On the Lp Novikovand Baum-Connes

conjectures

G. KasparovVanderbilt University

The right side of the Novikovand Baum-Connes conjectures is theK-theory of the reduced C∗-algebraC∗red(G) of the group G. This al-gebra is the completion of the alge-bra L1(G) acting on L2(G) by con-volution. If we complete the algebraL1(G) in the norm of the convolu-tion algebra on Lp(G) we will get theBanach algebra which I will denoteC×,pr (G). The K-theory of this alge-bra serves as the right side of the Lp-version of the Novikov and Baum-Connes conjectures. I will discussthe results on these conjectures fora certain class of groups.

This is a joint work with Guo-liang Yu. My student Fan Fei Chongalso participated in the Novikov con-jecture part.

The construction of the assem-bly map requires a technique ofasymptotic morphisms for Banachalgebras. The definition of asymp-totic morphisms for Banach alge-bras goes exactly in the same wayas for C∗-algebras. Also the com-position of asymptotic morphisms isdefined as for C∗-algebras. The firstnon-trivial point is the constructionof an asymptotic morphism corre-sponding to an extension of Banachalgebras: 0 → J → A → B → 0.Here we need a quasicentral contin-uous approximate unit for the idealJ . At this point, we will make a cer-tain assumption concerning our Ba-nach algebras.

Definition. A Banach algebrawill be called σ-unital if it possessesa countable bounded approximateunit contained in a closed subalge-bra isomorphic to C0(Y ) for somelocally compact space Y . In the casewith an action of a locally compactgroup G, we assume that the subal-gebra C0(Y ) is G-invariant and theG-action on C0(Y ) is induced by theG-action on Y .

(For the category of C∗-algeb-ras, this coincides with the usualdefinition of σ-unitality.)

Lemma. Let J be a closedtwo-sided ideal in a Banach algebraB. Assume that J is σ-unital andthe quotient B/J is separable. Thenthere is a quasicentral bounded ap-proximate unit in J with respect toB. In the case of a G-action onB which satisfies the assumption ofthe above definition, the approxi-mate unit can be chosen quasicen-tral with respect to G as well.

A Banach algebra B will becalled an Lp-algebra if it isometri-cally embeds as a closed subalge-bra into the algebra L(Lp(Z)) of allbounded linear operators on somespace Lp(Z), where Z is a measurespace. For such algebra, the op-posite algebra Bop embeds into thedual space Lq(Z) where 1/p+1/q =1.

Note that when W = lp(Z) fora discrete space Z with an atomicmeasure, the algebra of compactoperators K(W ) is clearly σ-unital.This is also true in the equivariantcase of a G-action induced by a G-action on Z (we assume G discretein this case).

A possibility to construct anasymptotic morphism out of an ex-act sequence of Banach algebras us-ing quasicentral approximate unitsas in the C∗-algebra case (theConnes-Higson construction) allowsto define E-theory for σ-unital Lp

Banach algebras and establish itsmain properties (product, Bott pe-riodicity, exact sequences). Notethat in the Lp case we use the al-gebra of compact operators on anLp space in the definition of the E-theory groups.

We will also need E-theory onthe category of C0(X)-algebras. Re-call that a Banach algebra A is aC0(X)-Banach algebra if C0(X) actson it as an algebra of central multi-pliers and C0(X) · A = A. On thecategory of C0(X)-Banach algebras,one defines RE-theory. Recall alsothat a G-Banach algebra is calledproper if it is a C0(X)-algebra withthe G-action on X being proper.Note that the group RE(X;A,B) =RE(X;A⊗C0(X), B ⊗C0(X)) canalso be defined as the group ofcontinuous families of E-theory ele-ments parametrized by the space X,like for KK-theory.

For any locally compact groupG and an Lp-algebra B ⊂ L(Lp(Z))we define the norm on the crossedproduct algebra Cc(G,B) as the op-erator norm for the action of this al-gebra on Lp(G×Z) by the formula:

(b · l)(t, z) =∫G

t−1(b(s)) · l(s−1t, z)ds

where b is a compactly supportedfunction on G with values in Band l ∈ Lp(G × Z). We definethe reduced crossed product alge-bra C×,pr (G,B) as the completion ofCc(G,B) in this norm.

In the special case when B =C, the algebra C×,pr (G) is justthe image of the algebra L1(G) inL(Lp(G)) acting on Lp(G) by convo-lution. If G is unimodular, the dualto the operator of convolution by anelement f(g) ∈ L1(G) on Lp(G) isthe convolution operator by f̄∗(g)on Lq(G), where f 7→ f∗ is the usualinvolution on L1(G). Therefore,C×,pr (G) ' C×,qr (G). By the in-terpolation theorem, since C×,pr (G)acts on both Lp(G) and Lq(G), italso acts on L2(G), so there is anatural homomorphism C×,pr (G) →C∗r (G). It is not clear however howthe K-theory groups of these alge-bras are related.

We will use the descent homo-morphism:

jG : EG(A,B)→

E(Cc(G,A), C×,pr (G,B)).

Let EG be the classifying spacefor proper actions of G. The left-hand side of the Baum-Connes con-jecture (with coefficients in an Lp-algebra B) is the inductive limitREG∗ (EG,B) = lim−→E

G∗ (C0(X), B)

over all G-proper, G-compact, lo-cally compact spaces X.

In particular, the left-hand sidefor the Lp Baum-Connes conjecturewith B = C is the same as for theconventional Baum-Connes conjec-ture.

The assembly map

µp : REG∗ (EG,B)→

K∗(C×,pr (G,B))

is defined as follows. We take a non-negative function c ∈ Cc(X) sat-isfying

∫Gc(g−1x)dg = 1 and de-

fine the idempotent element [c] ∈Cc(G,C0(X)) by

[c](g, x) = c(x)1/2c(g−1x)1/2.

For any a ∈ EG∗ (C0(X), B),we first apply the homomorphismjG to get an element jG(a) ∈E(Cc(G,C0(X)), C×,pr (G,B)), andthen take the E-theory product ofjG(a) with the idempotent [c].

We call the injectivity of µpthe Lp Novikov conjecture, and thebijectivity of µp the Baum-Connesconjecture (with coefficients in B).

Let us first discuss the Novikovconjecture.

Suppose that Γ is discrete andthat there exists a C0(X)-BanachΓ-algebra A, such that actuallyC0(X) ⊂ A, and X is Γ-proper.Assume that A is E-equivalent toS = C0(R), graded by even-oddfunctions, equivariantly with respectto any finite subgroup of Γ. Moreprecisely, we assume that there ex-ists a “Bott element” in the groupEΓ(S,A) which is invertible in E-theory when restricted to any finitesubgroup of Γ.

Then the Novikov conjecturecan be proved using the following di-agram:

REΓ∗ (EΓ, B⊗̂S) → K∗(C

×,pr (Γ, B⊗̂S))

↓ ↓REΓ∗ (EΓ, B⊗̂A) → K∗(C

×,pr (Γ, B⊗̂A)).

Here the horizontal arrows arethe assembly maps, the vertical ar-rows are the Bott maps.

We prove that the bottom hori-zontal arrow and the left vertical ar-row are isomorphisms. The injectiv-ity of the upper horizontal arrow fol-lows.

The main tool in the proof isthe Mayer-Vietoris exact sequencein E-theory.

Example when such algebra Aexists: Γ admits a coarse uniformembedding into a discrete lp-space.This is certainly satisfied in the as-sumptions of the next theorem.

Theorem. The Lp-version ofthe Baum-Connes conjecture withcoefficients is true for any count-able discrete group Γ which admitsan affine-isometric proper action onsome space lp(Z) so that the lin-ear part of this action is inducedby the action of Γ on Z. (Z is adiscrete countable set equipped withthe atomic measure.)

For any finite subset F ⊂ Z andVF = lp(F ), we consider Λ∗(VF ) asthe lp-space of all finite subsets ofF . We construct the inductive limitof the spaces Lp(Λ∗(VF )). Namely,if F1 ⊂ F2 are finite subsets of Zand α > 0, let fF2−F1 be the scalarfunction on VF2−F1

defined by:

fF2−F1(v) = exp(−α

n∑i=1

|xi|p/p),

where F2 − F1 = {zi| i = 1, · · · , n},v =

∑ni=1 xiδzi .

The isometric embedding:

Lp(Λ∗(VF1))→ Lp(Λ∗(VF2

))

is given by: ξ → ξ ⊗ (cnfF2−F1),

where the constant cn(p, α) is cho-sen in such a way that cnfF2−F1 hasLp-norm 1.

For any α > 0, we define theBanach space Bα as the inductivelimit of Lp(Λ∗(VF )). We define theBanach algebra Kα as the algebra ofall compact operators on the Banachspace Bα. To simplify notation wewill omit the coefficient algebra inthe statement of the Baum-Connesconjecture with coefficients.

Lemma. The right-handside of the Baum-Connes conjec-ture K∗(C

×,pr (Γ)) is isomorphic to

K∗(C×,pr (Γ,Kα)).

The image of the Baum-Connesmap can be described using local-ization algebras and finite propa-gation techniques (developed by G.Yu in the 90’s). The sketch ofthe argument below also uses quasi-projections, i.e. elements which sat-isfy the property ||P 2 − P || < δ forsome small δ and ||P || < c. Thenumbers δ and c depend on the con-text.

Propagation of operators on thespace Lp(Λ∗(VF )) is defined usingtheir support on Lp(VF ). Thepropagation of an element a ∈Cc(Γ,K(VF )) is defined as the supre-mum of propagations of all opera-tors a(g), g ∈ Γ.

The main part of the proofof the Lp Baum-Connes conjectureconsists of the technical construc-tion which allows to produce outof any element a ∈ K0(C×,pr (Γ)) aquasi-projection P which representsthe same element a in the groupK0(C×,pr (Γ,Kα)) and has arbitrar-ily small propagation. All elementsa ∈ K0(C×,pr (Γ,Kα)) with arbitrar-ily small propagation are in the im-age of the Baum-Connes map.

The crucial point in this con-struction is the use of the Bott-Diracoperator on the space Bα. Note thatq − 1 = (p− 1)−1. Let F be a finitesubset of Z, {ei}i∈F the standardbasis for VF , and xi the correspond-ing coordinate functions.

The Bott-Dirac operator isgiven by the formula:

Dα,t =∑i∈Z

tλi((∂/∂xi + αxp−1i )ext(ei)

+(−(q − 1)∂/∂xi + αxp−1i )int(ei)).

We omit the parameter t in the no-tation below, but it will play an im-portant role in the proof.

The kernel of the operator Dα

is the element of the space Bα de-fined as the inductive limit of func-tions fF2−F1

.An important remark is that

the operator D2α is accretive, i.e. for

any f ∈ Bα and f∗ in the dualspace B∗α, with ||f∗|| = ||f ||p−1

and < f, f∗ >= ||f ||p, one has:< D2

α(f), f∗ >≥ 0.

This allows to define the opera-tor (1 + D2

α,F )−1 which is boundedand compact. This property re-mains true on the inductive limitspace Bα if the sequence of positivenumbers {λi} → ∞. In addition,the sequence λi has to be specifi-cally chosen to insure the commuta-tion properties of the operator Dα

with the group action.Finally, when t→∞, the oper-

ator (1+D2α)−1 uniformly converges

to the projection onto the kernel ofthe operator Dα.

We give now a sketch of theproof of the main result. Let Pα bethe operator Dα going from Λev toΛod, and Qα = Dα/(1 + D2

α) goingin the opposite direction.

Lemma. The bounded op-erators PαQα − 1, QαPα − 1 andPα(QαPα− 1) can be approximatedin norm by operators with arbitrar-ily small propagation when α andt are large. The same is true forthe commutators of these operatorswith elements g ∈ Γ.

The idea of the proof: the pro-jection pα onto the kernel of Dα canbe approximated in norm by opera-tors with arbitrarily small propaga-tion when t > α is large enough.

Let Pα,n and Qα,n be respec-tively the direct sums of n copies ofPα and Qα. Given an element inK0(C×,pr (Γ)), we represent it as aquasi-idempotent q in Mn(CΓ) forsome n satisfying

||q|| ≤ c0, ||q2 − q|| < 1/100.

Let Pα,n ⊗ I and Qα,n ⊗ Ibe the operators acting on the Ba-nach space Bα ⊗ lp(Γ). We definePα,n,q = q(Pα,n ⊗ I)q and Qα,n,q =q(Qα,n ⊗ I)q. We abbreviate Pα,n,qand Qα,n,q respectively by Pα,q andQα,q.

Let:

Fα,q = Pα,qQα,q,Gα,q = Qα,qPα,q,

Aα,q = Fα,q + (q − Fα,q)Fα,q,Bα,q = Pα,q(q −Gα,q)+(q − Fα,q)Pα,q(q −Gα,q),Cα,q = (q −Gα,q)Qα,q,Dα,q = (q −Gα,q)2.

We put:

q1 =

(Aα,q Bα,qCα,q Dα,q

),

q0 =

(q 00 0

).

Note that

q1 ∈ q0(M2n(C×,pr (Γ,Kα)+))q0

(where + means adjoining a unit tothe algebra), and

q1 − q0 ∈M2n(C×,pr (Γ,Kα)).

We define ψα([q]) as the K-theory difference class [q1] − [q0] inK0(C×,pr (Γ,Kα)).

Proposition. The opera-tor q1 − q0 can be approximatedin norm by a bounded operatorin M2n(C×,pr (Γ,Kα)) with normbounded by a universal constant andwith arbitrarily small propagationwhen α is large enough and t >> α.

Remark. In the language oflocalization algebras this means thatq1 − q0 is in the image of the corre-sponding Baum-Connes map.

Denote by φα the isomorphismK0(Cpr (Γ,Kα)) ' K0(Cpr (Γ)).

Proposition. In the assump-tions of the main theorem, φα ◦ ψαis the identity homomorphism.

The proof uses the homotopy ofthe affine action of Γ to the linear ac-tion (by contraction of the 1-cocycleto 0).

Corollary. φα([q1]−[q0]) = [q].

This implies that q is in the im-age of the Baum-Connes map.