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Documenta Math. 275 Secondary Invariants for Frechet Algebras and Quasihomomorphisms Denis Perrot Received: March 3, 2008 Communicated by Joachim Cuntz Abstract. A Frechet algebra endowed with a multiplicatively con- vex topology has two types of invariants: homotopy invariants (topo- logical K-theory and periodic cyclic homology) and secondary in- variants (multiplicative K-theory and the non-periodic versions of cyclic homology). The aim of this paper is to establish a Riemann- Roch-Grothendieck theorem relating direct images for homotopy and secondary invariants of Frechet m-algebras under finitely summable quasihomomorphisms. 2000 Mathematics Subject Classification: 19D55, 19K56, 46L80, 46L87. Keywords and Phrases: K-theory, bivariant cyclic cohomology, index theory. 1 Introduction For a noncommutative space described by an associative Frechet algebra A over C, we distinguish two types of invariants. The first type are (smooth) homotopy invariants, for example topological K-theory [27] and periodic cyclic homology [5]. The other type are secondary invariants; they are no longer stable under homotopy and carry a finer information about the “geometry” of the space A . Typical examples of secondary invariants are algebraic K-theory [29] (which will not be used here), multiplicative K-theory [17] and the unstable versions of cyclic homology [18].
Voir
- chern character
- invariants appear
- secondary invariants
- algebras
- constructions involving
- unstable versions
- algebra situation
- baum-connes construction
- index theorems
- assembly maps
Publié par
Langue
English
Documenta Math.
Secondary
Invariants for Frechet Algebras
and
Quasihomomorphisms
Denis Perrot
Received: March 3, 2008
Communicated by Joachim Cuntz
Abstract.FAlgebraenr´echetaumahpitlewodtiwdcolyn-caliveti vex topology has two types of invariants: homotopy invariants (topo-logicalK-theory and periodic cyclic homology) and secondary in-variants (multiplicativeK-theory and the non-periodic versions of cyclic homology). The aim of this paper is to establish a Riemann-Roch-Grothendieck theorem relating direct images for homotopy and secondaryinvariantsofFr´echetm-algebras under finitely summable quasihomomorphisms.
2000 Mathematics Subject Classification: 19D55, 19K56, 46L80, 46L87. Keywords and Phrases:K-theory, bivariant cyclic cohomology, index theory.
1 Introduction
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ForanoncommutativespacedescribedbyanassociativeFre´chetalgebraA overC first type are (smooth) The, we distinguish two types of invariants. homotopy invariants, for example topologicalK-theory [27] and periodic cyclic homology [5]. The other type are secondary invariants; they are no longer stable under homotopy and carry a finer information about the “geometry” of the spaceA. Typical examples of secondary invariants are algebraic K-theory [29] (which will not be used here), multiplicativeK-theory [17] and the unstable versions of cyclic homology [18]. The aim of this paper is to define push-forward maps for homotopy and secondary invariants between two Fr´echetalgebrasAandB, induced by a smooth finitely summable quasiho-momorphism [8]. The compatibility between the different types of invariants is expressed through a noncommutative Riemann-Roch-Grothendieck theorem
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(Theorem 6.3). The present paper is the first part of a series on secondary characteristic classes. In the second part we will show how to obtainlocal formulasfor push-forward maps, following a general principle inspired by renormalization which establishes the link with chiral anomalies in quan-tum field theory [25]; in order to keep a reasonable size to the present paper, these methods will be published in a separate survey with further examples [26].
WedealwithFr´echetalgebrasendowedwithamultiplicativelyconvextopology, orFr´echetm These-algebras for short. algebras can be presented as inverse lim-its of sequences of Banach algebras, and as a consequence many constructions validforBanachalgebrascarryoverFr´echetm-algebras. In particular Phillips [27] defines topologicalK-theory groupsKntop(A) for any such algebraAand n∈Z fundamental properties of interest for us are (smooth) homotopy. The invariance and Bott periodicity, i.e.Knp+2to(A) =∼Kntop(A). Hence there are es-sentially two topologicalKt-Fy´rceehtheorygroupsforanm-algebra,K0top(A) whose elements are roughly represented by idempotents in the stabilization of Aby the algebraKof ”smooth compact operators”, andKtop1(A) whose elementsarerepresentedbyinvertibles.Fre´chetm-algebras naturally arise in many situations related to differential geometry, commutative or not, and the formulation of index problems. In the latter situation one usually encounters an algebraI´rceehtof”fibammpoeletinusylor,faFusaters”orm-algebra provided with a continuous trace on itsp-th power for somep≥1. A typ-ical example is the Schatten classI=Lp(H) ofp-summable operators on an infinite-dimensional separable Hilbert spaceH.Acan be stabilized by the completed projective tensor produIˆ⊗Aand its ory ct topologicalK-the Kntop(I⊗ˆA Other important topolog-) is the natural receptacle for indices. ical invariants ofA(as a locally convex algebra) are provided by the peri-odic cyclic homology groupsH Pn(A), which is the correct version sharing the properties of smooth homotopy invariance and periodicity mod 2 with topolog-icalK For any finitely summable algebra-theory [5].Ithe Chern character Kntop(Iˆ⊗A)→H Pn(A) allows to obtain cohomological formulations of index theorems. If one wants to go beyond differential topology and detectsecondaryinvari-ants as well, which are no longer stable under homotopy, one has to deal with algebraicK In-theory [29] and the unstable versions of cyclic homology [18]. principle the algebraicK-theory groupsKanlg(A) defined for anyn∈Zpro-vide interesting secondary invariants for any ringA, but are very hard to calculate. It is also unclear if algebraicK-theory can be linked to index theory in a way consistent with topologicalK-theory, and in particular if it is possible to construct direct images of algebraicK-theory classes in a reasonable con-text. Instead, we will generalize slightly an idea of Karoubi [16, 17] and define foranyFr´echetm-algebraAthe multiplicativeK-theory groupsM KnI(A), n∈Zteedni,sulyabmmFrlech´ebdexigayfinevetinm-algebraI. Depending on the parity of the degreen, multiplicativeK-theory classes are represented by idempotents or invertibles in certain extensions ofIˆ⊗A, together with
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a transgression of their Chern character in certain quotient complexes. Mul-tiplicativeK-theory is by definition a mixture of the topologicalK-theory Kntop(Iˆ⊗A) and the non-periodic cyclic homologyH Cn(A provides a). It “good” approximation of algebraicK In-theory but is much more tractable. addition, the Jones-Goodwillie Chern character innegative cyclic homology Kanlg(A)→H Nn(A) factors through multiplicativeK-theory. The precise relations between topological, multiplicativeK-theory and the various versions of cyclic homology are encoded in a commutative diagram whose rows are long exact sequences of abelian groups
Ktn1o+p(I⊗ˆA)
H Cn−1(A)δM KnI(A)
Kntop(I⊗ˆA) (1)
e H Pn+1(A)SH Cn−1(A)BH Nn(A)IH Pn(A) The particular caseI=Cwas already considered by Karoubi [16, 17] after the construction by Connes and Karoubi of regulator maps on algebraicK-theory [6]. The incorporation of a finitely summable algebraIis rather straightfor-ward. This diagram describes the primary and secondary invariants associated to the noncommutative “manifold”A mention that the restriction to. We Fr´echetm-algebras is mainly for convenience. these constructions In principle could be extended to all locally convex algebras overC, however the subsequent results, in particular the proof of the Riemann-Roch-Grothendieck theorem would become much more involved.
If nowAandBatwret´rFoehcem-algebras, it is natural to consider the ad-equate “morphisms” mapping the primary and secondary invariants fromA toB. LetIbe apecheeFr´tamlbs-mum-algebra. By analogy with Cuntz’ description of bivariantK-theory forC∗-algebras [8], ifE⊲Iˆ⊗Bdenotes a Fr´echetm-algebra containingI⊗ˆBas a (not necessarily closed) two-sided ideal, we define ap-summable quasihomomorphismfromAtoBas a contin-uous homomorphism ρ:A→Es⊲Is⊗ˆB, whereEsandIsare certainZ2-graded algebras obtained fromEandIby a standard procedure. Quasihomomorphisms come equipped with a parity (even or odd) depending on the construction ofEsandIs. In general, we may suppose that the parity ispmod 2. We say thatIis multiplicative if it is provided with a homomorphism⊠:Iˆ⊗I→I, possibly defined up to adjoint action of multipliers onI, and compatible with the trace. A basic example of multiplicativep-summable algebra is, once again, the Schatten classLp(H). Then it is easy to show that such a quasihomomorphism induces a pushforward map in topologicalK-theoryρ!:Kntop(I⊗ˆA)→Ktno−pp(Iˆ⊗B), whose degree coincides with the parity of the quasihomomorphism. This is what one expects from bivariantK Our goal is to extend this-theory and is not really new.
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map to the entire diagram (1). Direct images for the unstable versions of cyclic homology are necessarily induced by a bivariantnon-periodiccyclic cohomology class chp(ρ)∈H Cp(A,B bivariant Chern character exists only under). This certain admissibility properties about the algebraE(note that it is sufficient for Ito be (p instead of+ 1)-summablep-summable). In particular, the bivariant Chern caracter constructed by Cuntz for any quasihomomorphism in [9, 10] cannot be used here because it provides a bivariant periodic cyclic cohomology class, which does not detect the secondary invariants ofAandB give. We the precise definition of an admissible quasihomomorphism and construct the bivariant Chern character chp(ρ) in section 3, on the basis of previous works [23]. An analogous construction was obtained by Nistor [20, 21] or by Cuntz and Quillen [12]. However the bivariant Chern character of [23] is related to other constructions involving the heat operator and can be used concretely for establishing local index theorems, see for example [24]. The pushforward map in topologicalK-theory combined with the bivariant Chern character leads to a pushforward map in multiplicativeK-theoryρ!:M KnI(A)→M KnI−p(B). Our first main result is the following non-commutative version of the Riemann-Roch-Grothendieck theorem (see Theorem 6.3 for a precise statement):
Theorem 1.1Letρ:A→Es⊲Is⊗ˆBbe an admissible quasihomomorphism of paritypmod 2. that SupposeIis(p+ 1)-summable in the even case and p Then one has a graded-commutative diagram-summable in the odd case.
Knp+1to(Iˆ⊗A)H Cn−1(A)
ρ!
Ktnop+1−p(Iˆ⊗B)
chp(ρ)
H Cn−1−p(B)
M KnI(A)Ktnop(I⊗ˆA) ρ!ρ! M KnI−p(B)Knto−pp(I⊗ˆB)
compatible with the cyclic homologySBIexact sequences after taking the Chern charactersK∗top(Iˆ⊗)→H P∗andM K∗I→H N∗. At this point it is interesting to note that the pushforward mapsρ!and the bivariant Chern character chp(ρ) enjoy some invariance properties with respect to equivalence relations among quasihomomorphisms. Two types of equivalence relations are defined: smooth homotopy and conjugation by invertibles. The second relation corresponds to “compact perturbation” in KasparovKK-theory forC∗ the latter situation, the In-algebras [2].M2-stable version of conjugation essentially coincide with homotopy, at least for separableAandσ-unitalB.ecr´rFFoeblgtahevewohsar,reM2-stable conjugation isstrictly strongerthan homotopy as an equivalence relation. ThisisindeedinthecontextofFre´chetalgebrasthatsecondaryinvariants appear. The pushforward maps in topologicalK-theory and periodic cyclic homology are invariant under homotopy of quasihomomorphisms. The maps in multiplicativeK-theory and the non-periodic versions of cyclic homology H C∗andH N∗are only invariant under conjugation and not homotopy. Also
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note that in contrast with theC∗-algebra situation, the Kasparov product of two quasihomomorphismsρ:A→Es⊲Isˆ⊗Bandρ′:B→Fs⊲Isˆ⊗C is not defined as a quasihomomorphism fromAtoC. The various bivariant K-theories constructed form-algebras [9, 10] or even for general bornological algebras [11] cannot be used here, again because they are homotopy invariant by construction. We leave the construction of a bivariantK-theory compatible with secondary invariants as an open problem.
In the last part of the paper we illustrate the Riemann-Roch-Grothendieck theorem by constructing assembly maps for certain crossed product algebras. If ΓisadiscretegroupactingonaFr´echetm-algebraA, under certain conditions the crossed productA⋊Γisaganiae´rFtehcm-algebra and one would like to obtain multiplicativeK-theory classes out of a geometric model inspired by the Baum-Connes construction [1]. Thus letP→ΓMbe a principal Γ-bundle over a compact manifoldM, and denote byAPthe algebra of smooth sections of the associatedA-bundle. IfDis aK-cycle forMrepresented by a pseudodifferential operator, we obtain a quasihomomorphism fromAPto A⋊Γ and hence a map
M KnI(AP)→M KI−p(A⋊Γ) n
for suitablepand Schatten idealI general this map cannot exhaust. In the entire multiplicativeK-theory of the crossed product but nevertheless interesting secondary invariants arise in this way. In the case whereAis the algebra of smooth functions on a compact manifold,APis commutative and its secondary invaiants are closely related to (smooth) Deligne cohomology. From this point of view the pushforward map in multiplicativeK-theory should be considered as a non-commutative version of “integrating Deligne classes along the fibers” of a submersion. We perform the computations for the simple example provided by the noncommutative torus.
The paper is organized as follows. In section 2 we review the Cuntz-Quillen formulation of (bivariant) cyclic cohomology [12] in terms of quasi-free exten-sions formis new but we take the opportunity to fix the -algebras. Nothing notations and recall a proof of generalized Goodwillie theorem. In section 3 we define quasihomomorphisms and construct the bivariant Chern character. The formulas are identical to those found in [23] but in addition we carefully es-tablish their adic properties and conjugation invariance. In section 4 we recall Phillips’ topologicalKyforheorchetFr´et-m-algebras, and introduce the peri-odic Chern characterKtnop(Iˆ⊗A)→H Pn(A) whenIis a finitely summable algebra. The essential point here is to give explicit and simple formulas for subsequent use. Section 5 is devoted to the definition of the multiplicative K-theory groupsM KnI(A) and the proof of the long exact sequence relating them with topologicalK also construct the-theory and cyclic homology. We negative Chern characterM KnI(A)→H Nn(A) and show the compatibility
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with theSBI images of topological and multiplica- Directexact sequence. tiveK-theory under quasihomomorphisms are constructed in section 6 and the Riemann-Roch-Grothendieck theorem is proved. The example of assembly maps and crossed products is treated in section 7.
2 Cyclic homology
Cyclic homology can be defined for various classes of associative algebras over C, in particular complete locally convex algebras. us, a locally convex For algebraAtopology induced by a family of continuous seminormshas a p: A→R+, for which the multiplicationA×A→Ais jointly continuous. Hence for any seminormpthere exists a seminormqsuch thatp(a1a2)≤ q(a1)q(a2),∀ai∈A. For technical reasons however we shall restrict ourselves tomultiplicatively convexwhose topology is generated by a familyalgebras [5], of submultiplicative seminorms
p(a1a2)≤p(a1)p(a2)∀ai∈A. A complete multiplicatively convex algebra is calledm-algebra, and may equiv-alently be described as a projective limit of Banach algebras. The unitalization A+=C⊕Aof anm-algebraAis again anm-algebra, for the seminorms p˜(λ1 +a) =|λ|+p(a),∀λ∈C, a∈A. In the same way, ifBis another m-algebra, the direct sumA⊕Bis anm-algebra for the topology generated by the seminorms (p⊕q)(a, b) =p(a) +q(b), wherepis a seminorm onAandq a seminorm onB the algebraic tensor product. Also,A⊗Bmay be endowed with the projective topology induced by the seminorms n n (p⊗q)(c) = infnXp(ai)q(bi) such thatc=Xai⊗bi∈A⊗Bo.(2) i=1i=1 ThecompletionAˆ⊗B=Aˆ⊗πBof the algebraic tensor product under this family of seminorms is the projective tensor product of Grothendieck [14], and is again anm-algebra. Cyclic homology, cohomology and bivariant cyclic cohomology form-algebras can be defined either within the cyclic bicomplex formalism of Connes [5], or theX-complex of Cuntz and Quillen [12]. We will make an extensive use of both formalisms throughout this paper. In general, we suppose that all linear maps or homomorphims betweenm-algebras are continuous, tensor products are completed projective tensor products, and extensions ofm-algebras 0→ I→R→A→0 always admit a continuous linear splittingσ:A→R.
2.1 Cyclic bicomplex
Non-commutative differential forms.LetAbe anm-algebra. The space of non-commutative differential forms overAis the algebraic direct sum ˆ ΩA=Ln≥0ΩnAof then-forms subspaces ΩnA=A+⊗ˆA⊗nforn≥1
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and Ω0A=A, whereA+is the unitalization ofA of the subspaces. Each ΩnA is customary Itis complete but we do not complete the direct sum. to use the differential notationa0da1. . . dan(resp.da1. . . dan) for the string a0⊗a1. . .⊗an(resp. 1⊗a1. . .⊗an). A continuous differentiald: ΩnA→ Ωn+1Ais uniquely specified byd(a0da1. . . dan) =da0da1. . . danandd2= 0. A continuous and associative product ΩnA×ΩmA→Ωn+mAis defined as usual and fulfills the Leibniz ruled(ω1ω2) =dω1ω2+ (−)|ω1|ω1dω2, where|ω1| is the degree ofω1. This turns ΩAinto a differential graded (DG) algebra. On ΩA of all, the Hochschild boundary Firstare defined various operators. mapb: Ωn+1A→ΩnAreadsb(ωda) = (−)n[ω, a] forω∈ΩnA, andb= 0 on Ω0A=A easily shows that. Onebis continuous andb2= 0, hence ΩAis a complex graded overN. The Hochschild homology ofA(with coefficients in the bimoduleA) is the homology of this complex:
H Hn(A) =Hn(ΩA, b),∀n∈N.
(3)
Then the Karoubi operatorκ: ΩnA→ΩnAis defined by 1−κ=db+bd. Thereforeκis continuous and commutes withbandd. One hasκ(ω da) = (−)nda ωfor anyω∈ΩnAanda∈A. The last operator is Connes’B: ΩnA→Ωn+1A +, equal to (1κ+. . .+κn)don ΩnA is also continuous. It and verifiesB2= 0 =Bb+bBandBκ=κB=B. Thus ΩAendowed with the two anticommuting differentials (b, B splits as a direct It) becomes a bicomplex. sum ΩA= ΩA+⊕ΩA−odd degree differential forms, hence is aof even and Z2-graded complex for the total boundary mapb+B. However its homology is trivial [18]. The various versions of cyclic homology are defined using the natural filtrations on ΩA Cuntz and Quillen [12], we define the. Following Hodge filtrationon ΩAas the decreasing family ofZ2-graded subcomplexes for the total boundaryb+B
FnΩA=bΩn+1A⊕MΩkA,∀n∈Z, k>n with the convention thatFnΩA= ΩAforn < completion of Ω0. TheAis defined as the projective limit ofZ2-graded complexes
ΩbA= li←m−ΩAFnΩA=YΩnA nn≥0
.
(4)
b b b Hence ΩA= Ω+A⊕Ω−Ais aZ2-graded complex endowed with the total boundary mapb+bB is its. Itbelf filtered by the decreasing family ofZ2-graded subcomplexesFnΩA= Ker(ΩA→ΩAFnΩA), which may be written
FnbbΩn+1A⊕YΩkA,∀n∈Z. ΩA= k>n
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b b In particular the quotient ΩAFnΩAis aZ2-graded complex isomorphic to ΩAFnΩA, explicitly n−1 bFnΩbA=MΩkA⊕ΩnAb(+16) ΩAΩnA),( k=0
b b and it vanishes forn < a topological vector space, Ω0. AsAFnΩAmay fail to be separated because the imageb(Ωn+1A) is not closed in general.
Definition 2.1In any degreen∈Z, theperiodic, non-periodicandnegative cyclic homologies are respectively the(b+B)-homologies
b H Pn(A) =Hn+2Z(ΩA), b b H Cn(A) =Hn+2Z(ΩAFnΩA),(7) H Nn(A) =HnZ(−1b +2FnΩA). HenceH Pn(A)∼=H Pn+2(A) is 2-periodic,H Cn(A) = 0 forn <0 and H Nn(A) =∼H Pn(A) forn≤ construction these cyclic homology groups0. By fit into a long exact sequence
. . .−→H Pn+1(A)−S→H Cn−1(A)−B→H Nn(A)−I→H Pn(A)−→. . .(8) whereSis induced by projection,Iby inclusion, and the connecting map cor-responds to the operatorB link between cyclic and Hochshild homology. The may be obtained throughnon-commutative de Rham homology[16], defined as b H Dn(A) :=Hn+2Z(ΩAFn+1ΩbA),∀n∈Z.(9) This yields for anyn∈Za short exact sequence ofZ2-graded complexes 0−→Gn(A)−→ΩbAFnbΩA−→ΩbAFn−1ΩbA−→0, whereGnis ΩnAbΩn+1Ain degreenmod 2, andbΩnAin degreen−1 mod 2. One hasHn+2Z(Gn) =H Hn(A) andHn−1+2Z(Gn) = 0, so that the associated six-term cyclic exact sequence in homology reduces to
0→H Dn−1(A)→H Cn−1(A)→H Hn(A)→H Cn(A)→H Dn−2(A)→0, and Connes’sSBIexact sequence [4] for cyclic homology is actually obtained by splicing together the above sequences for alln∈Z: . . .−→H Cn+1(A)−S→H Cn−1(A)−B→H Hn(A)−I→H Cn(A)−→. . .(10) Hence the non-commutative de Rham homology groupH Dn(A) may be identified with the image of the periodicity shiftS:H Cn+2(A)→H Cn(A). Clearly the exact sequence (8) can be transformed to (10) by taking the
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natural mapsH Pn(A)→H Cn(A) andH Nn(A)→H Hn(A). b Passing to the dual theory, let Hom(ΩA,C) be theZ2-graded complex of linear b b maps ΩA→Cwhich are continuous for the adic topology on ΩAinduced by the Hodge filtration. It is concretely described as the direct sum Hom(ΩA,C) =MHom(ΩnA,C), b n≥0
where Hom(ΩnA,C) is the space of continuous linear maps ΩnA→C. The b space Hom(ΩA,C) is endowed with the transposed of the boundary operator b b+Bon ΩA the periodic cyclic cohomology of. ThenAis the cohomology of this complex: H Pn(A) =Hn+2Z(Hom(ΩbA,C)).(11)
One defines analogously the non-periodic and negative cyclic cohomologies which fit into anIBSlong exact sequence.
2.2X-complex and quasi-free algebras
We now turn to the description of theX-complex. It first appeared in the coalgebra context in Quillen’s work [28], and subsequently was used by Cuntz and Quillen in their formulation of cyclic homology [12]. Here we recall the X-complex construction form-algebras. LetRbe anm space of non-commutative one-forms Ω-algebra. The1Ris aR-bimodule, hence we can take its quotient Ω1R♮by the subspace of commutators [R,Ω1R] =bΩ2R. Ω1R♮ However,may fail to be separated in general. it is automatically separated whenRisquasi-free order to avoid In, see below. confusions in the subsequent notations, we always write a one-formx0dx1∈ Ω1Rwith a bolddwhen dealing with theX-complex ofR latter is the. The Z2-graded complex [12]
X(R)♮⇄dΩ1 :R R♮, b
(12)
whereR=X+(R) is located in even degree and Ω1R♮=X−(R) in odd degree. The class of the generic element (x0dx1mod [,])∈Ω1R♮is usually denoted by♮x0dx1 map. The♮d:R→Ω1R♮thus sendsx∈Rto♮dx. Also, the Hochschild boundaryb: Ω1R→Rvanishes on the commutator subspace [R,Ω1R], hence passes to a well-defined mapb: Ω1R♮→R the im-. Explicitly age of♮x0dx1bybis the commutator [x0, x1 maps are continuous and]. These satisfy♮db= 0 andb◦♮d= 0, so that (X(R), ♮db) indeed defines aZ2-graded complex. We mention that everything can be formulated whenRitself is a Z2-graded algebra: we just have to replace everywhere the ordinary commuta-tors by graded commutators, and the differentials anticommute with elements of odd degree. In particular one getsb♮xdy= (−)|x|[x, y], where|x|is the
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