fashion. In w ork in progress we ha ve fou n d th at this stru cture arises in other examples and app ears to b e a q u ite general p henomenon. W e n eed some tec hn ical lemmas for the discussion in the next Section. A fun ction f from a complex domain Ω into a Banac h sp ace X is called holomorphic if it is complex diﬀerentiable in norm on Ω . Lemma 3.15. (1) L et B b e a C ∗ -algebr a and let T ∈ B + . The mapping z 7→ T z is holomorph ic (in op er ator norm) in the half-plane Re ( z ) > 0 . (2) L et B b e a von Neumann algebr a with faithful normal semiﬁnite tr ac e φ and let T ∈ B + b e in L (1 , ∞ ) ( B , φ ) . Then, the mapping z 7→ T z is holomor phic (in tr ac e norm) i n the half-plane R e ( z ) > 1 . (3) L et B , and T b e as in item (2) and let A ∈ B then the mapping z 7→ φ ( AT z ) is holomorp hic for Re ( z ) > 1 . Pr o of. T o see item (1) we assume with ou t loss of generalit y that || T || ≤ 1 . W e ﬁx z 0 ∈ C with Re ( z 0 ) > 0 , and ﬁx R > 0 with R < Re ( z 0 ) so that the circle C : z = z 0 + R e iθ for θ ∈ [0 , 2 π ] lies in the half-plane Re ( z ) > 0 . T emp orarily we ﬁx t 6 = 0 in the sp ectrum of T so that t ∈ (0 , 1] . No w with | z − z 0 | < (1 / 2) R we app ly th e complex v ersion of T a ylor’s theorem to the function z 7→ t z (see [Ahl, Theorem 8, p p125-6]) and get: t z − t z 0 z − z 0 − t z 0 Log ( t ) = f 2 ( z )( z − z 0 ) where f 2 ( z ) = 1 2 π i Z C t w dw ( w − z 0 ) 2 ( w − z ) . So w ith | z − z 0 | < (1 / 2) R we get the estimate: | f 2 ( z ) | ≤ max C | t w | · R R 2 · (1 / 2 ) R ≤ 2 t ( Re ( z 0 )+ R ) R 2 ≤ 2 R 2 . Therefore,     t z − t z 0 z − z 0 − t z 0 Log ( t )     ≤ 2 R 2 | z − z 0 | . Since th is is true f or all nonzero t in the sp ectrum of T we h a ve :     1 z − z 0 ( T z − T z 0 ) − T z 0 Log ( T )     ∞ ≤ 2 R 2 | z − z 0 | . That is d/dz ( T z ) = T z Log ( T ) for Re ( z ) > 0 with the limit existing in op erator norm. T o see item (2) we ﬁx z 0 with Re ( z 0 ) > 1 and th en ﬁx ǫ su ﬃcien tly small so that Re ( z 0 − (1 + ǫ )) = Re ( z 0 ) − (1 + ǫ ) > 0 . Then T (1+ ǫ ) is tr ace-class, and this factor conv erts the op erator norm limits 16 b elo w in to trace norm limits: || · || 1 lim z → z 0 1 z − z 0 ( T z − T z 0 ) = || · || 1 lim z → z 0 T (1+ ǫ ) 1 z − z 0 ( T z − (1+ ǫ ) − T z 0 − (1+ ǫ ) ) = T (1+ ǫ )  || · || ∞ lim z → z 0 1 z − z 0 ( T z − (1+ ǫ ) − T z 0 − (1+ ǫ ) )  = T (1+ ǫ )  || · || ∞ lim z → z 0 1 ( z − (1 + ǫ )) − ( z 0 − (1 + ǫ )) ( T z − (1+ ǫ ) − T z 0 − (1+ ǫ ) )  = T (1+ ǫ ) ( T z 0 − (1+ ǫ ) Log ( T ) = T z 0 Log ( T ) . Item (3) f ollo ws f rom item (2) and inequalit y (7): | φ ( AB ) | ≤ || A || ∞ || B || 1 if B is φ -trace-cla ss.  Lemma 3.16. In these mo dular sp e ctr al triples ( A , H , D ) for matric es over the Cuntz algebr as we have (1 + D 2 ) − s/ 2 ∈ L 1 ( M , τ ∆ ) for al l s > 1 and for x ∈ N , τ ∆ ( x (1 + D 2 ) − r / 2 ) is holomorp hic for Re ( r ) > 1 and we have for a ∈ O nc τ ∆ ([ D , π ( a )](1 + D 2 ) − r / 2 ) = 0 , for Re ( r ) > 1 . Pr o of. Since the eigen v alues f or D are p recisely the s et of int egers, and the pro jection Φ k on the eigenspace with eigen v alue k satisﬁes τ ∆ (Φ k ) = 1 , it is clear that (1 + D 2 ) − s/ 2 ∈ L 1 ( M , τ ∆ ) . No w , τ ∆ ( x (1 + D 2 ) − r / 2 ) = b τ (Ψ( x )(1 + D 2 ) − r / 2 ) is holomorphic for Re ( r ) > 1 by item (3) of the p revious lemma. T o see the last statemen t, w e observe that τ ∆ ([ D , π ( a )](1 + D 2 ) − r / 2 ) = τ ∆ (Ψ([ D , π ( a )])(1 + D 2 ) − r / 2 ) , so it suﬃces to see that Ψ([ D , π ( a )]) = 0 for a ∈ A = O nc . T o this end, let a = S α S ∗ β b e one of the linear generators of O nc . Then b y calculating the action of the op er ator D π ( S α S ∗ β ) on the linear generators S µ S ∗ ν of the Hilb ert space, H , we obtain: D π ( S α S ∗ β ) = ( | α | − | β | ) π ( S α S ∗ β ) + π ( S α S ∗ β ) D that is [ D , π ( S α S ∗ β )] = ( | α | − | β | ) π ( S α S ∗ β ) . More generally , [ D , π ( m X i =1 c i S α i S ∗ β i )] = m X i =1 c i ( | α i | − | β i | ) π ( S α i S ∗ β i ) . If w e app ly Ψ to this equation, we see that Ψ( π ( S α i S ∗ β i )) = π (Φ( S α i S ∗ β i )) = 0 whenever ( | α | − | β | ) 6 = 0 , and so the wh ole sum is 0. W e also observe that [ D , π ( a )] ∈ π ( O nc ) for all a ∈ O nc . This is not to o surpr ising since D is the generator of th e action γ of T on O n .  In the remaind er of this pap er we will sh ed some ligh t on the cohomologica l signiﬁcance of these mo dular sp ectral trip les. Just as ordin ary B ( H ) sp ectral triples repr esen t K -homology classes, [C, CPRS1], and semiﬁnite sp ectral triples represent K K -classes, [KNR], mo dular s p ectral triples pro vid e analytic represent ative s of some K -theoretic type data whic h we n o w describ e. 4. Modular K 1 In this S ection we introd uce elemen ts of A that w ill ha ve a well deﬁned pairing w ith our Dixmier functional b τ ω . F ollo wing [HR] we sa y that a u nitary (in vertible, pro jection,...) in M n ( A ) for s ome n is a u n itary (inv ertible, p ro jection,...) o ver A . Deﬁnition 4.1. L et A b e a unital ∗ -algebr a and σ : A → A an algebr a automorph i sm such that σ ( a ) ∗ = σ − 1 ( a ∗ ) . W e say that σ i s a regular aut omorphism , [KMT] . 17 Remark . T h e automorphism σ ( a ) := ∆ − 1 ( a ) of a mo dular sp ectral triple is regular. This follo ws from AXIOM IV of Lemma 3.2: ( σ ( a )) ∗ = (∆ − 1 ( a )) ∗ = S (∆ − 1 ( a )) = ∆( S ( a )) = ∆ ( a ∗ ) = σ − 1 ( a ∗ ) . Deﬁnition 4.2. L et u b e a unitary over the ∗ -algebr a A , and σ : A → A a r e gular automorph ism with ﬁxe d p oint algebr a F = A σ . We say that u satisﬁes the mo dular condition with r esp e ct to σ if b oth the op er ators uσ ( u ∗ ) and u ∗ σ ( u ) ar e matric es over the algebr a F . We denote by U σ the set of mo dular unitaries . Of c ourse, any unitary over F is a mo dular unitary. Here w e are thinking of the case σ ( a ) = ∆ − 1 ( a ), wher e ∆ is the mo du lar op erator for some weigh t on A . Again, to av oid confusion, we remind the reader that as op erators we h a ve : π ( σ ( a )) = π (∆ − 1 ( a )) = ∆ − 1 π ( a )∆ . Hence the terminology mo dular unitaries. F or unitaries in matrix algebras o ver A w e us e the regular automorphism σ ⊗ I d n to state the mo dular condition, where I d n is the iden tit y automorphism of M n ( C ). Example . F or S µ ∈ O nc w e wr ite P µ = S µ S ∗ µ . Th en for eac h µ, ν w e hav e a unitary u µ,ν =  1 − P µ S µ S ∗ ν S ν S ∗ µ 1 − P ν  . It is simple to chec k that this a self-adjoin t unitary satisfying the mo du lar condition. Deﬁnition 4.3. L et u t b e a c ontinuous p ath of mo dular u nitaries in the ∗ -sub algebr a A such that u t σ ( u ∗ t ) and u ∗ t σ ( u t ) ar e also con tin uous p aths i n F (this is not guar ante e d sinc e σ is not gener al ly b ounde d). Then we say that u t is a mo dular homotop y , and say that u 0 and u 1 ar e mo dula r homotopic . If u and v ar e mo dular unitaries, we say that u is equiv a le nt to v if ther e exi st k , m ≥ 0 so that u ⊕ 1 k is mo dular homotopic to v ⊕ 1 m . Lemma 4.4. The r elation deﬁne d ab ove is an e quivalenc e r elation. Mor e over, if u is a mo dular unitary and k ≥ 0 then 1 k ⊕ u is mo dular homotopic to u ⊕ 1 k . The binary op e r ation on e quivalenc e classes in U σ , giv e n by [ u ] + [ v ] := [ u ⊕ v ] is wel l-deﬁne d and ab elian. Pr o of. It is straigh tforward to show that this is an equiv alence relation. T o see that 1 k ⊕ u is m o dular homotopic to u ⊕ 1 k it suﬃces to do this for k = 1 . If u ∈ M m ( A ) is a mo du lar unitary then let x 0 ∈ M m +1 ( C ) b e the (bac kwa r d) sh ift matrix whose action on the standard basis of C m +1 is giv en b y x 0 ( e k ) = e k − 1 ( mod )( m + 1) . Then, x 0 (1 ⊕ u ) x ∗ 0 = ( u ⊕ 1) . Let { x t } b e a conin u ous p ath of scalar unitaries fr om x 0 to x 1 = 1 m +1 . Of course eac h x t ∈ M m +1 ( F ) as well. S in ce σ ( x t ) = x t , one easily c hec ks that { x t (1 ⊕ u ) x ∗ t } is a mo du lar h omotop y from u ⊕ 1 to 1 ⊕ u. T o see that addition is well-deﬁned, w e m ust sh o w that u ⊕ v is equiv alen t to ( u ⊕ 1 k ) ⊕ ( v ⊕ 1 m ) . But this equals u ⊕ (1 k ⊕ v ) ⊕ 1 m . By the pr evious argument this is equiv alen t to u ⊕ ( v ⊕ 1 k ) ⊕ 1 m whic h equals ( u ⊕ v ) ⊕ 1 k + m whic h is equiv alen t to ( u ⊕ v ) . T o see that addition of classes is ab elian let u, v b e mo dular un itaries. By adding on copies of the iden tity , w e can assu me th at u and v are b oth the same size m atrices. Hence, it su ﬃces to show that u ⊕ v is mo d ular homotopic to v ⊕ u. T o this end, let R t =  cos( t ) sin( t ) − sin( t ) cos( t )  for t ∈ [0 , π / 2]. Let w t = R t ( u ⊕ v ) R ∗ t . Then we h a ve w t =  cos 2 ( t ) u + sin 2 ( t ) v cos( t ) s in( t )( v − u ) cos( t ) sin ( t )( v − u ) cos 2 ( t ) v + s in 2 ( t ) u  . 18 Observe th at at t = 0 we h av e u ⊕ v and at t = π / 2 we hav e v ⊕ u . W e need to show that w t σ ( w ∗ t ) ∈ M 2 ( F ) for all t ∈ [0 , π / 2]. W rite ˆ u for σ ( u ∗ ) and similarly for v . Then w e compu te w t σ ( w ∗ t ) =  cos 2 ( t ) u + sin 2 ( t ) v cos( t ) s in( t )( v − u ) cos( t ) sin( t )( v − u ) cos 2 ( t ) v + s in 2 ( t ) u   cos 2 ( t ) ˆ u + sin 2 ( t ) ˆ v cos( t ) sin( t )( ˆ v − ˆ u ) cos( t ) sin( t )( ˆ v − ˆ u ) cos 2 ( t ) ˆ v + s in 2 ( t ) ˆ u  =  cos 2 ( t ) u ˆ u + sin 2 ( t ) v ˆ v cos( t ) sin( t )( v ˆ v − u ˆ u ) cos( t ) sin( t )( v ˆ v − u ˆ u ) cos 2 ( t ) v ˆ v + sin 2 ( t ) u ˆ u  and since b oth u ˆ u and v ˆ v lie in F , this is in M 2 ( F ). T he other half of the m o dular condition follo ws b y replacing u, v by u ∗ , v ∗ .  W e can no w also see why the usu al pro of that the inv erse of u is u ∗ in K 1 ( A ) is not a v ailable to us. Th is usual pro of is as follo w s. In the K 1 setting one uses: u ⊕ v = ( u ⊕ 1)(1 ⊕ v ) ∼ (1 ⊕ u )(1 ⊕ v ) = (1 ⊕ uv ), so that addition in K 1 arises from multiplic ation of un itaries, and hence [ u ] + [ u ∗ ] = [ uu ∗ ] = [1] = 0. Ho w ever, in the mo du lar setting, while th e homotop y from u ⊕ 1 to 1 ⊕ u is a mo dular h omotop y in U σ b y the last Lemma, the homotopy from ( u ⊕ 1)(1 ⊕ v ) to (1 ⊕ u )(1 ⊕ v ) is n ot in general. The m ultiplication on the righ t b y (1 ⊕ v ) breaks the mo d ular condition. In particular, th e pro d uct of tw o mo dular unitaries need not b e a mo dular unitary . Lemma 4.5. If u ∈ M k ( F ) is unitary then u ⊕ u ∗ ∼ 1 . Pr o of. There is a path w t from u ⊕ u ∗ to 1 throu gh un itaries in M k ( F ) and so w t σ ( w ∗ t ) = 1 for all t and hence we ﬁ nd u ⊕ u ∗ ∼ 1.  W e n ow formalise the ab ov e discussion. Compare th e follo wing w ith [HR, Deﬁn ition 4.8.1] Deﬁnition 4.6. L et K 1 ( A , σ ) b e the ab elian semigr oup of e quivalenc e classes of mo dular unitaries u over A u nder the e q u ivalenc e r elation u is equiv alent to v i f ther e exist k, m ≥ 0 so that u ⊕ 1 k is mo dular homoto pic to v ⊕ 1 m . The fol lowing r elations hold in K 1 ( A , σ ) 1) [1] = 0 , 2) [ u ] + [ v ] = [ u ⊕ v ] , 3) If u t , t ∈ [0 , 1] is a c ontinuous p aths of unitaries in M k ( A ) with u t σ ( u ∗ t ) and u ∗ t σ ( u t ) c ontinuous over F then [ u 0 ] = [ u 1 ] . Corollary 4.7. If u ∈ M k ( F ) then − [ u ] = [ u ∗ ] in K 1 ( A, σ ) . W e can make K 1 ( A, σ ) a group by th e Grothendiec k constru ction, but this is not needed here. The follo wing lemma is a clear d eparture f rom the situation in [PR] (it implies that the ‘ob vious’ map f r om K 0 ( M ( F, A ) ) to K 1 ( A, σ ) is not well- d eﬁned). Lemma 4.8. R e c al l, for al l p aths µ, ν with P µ = S µ S ∗ µ we have a mo dular unitary u µ,ν =  1 − P µ S µ S ∗ ν S ν S ∗ µ 1 − P ν  . Then ther e is a mo dular homotop y u µ,ν ∼ u ν,µ . Pr o of. W e do the homotop y in t wo steps. The ﬁrs t is giv en by conju gating u µ,ν b y the s calar un itary matrix  cos θ sin θ − sin θ cos θ  , θ ∈ [0 , π / 2] , 19 whic h tak es us to  1 − P ν − S ν S ∗ µ − S µ S ∗ ν 1 − P µ  . Then f or θ ∈ [0 , π ] we consid er  1 − P ν e iθ S ν S ∗ µ e − iθ S µ S ∗ ν 1 − P µ  . The reader w ill readily conﬁrm that th ese t wo h omotopies are mo d ular.  Example . More generally , if σ is a regular automorphism of a unital ∗ -algebra A with ﬁxed p oin t algebra F , v ∈ A is a partial isometry with range an d source pro jections in F , and furtherm ore if v σ ( v ∗ ) , v ∗ σ ( v ) lie in F , then u v =  1 − v ∗ v v ∗ v 1 − v v ∗  is a mo dular unitary o ver A , as the reader ma y c heck. The pro of of Lemma 4.8 applies to these unitaries to sho w that u v ∼ u v ∗ . Lemma 4.9. L et ( A , H , D ) b e our mo dular sp e ctr al triple r elative to ( N , τ ∆ ) and set F = A σ and σ : A → A . L et L ∞ (∆) = L ∞ ( D ) b e the von N eumann algebr a gener ate d by the sp e ctr al pr oje ctions of ∆ then L ∞ (∆) ⊂ Z ( M ) . L et u ∈ A b e a unitary, then π ( u ) Qπ ( u ∗ ) ∈ M and π ( u ∗ ) Qπ ( u ) ∈ M for al l sp e ctr al pr oje c tions Q of D , if and only if u is mo dular. That is, π ( u )∆ π ( u ∗ ) and π ( u ∗ )∆ π ( u ) (or π ( u ) D π ( u ∗ ) and π ( u ∗ ) D π ( u ) ) ar e b oth aﬃliate d to M if and only if u is mo dular. Pr o of. First, L ∞ (∆) is an ab elian algebra. By L emma 3.9 all the Φ k are in M and since the Φ k are also the sp ectral pro jections of ∆, we ha ve L ∞ (∆) is cont ained in the cen tr e. (Note that this extends the fact that D comm utes with π ( F ) = π ( A σ )). Next we observe that π ( u ) Qπ ( u ∗ ) is a pro jection in N . F or one d irection, supp ose u is m o dular, then w e h a ve ∆ − 1 π ( u ) Qπ ( u ∗ )∆ = ∆ − 1 π ( u )∆∆ − 1 Q ∆∆ − 1 π ( u ∗ )∆ Q ∈ M = π ( σ ( u )) Qπ ( σ ( u ∗ )) = π ( u ) π ( u ∗ σ ( u )) Qπ ( σ ( u ∗ )) = π ( u ) Qπ ( u ∗ σ ( u ) σ ( u ∗ )) , u ∗ σ ( u ) ∈ F = π ( u ) Qπ ( u ∗ ) . Hence π ( u ) Qπ ( u ∗ ) comm utes with ∆, and so is in M . Similarly , uσ ( u ∗ ) ∈ F implies that π ( u ∗ ) Qπ ( u ) ∈ M . On the other hand if π ( u ) Qπ ( u ∗ ) ∈ M then π ( u ) Qπ ( u ∗ ) = ∆ − 1 π ( u ) Qπ ( u ∗ )∆ = π ( σ ( u )) Qπ ( σ ( u ∗ )) and so we hav e Q = π ( u ∗ σ ( u )) Qπ ( σ ( u ∗ ) u ) = Q + [ π ( u ∗ σ ( u )) , Q ] π ( σ ( u ∗ ) u ) . As σ ( u ∗ ) u is inv ertible, we see that [ π ( u ∗ σ ( u )) , Q ] = 0. Since π ( u ∗ σ ( u )) ∈ π ( A ), and commutes with all Q , w e ha ve π ( u ∗ σ ( u )) ∈ M and so lies in π ( F ) = M ∩ π ( A ). That is, u ∗ σ ( u ) ∈ F . Similarly , π ( u ∗ ) Qπ ( u ) ∈ M implies th at uσ ( u ∗ ) ∈ F .  The f u ndamenta l asp ect of th e last lemma is th at mo dular unitaries conjugate ∆ to an op erator aﬃliated to M , and so u ∆ u ∗ comm utes with ∆ (and u D u ∗ comm utes w ith D ). W e will next s h o w that there is a pairin g b et we en (part of ) mo d u lar K 1 and mo du lar sp ectral trip les. T o do this, we are going to use the analytic formulae for sp ectral ﬂ o w in [CP2]. 20 5. An L (1 , ∞ ) local inde x formula In this Section we will couc h our results in terms of the notion of a mo dular sp ectral triple. That is w e will assume pr op erties (0) to (3) listed in Section 3 app ly . O f course at this time the only examples w e ha ve presen ted are the matrix algebras o ver the smo oth subalgebra O nc of the Cu n tz algebra. Ho w ever, w e k n o w from work in progress that there are other examples and hence it is w orth arguing directly from the general prop erties and av oiding th e explicit f orm u lae of the Cuntz example. 5.1. The sp ectral ﬂo w form ula: correction terms. Th e sp ectral ﬂow form u la of [CP2 ] is, a priori, complicated in our setting. This is b ecause we are computing the sp ectral ﬂow b etw een tw o op erators whic h a re not u nitarily equiv alent via a unitary in M . Th u s w e must consider η -t yp e correction terms. W e will also recognise that the sp ectral ﬂow we are calculating dep end s on the c hoice of trace φ on M and u se the n otation s f φ . W e n o w quote [CP2, Corollary 8.11]. Prop osition 5.1. L et ( A , H , D 0 ) b e an o dd unb ounde d θ -summable semiﬁnite sp e ctr al triple r elative to ( M , φ ) . F or any ǫ > 0 we deﬁne a one-form α ǫ on M 0 = D 0 + M sa by α ǫ ( A ) = r ǫ π φ ( Ae − ǫ D 2 ) for D ∈ M 0 and A ∈ T D ( M 0 ) = M sa . Then the inte gr al of α ǫ is i ndep endent of the pie c ewise C 1 p ath in M 0 and if {D t } t ∈ [ a,b ] is any pie c ewise C 1 p ath in M 0 then sf φ ( D a , D b ) = r ǫ π Z b a φ ( D ′ t e − ǫ D 2 t ) dt + 1 2 η ǫ ( D b ) − 1 2 η ǫ ( D a ) + 1 2 φ ([ke r ( D b )] − [k er ( D a )]) . wher e the f ol lowing inte gr al c onver ges f or al l ǫ > 0 η ǫ ( D ) = 1 √ π Z ∞ ǫ φ ( D e − t D 2 ) t − 1 / 2 dt. W e n ote that the η terms are measures of φ -sp ectral asymmetry . W e will sho w that for the pair D , τ ∆ w e use on the Cuntz algebra, and th e kind s of p erturb ations we consider, these η terms v anish. Moreo v er w e will show that the τ ∆ -dimension of the kernel of D is unc h anged by the particular type of p erturb ations we consider, so these correction terms w ill cancel. First we must show that w e are actually w orking w ith the right kind s of p erturbations, that is, elements in M sa . Notation . W e d en ote the densely deﬁ ned spatial homomorph ism on N , T 7→ ∆ − 1 T ∆ by σ i ( T ), so that for a ∈ A we h a ve π ( σ ( a )) = σ i ( π ( a )) . W e observe that M , and π ( A ) are in the d omain of σ i , and that M is exactly th e ﬁx ed p oin t subalgebra of σ i . Lemma 5.2. L et ( A , H , D ) b e a mo dular sp e ctr al triple. If u is a mo dular unitary, then π ( u )[ D , π ( u ∗ )] ∈ M sa . This is a k ey fact which al lows us to dir e ctly u se r e sults ab out semiﬁnite sp e ctr al ﬂow in ( M , τ ∆ ) fr om [C P 2] . Pr o of. W e ju s t compute th e action of σ i on π ( u )[ D , π ( u ∗ )] . As observ ed ab o v e th e op erator [ D , π ( u ∗ )] ∈ π ( A ) , and we easily calculate: σ i ( π ( u )[ D , π ( u ∗ )]) = π ( σ ( u ))[ D , π ( σ ( u ∗ ))] = π ( uu ∗ σ ( u ))[ D , π ( σ ( u ∗ ))] = π ( u )[ D , π ( u ∗ )] .  Remarks. In th e follo wing few pages we will sometimes abuse n otation and write a in place of π ( a ) for a ∈ A in order to m ak e our form u lae more readable. Whenever we do this, how ev er, we will u se σ i ( · ) = ∆ − 1 ( · )∆ th e s patial version of the algebra homomorph ism, σ . W e will generally use the spatial v ersion σ i when in the p resence of op erators not in π ( A ) . 21 Lemma 5.3. L et ( A , H , D ) b e our mo dular sp e ctr al triple for the Cuntz algebr a and let u b e a mo dular unitary. Then τ ∆ ([k er( D )] − [k er ( u D u ∗ )]) = τ ∆ ((1 − σ i ( u ∗ ) u )[k er ( D )]) = τ (1 − σ ( u ∗ ) u ) , and for al l ǫ > 0 , η ǫ ( u D u ∗ ) = τ ( σ ( u ∗ ) u ) η ǫ ( D ) . Pr o of. W e show the second equalit y ﬁrs t. By the σ i -in v ariance of τ ∆ and th e fact that σ i ( u ∗ ) u ∈ M w e hav e, using Lemma 3.11 in th e last equalit y: τ ∆ ( u D u ∗ e − t ( u D u ∗ ) 2 ) = τ ∆ ( u D e − t D 2 u ∗ ) = τ ∆ ( σ i ( u ) D e − t D 2 σ i ( u ∗ )) = ˜ τ ( u ∆ D e − t D 2 σ i ( u ∗ )) = ˜ τ (∆ D e − t D 2 σ i ( u ∗ ) u ) = ˜ τ (∆ σ i ( u ∗ ) u D e − t D 2 ) = τ ∆ ( σ i ( u ∗ ) u D e − t D 2 ) = τ ( σ ( u ∗ ) u ) τ ∆ ( D e − t D 2 ) . Th u s, w e ha ve η ǫ ( u D u ∗ ) = 1 √ π Z ∞ ǫ τ ( σ ( u ∗ ) u ) τ ∆ ( D e − t D 2 ) t − 1 / 2 dt = τ ( σ ( u ∗ ) u ) η ǫ ( D ) , as w as to b e shown. F or the kernel we simp ly observ e that [ke r ( u D u ∗ )] = u [ke r ( D )] u ∗ ∈ N , so that τ ∆ ([k er( u D u ∗ )]) = ˜ τ (∆ u [ker( D )] u ∗ ) = ˜ τ ( u [k er( D )]∆∆ − 1 u ∗ ∆) = τ ∆ ( σ i ( u ∗ ) u [k er( D )]) . Then, b y Lemma 3.11, τ ∆ ( σ i ( u ∗ ) u [k er( D )]) = τ ( σ ( u ∗ ) u ) τ ∆ ([k er( D )]) = τ ( σ ( u ∗ ) u ) · 1 .  If w e hav e a mo dular unitary for which we h a ve b oth η ǫ ( u v D u v ) − η ǫ ( D ) = 0 and φ ([k er D ]) − φ ([k er u D u ∗ ]) = 0, we ma y apply the Laplace transform technique discussed in [CP2, Section 9] to reduce the θ -summ able formula to the ﬁnitely summable formula . F or r > 0 this giv es us (8) sf φ ( D , u D u ∗ ) = 1 C 1 / 2+ r Z 1 0 φ ( u [ D , u ∗ ](1 + ( D + tu [ D , u ∗ ]) 2 ) − 1 / 2 − r ) dt. W e are no w in a p osition to apply the metho ds employ ed in the p ro of of the semiﬁn ite lo cal index form u la, [CPS2] or [CPRS 2], to compute an index pairing. 5.2. A lo cal index form ula for the Cuntz algebras. Lemma 5.4 (cf [CP 2]) . L et ( A , H , D ) b e our (1 , ∞ ) -summable mo dular sp e ctr al for the Cuntz algebr a triple for a matrix algebr a A over O nc . L et M = N σ b e the ﬁxe d p oint algebr a for the mo dular automorp hism gr oup. The functional α deﬁne d on the self adjoint elements M sa of M by α S ( T ) = b τ ( T (1 + ( D + S ) 2 ) − s/ 2 ) , T ∈ M sa for s > 1 is an exact one form on the tangent sp ac e to the aﬃne sp ac e M 0 = M sa + D of M sa p erturb ations of D . This fact is all that we need to calculate b τ -sp ectral ﬂo w along paths in the aﬃne s pace M 0 . W e will b e inte r ested in b τ -sp ectral ﬂ o w along the linear path joining D to D + u [ D , u ∗ ] where u is a unitary in E nd F ( X ) s u c h that u [ D , u ∗ ] ∈ M sa . Since mo dular un itaries, u satisfy these r equ iremen ts, we can no w pro du ce a form u la for sp ectral ﬂow wh ich is analogous to the lo cal ind ex formula in noncommuta tive geometry . W e remind the reader that τ ∆ = b τ ◦ Ψ where Ψ : N → M is th e canonical exp ectati on, s o that τ ∆ restricted to M is b τ . 22 Theorem 5.5. L et ( A , H , D ) b e the (1 , ∞ ) -summable, mo dular sp e ctr al triple for the Cuntz algebr a we have c onstructe d pr eviously. Then for any mo dular unitary such that the diﬀer enc e of e ta terms η ǫ ( u D u ∗ ) − η ǫ ( D ) and b τ ([k er D ]) − b τ ([k er u D u ∗ ]) vanishes, and for any Dixmier tr ac e b τ ˜ ω asso ciate d to b τ , we have sp e ctr al ﬂow as an actual limit sf b τ ( D , u D u ∗ ) = 1 2 lim s → 1+ ( s − 1) b τ ( u [ D , u ∗ ](1 + D 2 ) − s/ 2 ) = 1 2 b τ ˜ ω ( u [ D , u ∗ ](1 + D 2 ) − 1 / 2 ) = τ ◦ Φ( u [ D , u ∗ ]) . The functional on A ⊗ A deﬁne d by a 0 ⊗ a 1 7→ 1 2 lim s → 1 + ( s − 1) τ ∆ ( a 0 [ D , a 1 ](1 + D 2 ) − s/ 2 ) is a σ -twiste d b, B -c o cycle (se e the pr o of b elow f or the deﬁnition). Pr o of. First we ob s erv e that by [CP S 2, Lemma 6.1], the diﬀerence (1 + ( D + tu [ D , u ∗ ]) 2 ) − s/ 2 − (1 + D 2 ) − s/ 2 is unif orm ly b ounded in trace class norm for t ∈ [0 , 1] and s ∈ (1 , 4 / 3). Hence in the sp ectral ﬂ o w form u la (8), by th e simp le change of v ariable r = 1 / 2( s − 1) , we may write C s/ 2 sf b τ ( D , u D u ∗ ) = b τ ( u [ D , u ∗ ](1 + D 2 ) − s/ 2 ) + remainder . Where th e r emainder is b ounded as s → 1 + . Mu ltiplying th is equ ation by ( s − 1) / 2 and taking the limit as s → 1 + recalling that C s/ 2 = Γ(( s − 1) / 2)Γ( 1 / 2) Γ( s/ 2) so that ( s − 1) / 2 C s/ 2 → 1 , w e get: sf b τ ( D , u D u ∗ ) = 1 2 lim s → 1 + ( s − 1) b τ ( u [ D , u ∗ ](1 + D 2 ) − s/ 2 ) . No w by the pro of of Lemma 3.16, u [ D , u ∗ ] is in A = O nc and since it is also in M it is in F c and s o b y Prop osition 3.14 th is last limit equals τ ◦ Φ( u [ D , u ∗ ]) as claimed. T o see that we obtain a σ -t wisted co cycle, we denote by θ the functional θ ( a 0 , a 1 ) = 1 2 lim s → 1 + ( s − 1) τ ∆ ( a 0 [ D , a 1 ](1 + D 2 ) − s/ 2 ) , and observe that b y the pro of of Lemma 3.16 the elemen ts [ D , a 1 ] are in A an d so b y Pr op osition 3.14 w e see that not only do these limits exist, but in fact, θ ( a 0 , a 1 ) = τ ◦ Φ( a 0 [ D , a 1 ]) . By deﬁnition B σ θ ( a 0 ) = θ (1 , a 0 ) , and so by Lemma 3.16 ( B σ θ )( a 0 ) = lim s → 1+ ( s − 1) τ ∆ ([ D , a 0 ](1 + D 2 ) − 1 / 2 − r ) = 0 By deﬁnition, b σ θ ( a 0 , a 1 , a 2 ) = θ ( a 0 a 1 , a 2 ) − θ ( a 0 , a 1 a 2 ) + θ ( σ ( a 2 ) a 0 , a 1 ) = − τ ◦ Φ ( a 0 [ D , a 1 ] a 2 ) + τ ◦ Φ( σ ( a 2 ) a 0 [ D , a 1 ]) This is 0 by the KMS condition (see the Remark prior to Corollary 3.4) for the state ψ = τ ◦ Φ . Thus, b oth b σ θ = 0 and B σ θ = 0 , and w e’re d one.  Remark . Sp ectral ﬂo w in this s etting is indep endent of the path joinin g the endp oin ts of unb ounded self adjoin t op erators aﬃliated to M ho wev er it is n ot obvious that this is enough to sh o w that it is constan t on homotop y classes of mo du lar un itaries. This latter f act is true but the pro of is lengthy and w e defer it until we ha ve a fuller un derstanding of the stru cture of the mo dular unitaries. 23 Theorem 5.6. We let ( O nc ⊗ M 2 , H ⊗ C 2 , D ⊗ 1 2 ) b e the mo dular sp e ctr al triple of ( O nc ⊗ M 2 ) and u a mo dular unitary of the form u µ,ν =  1 − P µ S µ S ∗ ν S ν S ∗ µ 1 − P ν  . Then the sp e ctr al ﬂow is p ositive b eing given by sf τ ∆ ( D , u D u ∗ ) = ( | µ | − | ν | )( n −| ν | − n −| µ | ) ∈ ( n − 1) Z [1 / n ] Pr o of. Once we ha ve veriﬁed that the diﬀerence of eta terms and the diﬀerence of kernel corrections v anish , th is is just a computation. In fact, by Lemma 5.3, η ǫ ( u D u ∗ ) = τ ( σ ( u ∗ ) u ) η ǫ ( D ) = τ ( σ ( u ∗ ) u ) Z ∞ ǫ X k ∈ Z k e − tk 2 ! dt = 0 = Z ∞ ǫ X k ∈ Z k e − tk 2 ! dt = η ǫ ( D ) . F or the kernel corrections we use Lemma 5.3 and ﬁrst compu te 1 − σ ( u ∗ v ) u v , n oting th at σ ( v )(1 − v ∗ v ) = σ ( v ) σ (1 − v ∗ v ) = σ ( v − v v ∗ v ) = 0 . 1 − σ ( u ∗ v ) u v = 1 − σ ( u v ) u v =  v ∗ v − σ ( v ∗ ) v 0 0 v v ∗ − σ ( v ) v ∗  . F or τ (1 − σ ( u ∗ v ) u v ) w e use the KMS prop ert y of ψ = τ ◦ Φ: τ (1 − σ ( u ∗ v ) u v ) = τ ( v ∗ v − σ ( v ∗ ) v ) + τ ( v v ∗ − σ ( v ) v ∗ ) = τ ( v ∗ v − v v ∗ ) + τ ( v v ∗ − v ∗ v ) = 0 . Hence b oth the eta terms and k ern el corrections v anish, and the sp ectral ﬂo w can b e computed from the in tegral of the exact one form of Lemma 5.4. F or the computation we use a calculation in the pro of of Lemma 3.16 to get u µ,ν [ D ⊗ 1 2 , u µ,ν ] =  1 − P µ S µ S ∗ ν S ν S ∗ µ 1 − P ν   0 [ D , S µ S ∗ ν ] [ D , S ν S ∗ µ ] 0  =  1 − P µ S µ S ∗ ν S ν S ∗ µ 1 − P ν   0 ( | µ | − | ν | ) S µ S ∗ ν ( | ν | − | µ | ) S ν S ∗ µ 0  = ( | µ | − | ν | )  − P µ 0 0 P ν  . So u sing Theorem 5.5 and our pr evious compu tation of the Dixmier trace, Pr op osition 3.12, w e h a ve sf τ ∆ ( D , u µ,ν D u µ,ν ) = ( | µ | − | ν | ) τ ( P ν − P µ ) = ( | µ | − | ν | )( n −| ν | − n −| µ | ) . This num b er is alwa ys p ositiv e as the reader may c hec k , and is con tained in ( n − 1) Z [1 /n ], the in teger p olynomials in 1 /n all of w hose co eﬀﬁ cients ha ve a factor of ( n − 1).  Remarks. W e ob s erv e that sin ce th is unitary u µ,ν is self-adjoin t th e sp ectral ﬂow cannot b e inte r - preted simp ly as the ind ex of the T o eplitz compression of u µ,ν b y the non-negativ e sp ectral pr o jection of D ⊗ 1 2 : for one thing this “T o eplitz compression” is not in M and if it were in M its index w ould ha ve to b e 0. Next we use the viewp oin t pro vid ed by the noncomm u tativ e APS theory of [CPR]. This giv es a partial explanation of the numerical v alues of th e sp ectral ﬂ o w obtained for the Cuntz algebras. Corollary 5.7. L et ( O nc ⊗ M 2 , H ⊗ C 2 , D ⊗ 1 2 ) b e the mo dular sp e ctr al triple of the the or em and u a mo dular u nitary of the form u v , wher e v = S µ S ∗ ν so that v ∗ v = S ν S ∗ ν and v v ∗ = S µ S ∗ µ ar e b oth in F . L et ( X , D ⊗ 1 2 ) b e the Kasp ar ov mo dule for O n ⊗ M 2 , F ⊗ M 2 describ e d e arlier. Then fr om the p airing K 0 ( M ( F ⊗ M 2 , O n ⊗ M 2 )) × ( X , D ⊗ 1 2 ) → K 0 ( F ) we have the classes of the pr oje ctions Index( P v P v ∗ : v P v ∗ ( X ) → v v ∗ P ( X )) and In dex( P v ∗ P v : v ∗ P v ( X ) → v ∗ v P ( X )) ∈ K 0 ( F ) . 24 These two classes ar e ne gatives of e ach other in K 0 ( F ) , but sf τ ∆ ( D , u v D u v ) = τ ∆ (Index( P v P v ∗ : v P v ∗ X → v v ∗ P X )) + τ ∆ (Index( P v ∗ P v : v ∗ P v X → v ∗ v P X )) , wher e her e we apply τ ∆ to the diﬀer enc e of pr oje ctions deﬁning the i ndex as a diﬀer enc e of F -mo dules. Pr o of. In [CPR] Lemma 3.5 and Theorem 5.1, the authors u s ed th e follo w in g op er ators and indices: Index( P v P : v ∗ v P ( X ) → v v ∗ P ( X )) and Index( P v ∗ P : v v ∗ P ( X ) → v ∗ v P X ) . Eac h index is the exact negativ e of the other, and so if we ev aluate b oth with τ ∆ and add w e get exactly 0 . The K 0 ( F ) elemen ts giv en by the indices of these tw o op erators are the same as the ones considered in this Corollary . Ho we ver, th e p oin t of view of this C orollary is to consider map p ings f r om sa y the non-negativ e sp ectral sub space of v D v ∗ (i.e., v P v ∗ ( X )) to the non-negativ e sp ectral sub space of v v ∗ D (i.e., v v ∗ P ( X )). Here w e get quite a diﬀerent answer. Let v = S µ S ∗ ν , and m = | µ | − | ν | , m > 0; so that v v ∗ = S µ S ∗ µ . A sim p le computation on mon omials S α S ∗ β giv es us the k ey fact that : v Φ k v ∗ = v v ∗ Φ k + m for all k ∈ Z . T his easily implies that v P v ∗ = v v ∗ ( P k ≥ m Φ k ) ≤ v v ∗ P so that ( P v P v ∗ ) v P v ∗ = v P v ∗ and so k er ( P v P v ∗ ) = { 0 } . This also sh ows that: cok ernel ( P v P v ∗ : v P v ∗ ( X ) → v v ∗ P ( X )) = v v ∗ P ( X ) ⊖ v v ∗ ( X k ≥ m Φ k )( X ) = m − 1 X k =0 v v ∗ Φ k ( X ) . Similarly , v ∗ P v = P k ≥− m v ∗ v Φ k ≥ v ∗ v P so that ( P v ∗ P v ) v ∗ P v = v ∗ v P a n d so P v ∗ P v is onto v ∗ v P ( X ) . T h at is, coke r nel( P v ∗ P v ) = { 0 } . This also sho ws that k ernel ( P v ∗ P v : v ∗ P v ( X ) → v ∗ v P ( X )) = X k ≥− m v ∗ v Φ k ( X ) ⊖ v ∗ v P ( X ) = − 1 X k = − m v ∗ v Φ k ( X ) . T o see that these indices are negativ es in K 0 ( F ) it suﬃces to see th e equiv alence b et we en the t wo pro jections P m − 1 k =0 v v ∗ Φ k and P − 1 k = − m v ∗ v Φ k . Th is is obtained f r om our key fact ab o ve: ( v Φ k )(Φ k v ∗ ) = v v ∗ Φ k + m , (Φ k v ∗ )( v Φ k ) = v ∗ v Φ k . This is of course the Mu rra y-von Neumann equiv alence whic h τ ∆ do es not resp ect. Assume then that m > 0. Applying τ ∆ w e hav e τ ∆ (Index( P v P v ∗ )) = − mτ ( v v ∗ ) = − m n | µ | , while τ ∆ (Index( P v ∗ P v )) = mτ ( v ∗ v ) = m n | ν | . The case m < 0 is s imilar.  Remark . This Corollary mak es it clear that our n ew ind ex pairings are non-trivial precisely b ecause τ ∆ do es not indu ce a map on K 0 ( E nd 0 F ( X )). Of course b τ ω just b ecomes the trace on elemen ts of F , but τ ∆ is a weig ht on N and so on E nd 0 F ( X ), whic h is Morita equ iv alent to F . Ho w ever, s in ce τ ∆ is not a trace on N it do es not resp ect all Mu rra y-von Neumann equiv alences in N , and so do es not giv e a we ll-deﬁn ed map on K -theory . So we ma y think of the sp ectral ﬂ o w inv arian t asso ciated to u µ,ν as a measure of the failure of τ ∆ to resp ect the Murra y-von Neumann equiv alence b etw een Index( P v P v ∗ ) and − Index( P v ∗ P v ). 25 More generally w e h a v e sf τ ∆ ( D 2 , u v D 2 u v ) = Ind ex τ ∆ ( P 2 u v P 2 u v ) = In dex τ ∆  (1 − v v ∗ ) P + P v P v ∗ 0 0 (1 − v ∗ v ) P + P v ∗ P v  . Since P (1 − v v ∗ ) = (1 − v v ∗ ) P is an isomorp hism from (1 − v v ∗ ) P X to itself, and similarly f or (1 − v ∗ v ) P , w e see th at the index is precisely the su m of the indices of P v P v ∗ from v P v ∗ X to v v ∗ P X , and P v ∗ P v from v ∗ P v X to v ∗ v P X . Hence the sp ectral ﬂo w for mo d ular u n itaries of the f orm u v arises p recisely b ecause τ ∆ do es not induce a homomorphism on K 0 ( E nd 0 F ( X )). Our argumen ts here rely on the v anish ing of the diﬀerence of eta terms and k ern el corrections. In the general case these eta and ke r nel terms cont r ibute and ma y h a ve cohomologi cal signiﬁcance. W e will return to th is more general set u p in a fu ture work. 6. Conclud ing R emarks 6.1. Relativ e entrop y. In this sub section w e giv e a p hysical in terp retation of our index. Let u b e a mo dular unitary o ver O n . Recall that ψ is the state on O n deﬁned b y ψ = τ ◦ Φ . Let ψ u b e the state ψ ◦ Adu on O n deﬁned by ψ u ( a ) = ψ ( u ∗ au ) , a ∈ A . The m o dular group for ψ u is t → u ∆ it u ∗ t ∈ R . The relativ e ent r op y of a p air of KMS s tates on a v on Neumann algebra was introd u ced by Araki [Ar] (it uses explicitly a cyclic and separating v ector). The Hilb ert space H = L 2 ( O n , ψ ) has a cyclic and separating ve ctor for the action of O n . In fact th is v ector remains cyclic and separating for the wea k closure π ( O n ) ′′ in N of π ( O n ). It may b e thought of as the identit y elemen t in O n but we w ill use the notation Ω b ecause of the p oten tial for confus ion. F or a ∈ O n , ψ ( a ) = h Ω , π ( a )Ω i so that we ma y wr ite ψ ( T ) = h Ω , T Ω i for all T ∈ π ( O n ) ′′ . So we can regard ψ u and ψ as a pair of KMS states on π ( O n ) ′′ . Th en the r elativ e entrop y of ψ u and ψ is [Ar] S ( ψ u , ψ ) = −h Ω , log ( u ∆ u ∗ )Ω i . This can b e written as S ( ψ u , ψ ) := − ψ ( u (log ∆) u ∗ − log ∆) This is b ecause ∆Ω = Ω implies that (log ∆)(Ω) = 0 . Now w e can relate the relativ e entrop y for this pair of K MS states on the wea k closure of π ( O n ) to sp ectral ﬂo w f or the Cuntz algebra example when w e hav e a mo dular u nitary u . W e just use th e f ormula log ∆ = − (log n ) D and then b y Theorem 5.5 w e see that this relativ e ent r op y is ju s t (log n ) ψ ( u D u ∗ − D ) = (log n ) ψ ( u [ D , u ∗ ]) = (log n ) τ ◦ Φ( u [ D , u ∗ ]) = (log n ) sf ( D , u D u ∗ ) . That is, the relativ e entrop y is j ust log n times the sp ectral ﬂo w from D to u D u ∗ . W e r emark that the relativ e entrop y is alw ays p ositiv e [Ar]. 6.2. Manifold structures. In [PRS2] it w as shown that m any of th e (tracial) examples of s emiﬁnite sp ectral triples constructed for graph and k -graph algebras satisﬁed n atural generalisatio n s of Conn es’ axioms for n oncomm utativ e manif olds , [C1]. Muc h of the discussion of [PRS2] can b e applied ve r batim to the triple ( O nc , H , D ) constr u cted here. F or in stance the axiom of ﬁniteness is obvious, as is Morita equ iv alence (sp in c ), ﬁrs t order condition, regularit y (or QC ∞ ), and irredu cibilit y . The realit y , or sp in , condition can b e pr o ve d as in [PRS 2], and w e hav e pro ven the closedness condition in Lemma 3.16. 26 The c hief diﬀerences come from the summability/ d imension/absolute con tinuit y and crucial orient abil- it y conditions. W e h a ve a version of sum m abilit y satisﬁed s ince (1 + D 2 ) − 1 / 2 ∈ L (1 , ∞ ) ( M , b τ ) , and for a ∈ O nc nonzero and p ositiv e, lim s → 1 + ( s − 1) τ ∆ ( a (1 + D 2 ) − s/ 2 ) = lim s → 1 + ( s − 1) b τ (Ψ( a )(1 + D 2 ) − s/ 2 ) = 2 ψ ( a ) > 0 . Moreo v er, we h av e a twisted Ho c hschild cycle satisfying the (t wisted) orien tabilit y condition, and moreo v er it is giv en by the same formula as in the tracial case. This co cycle is c = 1 n n X j =1 S ∗ j ⊗ S j . W e hav e t wo prop erties to c h ec k: that it is indeed a co cycle, and that it is r epresent ed by the iden tity op erator on H . Applying the t w isted Ho c hschild b oundary gives b σ c = 1 n n X j =1 ( S ∗ j S j − σ ( S j ) S ∗ j ) = 1 n n X j =1 (1 − nS j S ∗ j ) = 1 − n X j =1 S j S ∗ j = 0 . This Ho c hschild cycle is r epresen ted on H by π ( c ) = 1 n n X j =1 S ∗ j [ D , S j ] = 1 n n X j =1 S ∗ j S j = 1 n n X j =1 1 = 1 . Hence c has the requ ired represen tation prop erties, and the replacemen t of the Ho chsc hild th eory with its t wisted analogue has provi d ed us with an orient ation cycle for the ‘mo dular sp ectral triple’ of the Cuntz alge b ra. Thus Cuntz algebras m a y b e a protot yp e for ‘t yp e I I I noncomm utativ e one dimensional manifolds’. 6.3. Outlo ok. Ther e are many u nresolv ed issues r aised b y these examples of an index theory for the KMS state on the Cuntz algebra. 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Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras

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