arXiv:1008.1212v1 [math.OA] 6 Aug 2010 aldteflainagbao ( of algebra foliation the called ati a none nta eri ofrnei brofc [ Oberwolfach in conference a in year that in announced was it fact ruodo h oito n sa is and foliation the of groupoid eoetesaeo evso ( of leaves of space the denote oue1,2008 10, Volume Proceedings Mathematics Clay erdcdtetx fhsasrc eo.A tapasi i repo his in appears ( it Let cam As theory spaces. cohomology foliated below. cyclic on introduce abstract theory to his motivations main of Connes’ text the reproduced s ngnrl ihysnua pc n nnnomttv geometry noncommutative space in quotient and space the singular replaces highly a general, in is, eataseslyelpi prtro ( on operator elliptic transversally a be K C ∞ 0 .Telclidxfruaadbyn 16 cohomology beyond cyclic References and Hopf formula index local 6. The modules 5. Cyclic 4. .From cohomology 3. Cyclic 2. Introduction 1. ( ( A ylccohomology Cyclic 2010 V, ,i neeeto the of element an is ), F in occi ooooyter ntecus fhswr on years. work 30 his past of course the the over in geometry theory cohomology cyclic to tions Abstract. ahmtc ujc Classification. Subject Mathematics hc ly h oeo h ler fsot ucin on functions smooth of algebra the of role the plays which ) K hmlg occi ooooy10 cohomology cyclic to -homology hr uvyo ylcCohomology Cyclic of Survey Short A eiae ihamrto n ffcint li Connes Alain to affection and admiration with Dedicated hsi hr uvyo oeapcso li ons contri Connes’ Alain of aspects some of survey short a is This a icvrdb li onsn ae hn18 n in and 1981 than later no Connes Alain by discovered was K V/ ter of -theory V, V, aodKhalkhali Masoud F V, F .Introduction 1. F .Ti pc,wt t aua utettopology, quotient natural its with space, This ). F ihannomttv algebra noncommutative a with .I stecnouinagbao h holonomy the of algebra convolution the is It ). Contents eacmatflae aiodadlet and manifold foliated compact a be ) C ∗ V, agba thsadnesubalgebra dense a has It -algebra. A rmr 83;Scnay1D5 60,18G30. 16T05, 19D55, Secondary 58B34; Primary 1 hssol ecmae ihtefamily the with compared be should This . F .Teaayi ne of index analytic The ). noncommutative c 00Msu Khalkhali Masoud 2010 D A index( , rmindex from e V/ = n usually one bu- 5 t n of one rt, .Ihave I ]. F C Let . ∗ ( V, D A V/ ) F 28 21 13 D F = ∈ 3 1 ) 2 MASOUD KHALKHALI index theorem [1] where the analytic index of a family of fiberwise elliptic operators is an element of the K-theory of the base. Connes realized that to identify this class by a cohomological expression it would be necessary to have a noncommutative analogue of the Chern character, i.e., a map from K0( ) to a, then unknown, cohomology theory for the noncommutative algebra .A This theory, now known as cyclic cohomology, would then play the role of theA noncommutative analogue of de Rham homology of currents for smooth manifolds. Its dual version, cyclic homology, corresponds, in the commutative case, to de Rham cohomology. Connes arrived at his definition of cyclic cohomology by a careful analysis of the algebraic structures deeply hidden in the (super)traces of products of commutators of operators. These expressions are directly defined in terms of an elliptic operator and its parametrix and give the index of the operator when paired witha K-theory class. In his own words [5]: “The transverse elliptic theory for foliations requires as a preliminary step a purely algebraic work, of computing for a noncommutative algebra the cohomology of the following complex: n-cochains are multilinear functions A ϕ(f 0,...,f n) of f 0,...,f n where ∈ A ϕ(f 1,...,f 0) = ( 1)nϕ(f 0,...,f n) − and the boundary is bϕ(f 0,...,f n+1) = ϕ(f 0f 1,...,f n+1) ϕ(f 0,f 1f 2,...,f n+1)+ − +( 1)n+1ϕ(f n+1f 0,...,f n). − The basic class associated to a transversally elliptic operator, for = the algebra of the foliation, is given by: A ϕ(f 0,...,f n)= Trace (εF [F,f 0][F,f 1] [F,f n]), f i ∈ A where 0 Q 1 0 F = , ε = , P 0 0 1 − and Q is a parametrix of P . An operation S : Hn( ) Hn+2( ) A → A is constructed as well as a pairing K( ) H( ) C A × A → where K( ) is the algebraic K-theory of A. It gives the index of the operator from its associatedA class ϕ. Moreover e, ϕ = e, Sϕ , so that the important group n to determine is the inductive limit Hp = Lim H ( ) for the map S. Using the → A tools of homological algebra the groups Hn( , ∗) of Hochschild cohomology with coefficients in the bimodule ∗ are easierA toA determine and the solution of the problem is obtained in two steps:A 1) the construction of a map B : Hn( , ∗) Hn−1( ) A A → A and the proof of a long exact sequence Hn( , ∗) B Hn−1( ) S Hn+1( ) I Hn+1( , ∗) → A A → A → A → A A → where I is the obvious map from the cohomology of the above complex to the Hochschild cohomology; ASHORTSURVEYOFCYCLICCOHOMOLOGY 3
2) the construction of a spectral sequence with E2 term given by the cohomology of the degree 1 differential I B on the Hochschild groups Hn( , ∗) and which converges strongly− to a graded◦ group associated to the inductive limit.A A This purely algebraic theory is then used. For = C∞(V ) one gets the de Rham homology of currents, and for the pseudo-torus,A i.e. the algebra of the Kronecker foliation, one finds that the Hochschild cohomology depends on the Diophantine na- 0 1 ture of the rotation number while the above theory gives Hp of dimension 2 and Hp of dimension 2, as expected, but from some remarkable cancellations.”
A full exposition of these results later appeared in two IHES preprints [6], and were eventually published as [9]. With the appearance of [9] one could say that the first stage of the development of noncommutative geometry and specially cyclic cohomology reached a stage of maturity. In the next few sections I shall try to give a quick and concise survey of some aspects of cyclic cohomology theory as they were developed in [9]. The last two sections are devoted to developments in the subject after [9] arising from the work of Connes. It is a distinct honor and a great pleasure to dedicate this short survey of cyclic cohomology theory as a small token of our friendship to Alain Connes, the originator of the subject, on the occasion of his 60th birthday. It inevitably only covers part of what has been done by Alain in this very important corner of noncommutative geometry. It is impossible to cover everything, and in particular I have left out many important topics developed by him including, among others, the Godbillon- Vey invariant and type III factors [8], the transverse fundamental class for foliations [8], the Novikov conjecture for hyperbolic groups [18], entire cyclic cohomology [10], and multiplicative characteristic classes [12]. Finally I would like to thank Farzad Fathi zadeh for carefully reading the text and for several useful comments, and Arthur Greenspoon who kindly edited the whole text.
2. Cyclic cohomology Cyclic cohomology can be defined in several ways, each shedding light on a different aspect of it. Its original definition [5, 9] was through a remarkable sub- complex of the Hochschild complex that we recall first. By algebra in this paper we mean an associative algebra over C. For an algebra let A Cn( ) = Hom( ⊗(n+1), C), n =0, 1,..., A A denote the space of (n + 1)-linear functionals on . These are our n-cochains. The Hochschild differential b : Cn( ) Cn+1( ) is definedA as A → A n (bϕ)(a ,...,a ) = ( 1)iϕ(a ,...,a a ,...,a ) 0 n+1 − 0 i i+1 n+1 i=0 +( 1)n+1ϕ(a a ,...,a ). − n+1 0 n The cohomology of the complex (C∗( ),b) is the Hochschild cohomology of with coefficients in the bimodule ∗. A A The following definition isA fundamental and marks the departure from Hochschild cohomology in [5, 9]: Definition 2.1. An n-cochain ϕ Cn( ) is called cyclic if ∈ A ϕ(a ,a ,...,a ) = ( 1)nϕ(a ,a ,...,a ) n 0 n−1 − 0 1 n 4 MASOUD KHALKHALI for all a ,...,a in . The space of cyclic n-cochains will be denoted by Cn( ). 0 n A λ A Just why, of all possible symmetry conditions on cochains, the cyclic property is a reasonable choice is at first glance not at all clear. Lemma 2.1. The space of cyclic cochains is invariant under the action of b, i.e., b Cn( ) Cn+1( ) for all n 0. λ A ⊂ λ A ≥ To see this one introduces the operators λ : Cn( ) Cn( ) and b′ : Cn( ) Cn+1( ) by A → A A → A n (λϕ)(a0,...,an) = ( 1) ϕ(an,a0,...,an−1), −n (b′ϕ)(a ,...,a ) = ( 1)iϕ(a ,...,a a ,...,a ), 0 n+1 − 0 i i+1 n+1 i=0 ′ ∗ and checks that (1 λ)b = b (1 λ). Since Cλ( ) = Ker(1 λ), the lemma is proved. − − A − We therefore have a subcomplex of the Hochschild complex, called the cyclic complex of : A (1) C0( ) b C1( ) b C2( ) b . λ A −→ λ A −→ λ A −→ Definition 2.2. The cohomology of the cyclic complex (1) is the cyclic cohomology of and will be denoted by HCn( ), n =0, 1, 2,.... A A And that is Connes’ first definition of cyclic cohomology. A cocycle for the cyclic cohomology group HCn( ) is called a cyclic n-cocycle on . It is an (n +1)- linear functional ϕ on whichA satisfies the two conditions: A A (1 λ)ϕ =0, and bϕ =0. − The inclusion of complexes (C∗( ), b) ֒ (C∗( ), b) (2) λ A → A induces a map I from cyclic cohomology to Hochschild cohomology: I : HCn( ) HHn( ), n =0, 1, 2,.... A −→ A A closer inspection of the long exact sequence associated to (2), yields Connes’ long exact sequence relating Hochschild cohomology to cyclic cohomology. This is however easier said than done. The reason is that to identify the cohomology of the quotient one must use another long exact sequence, and combine the two long exact sequences to obtain the result. To simplify the notation, let us denote the Hochschild and cyclic complexes by C and Cλ, respectively. Then (2) gives us an exact sequence of complexes (3) 0 C C π C/C 0. → λ → → λ → Its associated long exact sequence is (4) HCn( ) HHn( ) Hn(C/C ) HCn+1( ) −→ A −→ A −→ λ −→ A −→ n We need to identify the cohomology groups H (C/Cλ). To this end, consider the short exact sequence of complexes
(5) 0 C/C 1−λ (C,b′) N C 0, −→ λ −→ −→ λ −→ ASHORTSURVEYOFCYCLICCOHOMOLOGY 5 where the operator N is defined by
N =1+ λ + λ2 + + λn : Cn Cn. −→ The relations (1 λ)b = b′(1 λ), N(1 λ) = (1 λ)N = 0, and bN = Nb′ show that 1 λ and− N are morphisms− of− complexes− in (5). As for the exactness of (5), the only− nontrivial part is to show that ker (N) im (1 λ), which can be verified. Assuming is unital, the middle complex (C, b⊂′) in (5)− can be shown to be A n n−1 exact with a contracting homotopy s : C C defined by (sϕ)(a0,...,an−1)= n−1 →′ ′ ( 1) ϕ(a0,...,an−1, 1), which satisfies b s + sb = id. The long exact sequence associated− to (5) looks like (6) n n n n+1 n+1 H (C/C ) H ′ (C) HC ( ) H (C/C ) H ′ (C) −→ λ −→ b −→ A −→ λ −→ b −→ n Since Hb′ (C) = 0 for all n, it follows that the connecting homomorphism
(7) δ : HCn−1( ) Hn(C/C ) A → λ is an isomorphism for all n 0. Using this in (4), we obtain Connes’ long exact sequence relating Hochschild≥ and cyclic cohomology:
(8) HCn( ) I HHn( ) B HCn−1( ) S HCn+1( ) . −→ A −→ A −→ A −→ A −→ The operators B and S play a prominent role in noncommutative geometry. As we shall see, the operator B is the analogue of de Rham’s differential in the noncommutative world, while the periodicity operator S is closely related to Bott periodicity in topological K-theory. Remarkably, there is a formula for B on the level of cochains given by B = NB , where B : Cn Cn−1 is defined by 0 0 → B ϕ(a ,...,a )= ϕ(1,a ,...,a ) ( 1)nϕ(a ,...,a , 1). 0 0 n−1 0 n−1 − − 0 n−1 Using the relations (1 λ)b = b′(1 λ), (1 λ)N = N(1 λ)=0, bN = Nb′, and sb′ + b′s = 1, one shows− that − − −
(9) bB + Bb =0, and B2 =0.
Using the periodicity operator S, the periodic cyclic cohomology of is then defined as the direct limit of cyclic cohomology groups under the operatorA S:
HP i( ) := Lim HC2n+i( ), i =0, 1. A −→ A
Notice that since S has degree 2, there are only two periodic groups. These periodic groups have better stability properties compared to cyclic cohomology groups. For example, they are homotopy invariant, and they pair with K-theory. A much deeper relationship between Hochschild and cyclic cohomology groups is encoded in Connes’ (b,B)-bicomplex and the associated Connes spectral sequence that we shall briefly recall now. Consider the relations (9). The (b,B)-bicomplex 6 MASOUD KHALKHALI of a unital algebra , denoted by ( ), is the bicomplex A B A ...... C2( ) B C1( ) B C0( ) A −−−−→ A −−−−→ A b b B C1( ) C0( ) A −−−−→ A b C0( ) A As usual, there are two spectral sequences attached to this bicomplex. The fol- lowing fundamental result of Connes [9] shows that the spectral sequence obtained from filtration by rows converges to cyclic cohomology. Notice that the E1 term of this spectral sequence is the Hochschild cohomology of . A Theorem 2.1. The map ϕ (0,..., 0, ϕ) is a quasi-isomorphism of complexes → (C∗( ), b) (Tot ( ), b + B). λ A → B A This is a consequence of the vanishing of the E2 term of the second spectral sequence (filtration by columns) of ( ). To prove this, Connes considers the short exact sequence of b-complexes B A 0 Im B Ker B Ker B/Im B 0, −→ −→ −→ −→ and proves that ([9], Lemma 41), the induced map H (Im B) H (Ker B) b −→ b is an isomorphism. This is a very technical result. It follows that Hb(Ker B/Im B) vanishes. To take care of the first column one appeals to the fact that Im B Ker(1 λ) is the space of cyclic cochains. ≃ We− give an alternative proof of Theorem (2.1) above. To this end, consider the cyclic bicomplex ( ) defined by C A ...... C2( ) 1−λ C2( ) N C2( ) 1−λ A −−−−→ A −−−−→ A −−−−→ ′ b −b b 1−λ N 1−λ C1( ) C1( ) C1( ) A −−−−→ A −−−−→ A −−−−→ ′ b −b b 1−λ N 1−λ C0( ) C0( ) C0( ) A −−−−→ A −−−−→ A −−−−→ The total cohomology of ( ) is isomorphic to cyclic cohomology: C A Hn(Tot ( )) HCn( ), n 0. C A ≃ A ≥ This is a consequence of the simple fact that the rows of ( ) are exact except in degree zero, where their cohomology coincides with the cyclicC A complex (C∗( ),b). λ A ASHORTSURVEYOFCYCLICCOHOMOLOGY 7
So it suffices to show that Tot ( ) and Tot ( ) are quasi-isomorphic. This can be done by explicit formulas. ConsiderB A the chainC A maps I : Tot ( ) Tot ( ), I = id + Ns, B A → C A J : Tot ( ) Tot ( ), J = id + sN. C A → B A It can be directly verified that the following operators define chain homotopy equiv- alences: g : Tot (A) Tot (A), g = Ns2B , B → B 0 h : Tot (A) Tot ( ), h = s. C → C A To give an example of an application of the spectral sequence of Theorem (2.1), let me recall Connes’ computation of the continuous cyclic cohomology of the topological algebra = C∞(M), i.e., the algebra of smooth complex valued functions on a closed smoothA n-dimensional manifold M. This example is important since, apart from its applications, it clearly shows that cyclic cohomology is a noncommutative analogue of de Rham homology. The continuous analogues of Hochschild and cyclic cohomology for topological algebras are defined as follows [9]. Let be a topological algebra. A continuous cochain on is a jointly continuous multilinearA map ϕ : C. By workingA with just continuous cochains, as opposed to allA×A× ×A→ cochains, one obtains the continuous analogues of Hochschild and cyclic cohomology groups. In working with algebras of smooth functions (both in the commutative and noncommutative case), it is essential to use this continuous analogue. The topology of C∞(M) is defined by the sequence of seminorms f = sup ∂α f , α n, n | | | |≤ where the supremum is over a fixed, finite, coordinate cover for M. Under this topology, C∞(M) is a locally convex, in fact nuclear, topological algebra. Similarly one topologizes the space of p-forms on M for all p 0. Let ≥ p ΩpM := Homcont(Ω M, C) denote the continuous dual of the space of p forms on M. Elements of ΩpM are called de Rham p-currents. By dualizing the− de Rham differential d, we obtain a ∗ differential d : Ω∗M Ω∗−1M, and a complex, called the de Rham complex of currents on M: → ∗ ∗ ∗ Ω M d Ω M d Ω M d . 0 ←− 1 ←− 2 ←− The homology of this complex is the de Rham homology of M and we denote it by dR H∗ (M). It is easy to check that for any m-current C, closed or not, the cochain (10) ϕ (f ,f ,...,f ) := C, f df df , C 0 1 m 0 1 m is a continuous Hochschild cocycle on C∞(M). Now if C is closed, then one checks ∞ that ϕC is a cyclic m-cocycle on C (M). Thus we obtain natural maps (11) Ω M HHm (C∞(M)) m → cont and (12) Z M HCm (C∞(M)), m → cont 8 MASOUD KHALKHALI where Zm(M) ΩmM is the space of closed m-currents on M. For example, if M is oriented and⊂C represents its orientation class, then
(13) ϕC (f0,f1,...,fn)= f0df1 dfn, M which is easily checked to be a cyclic n-cocycle on . In [9], using an explicit resolution, Connes showsA that (11) is a quasi-isomorph- ism. Thus we have a natural isomorphism between space of de Rham currents on M and the continuous Hochschild cohomology of C∞(M): (14) HHi (C∞(M)) Ω M i =0, 1,... cont ≃ i To compute the continuous cyclic homology of , one first observes that under the isomorphism (14) the operator B corresponds toA the de Rham differential d∗. More precisely, for each integer n 0 there is a commutative diagram: ≥ Ω M Cn+1( ) n+1 −−−−→ A ∗ d B n Ωn M C ( ) −−−−→ A where (C)= ϕC and ϕC is defined by (10). Then, using the spectral sequence of Theorem (2.1) and the isomorphism (14), Connes obtains [9]: (15) HCn (C∞(M)) Z (M) HdR (M) HdR(M), cont ≃ n ⊕ n−2 ⊕ ⊕ k where k = 0 if n is even and k = 1 is n is odd. For the continuous periodic cyclic cohomology he obtains (16) HP k (C∞(M)) HdR (M), k =0, 1. cont ≃ 2i+k i We shall also briefly recall Connes’ computation of the Hochschild and cyclic cohomology of smooth noncommutative tori [9]. This result already appeared in Connes’ Oberwolfach report [5]. When θ is rational, the smooth noncommutative ∞ 2 torus θ can be shown to be Morita equivalent to C (T ), the algebra of smooth functionsA on the 2-torus. One can then use Morita invariance of Hochschild and cyclic cohomology to reduce the computation of these groups to those for the algebra C∞(T 2). This takes care of rational θ. So we can assume θ is irrational and we denote the generators of θ by U and V with the relation VU = λUV , where λ = e2πiθ. A Recall that an irrational number θ is said to satisfy a Diophantine condition if 1 λn −1 = O(nk) for some positive integer k. | − | Proposition 2.1. ([9]) Let θ / Q. Then 0 ∈ a) One has HH ( θ)= C, A i b) If θ satisfies a Diophantine condition then HH ( θ) is 2-dimensional for i=1 and is 1-dimensional for i =2, A i c) If θ does not satisfy a Diophantine condition, then HH ( θ) are infinite dimen- sional non-Hausdorff spaces for i =1, 2. A Remarkably, for all values of θ, the periodic cyclic cohomology is finite dimen- sional and is given by HP 0( )= C2, HP 1( )= C2. Aθ Aθ ASHORTSURVEYOFCYCLICCOHOMOLOGY 9
An explicit basis for these groups are given by cyclic 1-cocycles
ϕ1(a0,a1)= τ(a0δ1(a1)), and ϕ1(a0,a1)= τ(a0δ2(a1)) and cyclic 2-cocycles ϕ(a ,a ,a )= τ(a (δ (a )δ (a ) δ (a )δ (a ))), and Sτ, 0 1 2 0 1 1 2 2 − 2 1 1 2 where δ ,δ : are the canonical derivations defined by 1 2 Aθ → Aθ m n m n m n m n δ1( amnU V )= mamnU V , δ2(U V )= namnU V , and τ : θ C is the canonical trace. A noncommutativeA → generalization of formulas like (13) was introduced in [9] and played an important role in the development of cyclic cohomology theory in general. It gives a geometric meaning to the notion of a cyclic cocycle over an algebra and goes as follows. Let (Ω, d) be a differential graded algebra. A closed graded trace of dimension n on (Ω, d) is a linear map
:Ωn C −→ such that
dω =0, and [ω1, ω2]=0, n−1 i j for all ω in Ω , ω1 in Ω , ω2 in Ω and i + j = n. An n dimensional cycle over an algebra is a triple (Ω, , ρ), where is an n-dimensional closed graded trace on A (Ω, d) and ρ : A Ω0 is an algebra homomorphism. Given a cycle (Ω, , ρ) over → , its character is the cyclic n-cocycle on defined by A A
(17) ϕ(a0,a1,...,an)= ρ(a0)dρ(a1) dρ(an). Conversely one shows that all cyclic cocycles are obtained in this way. Once one has the definition of cyclic cohomology, it is not difficult to formulate a dual notion of cyclic homology and a pairing between the two. Let Cn( ) = ⊗(n+1) ′ A . The analogues of the operators b,b and λ are easily defined on C∗( ) andA are usually denoted by the same letters, as we do here. For example bA: C ( ) C ( ) is defined by n A → n−1 A n−1 (18) b(a a ) = ( 1)i(a a a a ) 0 ⊗ ⊗ n − 0 ⊗ ⊗ i i+1 ⊗ ⊗ n i=0 (19) + ( 1)n(a a a ). − n 0 ⊗ ⊗ n−1 Let Cλ( ) := C ( )/Im(1 λ). n A n A − The relation (1 λ)b′ = b(1 λ) shows that the operator b is well defined on λ − λ − C∗ ( ). The complex (C∗ ( ), b) is called the homological cyclic complex of and its homology,A denoted by HCA ( ),n = 0, 1,... , is the cyclic homology of A. The n A A evaluation map ϕ, (a0 an) ϕ(a0,...,an) clearly defines a degree zero ∗ ⊗ ⊗ → pairing HC ( ) HC∗( ) C. Many results of cyclic cohomology theory, such as Connes’ long exactA ⊗ sequenceA → and spectral sequence, and Morita invariance, continue to hold for cyclic homology theory with basically the same proofs. Another important idea of Connes in the 1980’s was the introduction of entire cyclic cohomology of Banach algebras [10]. This allows one to deal with algebras of 10 MASOUD KHALKHALI functions on infinite dimensional (noncommutative) spaces such as those appearing in constructive quantum field theory. These algebras typically don’t carry finitely summable Fredholm modules, but in some cases have so-called θ-summable Fred- holm modules. In [10] Connes extends the definition of Chern character to such Fredholm modules with values in entire cyclic cohomology. After the appearance of [9], cyclic (co)homology theory took on many lives and was further developed along distinct lines, including a purely algebraic one, with a big impact on algebraic K-theory. The cyclic cohomology of many algebras was later computed including the very important case of group algebras [2] and groupoid algebras. For many of these more algebraic aspects of the theory we refer to [33, 32] and references therein.
3. From K-homology to cyclic cohomology As I said in the introduction, Connes’ original motivation for the development of cyclic cohomology was to give a receptacle for a noncommutative Chern character map on the K-homology of noncommutative algebras. The cycles of K-homology can be represented by, even or odd, Fredholm modules. Here we just focus on the odd case, and we refer to [9, 13] for the even case. Given a Hilbert space , let ( ) denote the algebra of bounded linear operators on , and ( ) denoteH the algebraL H of compact operators. Also, for 1 p< , let pH( ) denoteK H the Schatten ideal of p-summable operators. By definition,≤ T∞ pL( )H if T p is a trace class operator. ∈ L H | | Definition 3.1. An odd Fredholm module over a unital algebra is a pair ( , F ) where A H 1. is a Hilbert space endowed with a representation H π : ( ), A −→ L H 2. F ( ) is a bounded selfadjoint operator with F 2 = I, 3. For∈ all L Ha we have ∈ A (20) [F, π(a)] = F π(a) π(a)F ( ). − ∈ K H A Fredholm module ( , F ) is called p-summable if, instead of (20), we have the stronger condition: H (21) [F, π(a)] p( ) ∈ L H for all a . To give∈ A a simple example, let A = C(S1) be the algebra of continuous functions 2 1 on the circle and let A act on = L (S ) as multiplication operators. Let F (en)= H 2πinx en if n 0 and F (en) = en for n < 0, where en(x) = e ,n Z, denotes the standard≥ orthonormal basis− of . Clearly F is selfadjoint and F 2 =∈I. To show that [F, π(f)] is a compact operatorH for all f C(S1), notice that if f = a e ∈ |n|≤N n n is a finite trigonometric sum then [F, π(f)] is a finite rank operator and hence is compact. In general we can uniformly approximate a continuous function by a trigonometric sum and show that the commutator is compact for any continuous f. This shows that ( , F ) is an odd Fredholm module over C(S1). This Fredholm module is not p-summableH for any 1 p < . If we restrict it to the subalgebra C∞(S1) of smooth functions, then it≤ can be∞ checked that ( , F ) is in fact p- summable for all p> 1, but is not 1-summable even in this case.H ASHORTSURVEYOFCYCLICCOHOMOLOGY 11
Now let me describe Connes’ noncommutative Chern character from K-homology to cyclic cohomology. Let ( , F ) be an odd p-summable Fredholm module over an algebra . For any odd integerH 2n 1 such that 2n p, Connes defines a cyclic (2n 1)-cocycleA ϕ on by [9] − ≥ − 2n−1 A (22) ϕ (a ,a ,...,a ) = Tr (F [F, a ][F, a ] [F, a ]), 2n−1 0 1 2n−1 0 1 2n−1 where Tr denotes the operator trace and instead of π(a) we simply write a. Notice that by our p-summability assumption, each commutator is in p( ) and hence, by H¨older inequality for Schatten class operators, their productLisH in fact a trace class operator as soon as 2n p. One checks by a direct computation that ϕ2n−1 is a cyclic cocycle. ≥ The next proposition shows that these cyclic cocycles are related to each other via the periodicity S-operator of cyclic cohomology. This is probably how Connes came across the periodicity operator S in the first place. Proposition 3.1. For all n with 2n p we have ≥ Sϕ = (n + 1 ) ϕ . 2n−1 − 2 2n+1 By rescaling ϕ2n−1’s, one obtains a well defined element in the periodic cyclic cohomology. The (unstable) odd Connes-Chern character Ch2n−1 = Ch2n−1( , F ) of an odd finitely summable Fredholm module ( , F ) over is defined by rescalingH H A the cocycles ϕ2n−1 appropriately. Let Ch2n−1 (a ,...,a ) := ( 1)n2(n 1 ) 1 Tr (F [F,a ][F,a ] [F,a ]). 0 2n−1 − − 2 2 0 1 2n−1 Definition 3.2. The Connes-Chern character of an odd p-summable Fredholm module ( , F ) over an algebra is the class of the cyclic cocycle Ch2n−1 in the odd periodicH cyclic cohomology groupA HP odd( ). A By the above Proposition, the class of Ch2n−1 in HP odd( ) is independent of the choice of n. A
Let us compute the character of the Fredholm module of the above Example ∞ 1 1 with = C (S ). By the above definition, Ch ( , F ) = [ϕ1] is the class of the followingA cyclic 1-cocycle in HP odd( ): H A ϕ1 (f0,f1) = Tr (F [F,f0][F,f1]). One can identify this cyclic cocycle with a local formula. We claim that 4 ϕ1 (f0,f1)= f0df1, for all f0,f1 . 2πi ∈ A
By linearity, It suffices to check this relation for basis elements f0 = em,f1 = en for all m,n Z, which is easy to do. The duality,∈ that is, the bilinear pairing, between K-theory and K-homology is defined through the Fredholm index. More precisely there is an index pairing between odd (resp. even) Fredholm modules over and the algebraic K-theory alg A group K1 ( ) (resp. K0( )). We shall describe it only in the odd case at hand. Let ( , F ) beA an odd FredholmA module over and let U × be an invertible H F +1 A ∈ A element in . Let P = 2 : be the projection operator defined by F . One checksA that the operator H → H P UP : P P H → H 12 MASOUD KHALKHALI is a Fredholm operator. In fact, using the compactness of commutators [F,a], one checks that PU −1P is an inverse for P UP modulo compact operators, which of course implies that P UP is a Fredholm operator. The index pairing is then defined as ( , F ), [U] := index (P UP ), H where the index on the right hand side is the Fredholm index. If the invertible U happens to be in Mn( ) we can apply this definition to the algebra Mn( ) by n A A noticing that ( C , F 1) is a Fredholm module over Mn( ) in a natural way. The resulting mapH⊗ can be⊗ shown to induce a well defined additiveA map ( , F ), : Kalg( ) C. H − 1 A → Notice that this map is purely topological in the sense that to define it we did not have to impose any finite summability, i.e., smoothness, condition on the Fredholm module. 1 Going back to our example and choosing f : S GL1(C) a continuous func- 1 alg 1 → tion on S representing an element of K1 (C(S )), the index pairing [( , F )], [f] = index(P fP ) can be explicitly calculated. In fact in this case a simple H homotopy argument gives the index of the Toeplitz operator P fP : P P in terms of the winding number of f around the origin: H → H [(H, F )], [f] = W (f, 0). − Of course, when f is smooth the winding number can be computed by integrating 1 dz the 1-form 2πi z over the curve defined by f: 1 1 W (f, 0) = f −1df = ϕ(f −1,f) 2πi 2πi where ϕ is the cyclic 1-cocycle on C∞(S1) defined by ϕ(f,g) = fdg. This is a special case of a very general index formula proved by Connes [9] in a fully noncommutative situation: Proposition 3.2. Let ( , F ) be an odd p-summable Fredholm module over an algebra and let 2n 1 Hbe an odd integer such that 2n p. If u is an invertible elementA in A then − ≥ ( 1)n index (P uP )= − ϕ (u−1,u,...,u−1,u), 22n 2n−1
where the cyclic cocycle ϕ2n−1 is defined by ϕ (a ,a ,...,a ) = Tr (F [F,a ][F,a ] [F,a ]). 2n−1 0 1 2n−1 0 1 2n−1 The above index formula can be expressed in a more conceptual manner once Connes’ Chern character in K-theory is introduced. In [4, 9], Connes shows that the Chern-Weil definition of Chern character on topological K-theory admits a vast generalization to a noncommutative setting. For a noncommutative algebra and each integer n 0, he defined pairings between cyclic cohomology and K-theory:A ≥ (23) HC2n( ) K ( ) C, HC2n+1( ) Kalg( ) C. A ⊗ 0 A −→ A ⊗ 1 A −→ These pairings are compatible with the periodicity operator S in cyclic cohomology in the sense that [ϕ], [e] = S[ϕ], [e] , ASHORTSURVEYOFCYCLICCOHOMOLOGY 13 for all cyclic cocycles ϕ and K-theory classes [e], and thus induce a pairing HP i( ) Kalg( ) C, i =0, 1 A ⊗ i A −→ between periodic cyclic cohomology and K-theory. We briefly recall its definition. Let ϕ be a cyclic 2n-cocycle on and let A 2n e Mk( ) be an idempotent representing a class in K0( ). The pairing HC ( ) K∈( ) A C is defined by A A ⊗ 0 A −→ (24) [ϕ], [e] = (n!)−1 ϕ˜(e,...,e), whereϕ ˜ is the ‘extension’ of ϕ to M ( ) defined by the formula k A (25)ϕ ˜(m a ,...,m a ) = tr(m m )ϕ(a ,...,a ). 0 ⊗ 0 2n ⊗ 2n 0 2n 0 2n It can be shown thatϕ ˜ is a cyclic n-cocycle as well. The formulas in the odd case are as follows. Given an invertible matrix u alg ∈ Mk( ), representing a class in K1 ( ), and an odd cyclic (2n 1)-cocycle ϕ on , theA pairing is given by A − A 2−(2n+1) (26) [ϕ], [u] := ϕ˜(u−1 1,u 1,...,u−1 1,u 1). (n 1 ) 1 − − − − − 2 2 Any cyclic cocycle can be represented by a normalized cocycle for which ϕ(a0,...,an)= 0 if ai = 1 for some i. When ϕ is normalized, formula (26) reduces to a particularly simple form: 2−(2n+1) (27) [ϕ], [u] = ϕ˜(u−1,u,...,u−1,u). (n 1 ) 1 − 2 2 2n−1 alg 2n−1 Using the pairing HC ( ) K1 ( ) C and the definition of Ch (H, F ), the above index formula in PropositionA ⊗ (3.2)A → can be written as (28) index (P uP )= Ch2n−1 (H, F ), [u] , or in its stable form index (P uP )= Chodd (H, F ), [u] . This equality amounts to the equality Topological Index = Analytic Index in a fully noncommutative setting. An immediate consequence of the index formula (28) is an integrality theorem for numbers defined by the right hand side of (28). This should be compared with classical integrality results for topological invariants of manifolds that are established through the Atiyah-Singer index theorem. An early nice application was Connes’ proof of the idempotent conjecture for group C∗-algebras of free groups in [9]. Among other applications I should mention a mathematical treatment of integral quantum Hall effect, and most recently to quantum computing in the work of Mike Freedman and collaborators [29].
4. Cyclic modules With the introduction of the cyclic category Λ in [7], Connes took another major step in conceptualizing and generalizing cyclic cohomology far beyond its original inception. We already saw in the last section three different definitions of the cyclic cohomology of an algebra through explicit complexes. The original motivation of [7] was to define the cyclic cohomology of algebras as a derived functor. Since the category of algebras and algebra homomorphisms is not an additive category, 14 MASOUD KHALKHALI the standard abelian homological algebra is not applicable here. Let k be a unital commutative ring. In [7], an abelian category Λk of cyclic k-modules is defined that can be thought of as the ‘abelianization’ of the category of k-algebras. Cyclic cohomology is then shown to be the derived functor of the functor of traces, as we shall explain in this section. More generally Connes defined the notion of a cyclic object in an abelian category and its cyclic cohomology [7]. Later developments proved that this extension of cyclic cohomologywas of great significance. Apart from earlier applications, we should mention the recent work [16] where the abelian category of cyclic modules plays a role similar to that of the category of motives for noncommutative geometry. Another recent example is the cyclic cohomology of Hopf algebras [20, 21, 30, 31], which cannot be defined as the cyclic cohomology of an algebra or a coalgebra but only as the cyclic cohomology of a cyclic module naturally attached to the given Hopf algebra and a coefficient system (see the last section for more on Hopf cyclic cohomology). Let us briefly sketch the definition of the cyclic category Λ. Recall that the simplicial category ∆ is a small category whose objects are the totally ordered sets [n]= 0 < 1 <