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A L

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L’INSTITUT FOURIER

Hervé OYONO-OYONO & Guoliang YU On quantitative operatorK-theory

Tome 65, n^o2 (2015), p. 605-674.

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ON QUANTITATIVE OPERATOR K-THEORY

by Hervé OYONO-OYONO & Guoliang YU (*)

Abstract. — In this paper, we develop a quantitativeK-theory for filtered C^∗-algebras. Particularly interesting examples of filteredC^∗-algebras include group C^∗-algebras, crossed productC^∗-algebras and Roe algebras. We prove a quantitative version of the six term exact sequence and a quantitative Bott periodicity. We apply the quantitativeK-theory to formulate a quantitative version of the Baum- Connes conjecture and prove that the quantitative Baum-Connes conjecture holds for a large class of groups.

Résumé. — Dans cet article, nous développons une K-théorie quantitative pour lesC^∗-algèbres filtrées. Parmi les exemples les plus intéressants de tellesC^∗- algèbres figurent les algèbres de Roe, lesC^∗-algèbres de groupes et lesC^∗-algèbres de produits croisés. Nous établissons une version quantitative de la suite exacte à six termes en K-théorie ainsi que de la périodicité de Bott. Nous formulons en utilisant la K-théorie quantitative une version quantitative de la conjecture de Baum-Connes. Nous montrons que cette conjecture de Baum-Connes quantitative est vérifiée pour une large classe de groupes.

Introduction

The receptacles of higher indices of elliptic differential operators are K-theory of C^∗-algebras which encode the (large scale) geometry of the underlying spaces. The following examples are important for purpose of applications to geometry and topology.

• K-theory of groupC^∗-algebras is a receptacle for higher index theory of equivariant elliptic differential operators on covering spaces [1, 2, 5, 11];

Keywords:Baum-Connes Conjecture, Coarse Geometry, Group and Crossed product C^∗-algebras, Novikov Conjecture, Operator AlgebraK-theory, Roe Algebras.

Math. classification:19K35,46L80,58J22.

(*) Oyono-Oyono is partially supported by the ANR “Kind” and Yu is partially supported by a grant from the U.S. National Science Foundation.

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• K-theory of crossed productC^∗-algebras and more generally grou- poid C^∗-algebras for foliations serve as receptacles for longitudi- nally elliptic operators [3, 4];

• the higher indices of elliptic operators on noncompact complete Riemannian manifolds live inK-theory of Roe algebras [15].

The local nature of differential operators implies that these higher indices can be defined in term of idempotents and invertible elements with finite propagation. Using homotopy invariance of theK-theory for C^∗-algebras, these higher indices give rise to topological invariants.

In the context of Roe algebras, a quantitative operator K-theory was introduced to compute the higher indices of elliptic operators for noncompact spaces with finite asymptotic dimension [19]. The aim of this paper is to develop a quantitative K-theory for general C^∗-algebras equipped with a filtration. The filtration structure allows us to define the concept of propagation. Examples of C^∗-algebras with filtrations include group C^∗- algebras, crossed productC^∗-algebras and Roe algebras. The quantitative K-theory forC^∗-algebras with filtrations is then defined in terms of homotopy classes of quasi-projections and quasi-unitaries with propagation and norm controls. We introduce controlled morphisms to study quantitative operator K-theory. In particular, we derive a quantitative version of the six term exact sequence. In the case of crossed product algebras, we also define a quantitative version of the Kasparov transformation compatible with Kasparov product. We end this paper by using the quantitativeK- theory to formulate a quantitative version of the Baum-Connes conjecture and prove it for a large class of groups.

This paper is organized as follows: In section 1, we collect a few notations and definitions including the concept of filtered C^∗-algebras. We use the concepts of almost unitary and almost projection to define a quantitative K-theory for filteredC^∗-algebras and we study its elementary properties. In section 2, we introduce the notion of controlled morphism in quantitative K-theory. Section 3 is devoted to extensions of filtered C^∗-algebras and to a controlled exact sequence for quantitativeK-theory. In section 4, we prove a controlled version of the Bott periodicity and as a consequence, we obtain a controlled version of the six-term exact sequence inK-theory.

In section 5, we applyKK-theory to study the quantitativeK-theory of crossed productC^∗-algebras and discuss its application to K-amenability.

Finally in section 8, we formulate a quantitative Baum-Connes conjecture and prove the quantitative Baum-Connes conjecture for a large class of groups.

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1. Quantitative K-theory

In this section, we introduce a notion of quantitative K-theory for C^∗- algebras with a filtration. Let us fix first some notations aboutC^∗-algebras we shall use throughout this paper.

• IfB is a C^∗-algebra and if b1, . . . , bk are respectively elements of Mn₁(B), . . . , Mn_k(B), we denote by diag(b1, . . . , bk) the block di- agonal matrix





 b1

. .. bk





ofMn₁+···+n_k(B).

• IfX is a locally compact space andB is aC^∗-algebra, we denote byC0(X, B) the C^∗-algebra of B-valued continuous functions on X vanishing at infinity. The special cases ofX = (0,1],X = [0,1), X = (0,1) and X = [0,1], will be respectively denoted by CB, B[0,1), SB andB[0,1].

• For a separable Hilbert spaceH, we denote byK(H) theC^∗-algebra of compact operators onH.

• IfAandBareC^∗-algebras, we will denote byA⊗B their spatial tensor product.

1.1. FilteredC^∗-algebras

Definition 1.1. — A filtered C^∗-algebra A is a C^∗-algebra equipped with a family(Ar)r>0 of closed linear subspaces indexed by positive numbers such that:

• Ar⊂Ar⁰ ifr6r⁰;

• A_r is stable by involution;

• Ar·Ar⁰ ⊂Ar+r⁰;

• the subalgebra [

r>0

A_r is dense inA.

IfA is unital, we also require that the identity 1 is an element ofAr for every positive numberr. The elements ofA_rare said to havepropagationr.

• LetAand A⁰ be respectivelyC^∗-algebras filtered by (Ar)r>0 and (A⁰_r)r>0. A homomorphism of C^∗ -algebras φ : A−→A⁰ is a filtered homomorphism (or a homomorphism of filteredC^∗-algebras) ifφ(A_r)⊂A⁰_r for any positive numberr.

• IfAis a filteredC^∗-algebra andX is a locally compact space, then C₀(X, A) is aC^∗-algebra filtered by (C₀(X, A_r))_r>0. In particular the algebrasCA,A[0,1],A[0,1) andSAare filteredC^∗-algebras.

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• IfAis a non unital filteredC^∗-algebra, then its unitarizationAeis filtered by (Ar+C)r>0. We define forAnon-unital the homomorphism

ρ_A:Ae→C;a+z7→z fora∈Aandz∈C.

Prominent examples of filteredC^∗-algebra are provided by Roe algebras associated to proper metric spaces, i.e. metric spaces such that closed balls of given radius are compact. Recall that for such a metric space (X, d), a X-module is a Hilbert space HX together with a ∗-representation ρX of C0(X) inHX (we shall writef instead ofρX(f)). If the representation is non-degenerate, theX-module is said to be non-degenerate. AX-module is called standard if no non-zero function ofC0(X) acts as a compact operator onH_X.

The following concepts were introduced by Roe in his work on index theory of elliptic operators on noncompact spaces [15].

Definition 1.2. — Let H_X be a standard non-degenerate X-module and letT be a bounded operator onHX.

(i) The support ofT is the complement of the open subset ofX×X {(x, y)∈X×X s.t. there existf andg inC0(X)satisfying

f(x)6= 0, g(y)6= 0andf ·T·g= 0}.

(ii) The operator T is said to have finite propagation (in this case propagation less thanr) if there exists a realrsuch that for anyx andyinX withd(x, y)> r, then(x, y)is not in the support ofT. (iii) The operator T is said to be locally compact iff·T andT·f are compact for any f in C0(X). We then define C[X] as the set of locally compact and finite propagation bounded operators ofH_X, and for everyr >0, we defineC[X]ras the set of elements ofC[X] with propagation less thanr.

We clearly have C[X]_r·C[X]_r⁰ ⊂ C[X]_r+r⁰. We can check that up to (non-canonical) isomorphism,C[X] does not depend on the choice ofHX. Definition 1.3. — The Roe algebraC^∗(X)is the norm closure ofC[X] in the algebra L(H_X) of bounded operators on H_X. The Roe algebra in then filtered by(C[X]r)r>0.

Although C^∗(X) is not canonically defined, it was proved in [9] that up to canonical isomorphisms, its K-theory does not depend on the choice of a non-degenerate standardX-module. Furthermore,K∗(C^∗(X)) is the

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natural receptacle for higher indices of elliptic operators with support on X [15].

IfXhas bounded geometry, then the Roe algebra admits a maximal version [7] filtered by (C[X]r)r>0. Other important examples are reduced and maximal crossed product of aC^∗-algebra by an action of a discrete group by automorphisms. These examples will be studied in detail in Section 5.

1.2. Almost projections/unitaries

LetAbe a unital filteredC^∗-algebra. For any positive numbersrandε, we call

• an elementuinAanε-r-unitary ifubelongs toA_r,ku^∗·u−1k< ε andku·u^∗−1k< ε. The set ofε-r-unitaries onAwill be denoted by U^ε,r(A).

• an elementpinAanε-r-projection ifpbelongs toAr,p=p^∗ and kp²−pk < ε. The set of ε-r-projections onA will be denoted by P^ε,r(A).

Forninteger, we set U^ε,r_n (A) = U^ε,r(M_n(A)) and P^ε,r_n (A) = P^ε,r(M_n(A)).

For any unital filteredC^∗-algebra A, any positive numbersεandr and any positive integern, we consider inclusions

P^ε,r_n (A),→P^ε,r_n+1(A);p7→

p 0 0 0

and

U^ε,r_n (A),→U^ε,r_n+1(A);u7→

u 0 0 1

. This allows us to define

U^ε,r_∞(A) = [

n∈N

U^ε,r_n (A) and

P^ε,r_∞(A) = [

n∈N

P^ε,r_n (A).

Remark 1.4. — Letrandεbe positive numbers withε <1/4;

(i) Ifpis anε-r-projection inA, then the spectrum ofpis included in 1−√

1+4ε 2 ,¹⁻

√1−4ε 2

∪

1+√ 1−4ε 2 ,¹⁺

√1+4ε 2

and thuskpk<1 +ε.

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(ii) Ifuis anε-r-unitary inA, then

1−ε <kuk<1 +ε/2, 1−ε/2<ku⁻¹k<1 +ε,

ku^∗−u⁻¹k<(1 +ε)ε.

(iii) Let κ_0,ε:R→Rbe a continuous function such that

• κ0,ε(t) = 0 ift6¹⁻

√1−4ε

2 ;

• κ0,ε(t) = 1 ift>¹⁺

√1−4ε

2 .

Ifpis anε-r-projection inA, thenκ_0,ε(p) is a projection such that kp−κ0,ε(p)k<2εwhich moreover does not depends on the choice ofκ_0,ε. From now on, we shall denote this projection by κ₀(p).

(iv) Ifuis anε-r-unitary inA, setκ1(u) =u(u^∗u)^−1/2. Thenκ1(u) is a unitary such thatku−κ₁(u)k< ε.

(v) Ifpis anε-r-projection in Aand qis a projection in Asuch that kp−qk<1−2ε, thenκ0(p) andqare homotopic projections [18, Chapter 5].

(vi) Ifuandvareε-r-unitaries inA, thenuvis anε(2 +ε)-2r-unitary inA.

Definition 1.5. — LetAbe aC^∗-algebra filtered by(Ar)r>0.

• Let p0 and p1 be ε-r-projections. We say thatp0 and p1 are ho- motopicε-r-projections if there exists anε-r-projectionqinA[0,1]

such thatq(0) =p₀ and q(1) =p₁. In this case,q is called a homotopy ofε-r-projections inAand will be denoted by(qt)_t∈[0,1].

• IfA is unital, letu₀ and u₁ be ε-r-unitaries. We say that u₀ and u1 are homotopic ε-r-unitaries if there exists an ε-r-unitary v in A[0,1]such thatv(0) =u₀ andv(1) =u₁. In this case,v is called a homotopy ofε-r-unitaries inAand will be denoted by(vt)_t∈[0,1]. Example 1.6. — Let p be an ε-r-projection in a unital filtered C^∗- algebra A. Set ct = cosπt/2 and st = sinπt/2 for t ∈ [0,1] and let us considerer the homotopy of projections (ht)_t∈[0,1] withht=

c²_t c_ts_t ctst s²_t

in M₂(C) between diag(1,0) and diag(0,1). Set (q_t)_t∈[0,1] = (diag(p,0) + (1−p)⊗ht)_t∈[0,1]. Sinceq²_t −qt=s²_t(p²−p)⊗I2, we see that (qt)_t∈[0,1]

is a homotopy of ε-r-projections between diag(1,0) and diag(p,1−p) in M2(A).

Next result will be used quite extensively throughout the paper and is fairly easy to prove.

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Lemma 1.7. — LetAbe aC^∗-algebra filtered by(A_r)r>0.

(i) If p is an ε-r-projection in A and q is a self-adjoint element of Ar such thatkp−qk < ^ε−kp₄²^−pk, then q is anε-r-projection. In particular, if pis an ε-r-projection in A and ifq is a self-adjoint element inAr such thatkp−qk< ε, thenqis a5ε-r-projection in Aandpandqare connected by a homotopy of5ε-r-projections.

(ii) If A is unital and if u is an ε-r-unitary and v is an element of Ar such that ku−vk < ^ε−ku^∗₃^u−1k, then v is an ε-r-unitary. In particular, if u is an ε-r-unitary and v is an element of Ar such that ku−vk < ε, thenv is an 4ε-r-unitary inA anduand v are connected by a homotopy of4ε-r-unitaries.

(iii) Ifpis a projection inA andqis a self-adjoint element ofAr such thatkp−qk< ^ε₄, thenqis anε-r-projection.

(iv) IfAis unital and if uis a unitary inAandv is an element of Ar

such thatku−vk<^ε₃, thenv is anε-r-unitary.

Corollary 1.8. — Let u be an ε-r-unitary in a unital filtered C^∗- algebra A, then diag(u, u^∗) and I2 are homotopic as 3ε-2r-unitaries in M₂(A).

Proof. — According to point (vi) of Remark 1.4 and with notations of Example 1.6, we see that diag(1, u) ^c_s^t_t^−s_c_t^t

·diag(1, u^∗)·(_−s^c^t_t^s_c^t_t)

t∈[0,1]is a homotopy of 3ε-2r-unitaries between diag(u, u^∗) and diag(uu^∗,1). Since kuu^∗−1k< ε, we deduce from Lemma 1.7 thatuu^∗ and 1 are homotopic

3ε-2r-unitaries.

Lemma 1.9. — There exists a numberλ >4such that for any positive numberε withε < 1/λ, any positive real r, anyε-r-projectionpand ε-r- unitaryW in a filtered unitalC^∗-algebraA, the following assertions hold:

(i) W pW^∗ is aλε-3r-projection ofA;

(ii) diag(W pW^∗,1)anddiag(p,1)are homotopicλε-3r-projections.

Proof. — The first point is straightforward to check from Remark 1.4.

For the second point, with notations of Example 1.6, use the homotopy of ε-r-unitaries

_{W c}2

t+s²_t (W−1)s_tc_t (W−1)stc_t W s²_t+c²_t

t∈[0,1]= ^c_s^t_t^−s_c_t^t

·diag(W,1)·(_−s^c^t_t^s_c^t_t)

t∈[0,1]

to connect by conjugation diag(W pW^∗,1) to diag(p, W W^∗) and then con-

nect to diag(p,1) by a ray.

Recall that if two projections in a unital C^∗-algebra are close enough in norm, then there are conjugated by a canonical unitary. To state a

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similar result in term ofε-r-projections andε-r-unitaries, we will need the definition of a control pair.

Definition 1.10. — A control pair is a pair (λ, h), where

• λ >1;

• h: (0,_4λ¹)→ (1,+∞);ε7→ h_ε is a map such that there exists a non-increasing mapg: (0,_4λ¹ )→(0,+∞), withh6g.

Lemma 1.11. — There exists a control pair(λ, h)such that the following holds:

for every positive number r, any ε in (0,_4λ¹) and any ε-r-projectionsp and q of a filtered unital C^∗-algebra A satisfying kp−qk < 1/16, there exists anλε-hεr-unitaryW in Asuch thatkW pW^∗−qk6λε.

Proof. — We follow the proof of [18, Proposition 5.2.6]. If we set z= (2κ₀(p)−1)(2κ₀(q)−1) + 1,

• then

kz−2k 6 2kκ0(p)−κ₀(q)k 6 8ε+ 2kp−qk and hencez is invertible forε <1/16.

• Moreover, if we setU =z|z⁻¹|and sincezκ0(q) =κ0(p)z, then we haveκ₀(q) =U κ₀(p)U^∗.

Let us define z⁰ = (2p−1)(2q−1) + 1. Then we have kz−z⁰k 6 9ε and kz⁰k 6 3. If ε is small enough, then kz^0∗z⁰−4k 6 2 and hence the spectrum ofz^0∗z⁰ is in [2,6]. Let us consider the expansion in power serie P

k∈Nakt^k of t 7→(1 +t)^−1/2 on (0,1) and let nε be the smallest integer such thatP

n_ε6k|ak|/2^k6ε. Let us set thenW =z⁰/2Pn_ε

k=0ak(^z^0∗^z₄⁰⁻⁴)^k. Then for a suitableλ(not depending on A, p, qor ε), we get thatW is a λε-(4nε+ 2)r-unitary which satisfies the required condition.

Remark 1.12. — The order ofhwhenεgoes to zero in Lemma 1.11 is Cε^−3/2for some constantC.

1.3. Definition of quantitativeK-theory

For a unital filtered C^∗-algebra A, we define the following equivalence relations on P^ε,r_∞(A)×Nand on U^ε,r_∞(A):

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• if p and q are elements of P^ε,r_∞(A), l and l⁰ are positive integers, (p, l)∼(q, l⁰) if there exists a positive integerk and an ele- menthof P^ε,r_∞(A[0,1]) such thath(0) = diag(p, I_k+l⁰) andh(1) = diag(q, Ik+l).

• ifuandvare elements of U^ε,r_∞(A),u∼v if there exists an element hof U^3ε,2r_∞ (A[0,1]) such thath(0) =uand h(1) =v.

Ifpis an element of P^ε,r_∞(A) andl is an integer, we denote by [p, l]ε,r the equivalence class of (p, l) modulo ∼and ifu is an element of U^ε,r_∞(A) we denote by [u]_ε,r its equivalence class modulo∼.

Definition 1.13. — Letrandεbe positive numbers withε <1/4. We define:

(i) K₀^ε,r(A) = P^ε,r_∞(A)×N/∼forAunital and

K₀^ε,r(A) ={[p, l]ε,r∈P^ε,r( ˜A)×N/∼ such that dimκ0(ρA(p)) =l}

forAnon unital.

(ii) K₁^ε,r(A) = U^ε,r_∞( ˜A)/∼(withA= ˜A ifAis already unital).

Remark 1.14. — We shall see in Lemma 1.23 that as it is the case for K-theory, K_∗^ε,r(•) can indeed be defined in a uniform way for unital and non-unital filteredC^∗-algebras.

It is straightforward to check that for any unital filtered C^∗-algebraA, ifpis anε-r-projection in A anduis an ε-r-unitary inA, then diag(p,0) and diag(0, p) are homotopicε-r-projections in M2(A) and diag(u,1) and diag(1, u) are homotopicε-r-unitaries in M₂(A). Thus we obtain the following:

Lemma 1.15. — Let A be a filtered C^∗-algebra. Then K₀^ε,r(A) and K₁^ε,r(A)are equipped with a structure of abelian semi-group such that

[p, l]_ε,r+ [p⁰, l⁰]ε,r= [diag(p, p⁰), l+l⁰]ε,r

and

[u]_ε,r+ [u⁰]_ε,r= [diag(u, v)]_ε,r,

for any[p, l]ε,rand[p⁰, l⁰]ε,rinK₀^ε,r(A)and any[u]ε,rand[u⁰]ε,rinK₁^ε,r(A).

According to Example 1.6, for every unital filtered C^∗-algebra A, any ε-r-projectionpin M_n(A) and any integer l withn>l, we see that [I_n− p, n−l]ε,r is an inverse for [p, l]ε,r. In the same way, using Corollary 1.8, we get that for anyε-r-unitaryuin M_n(A), then [diag(u, u^∗)]_ε,r = [1]_ε,r. Hence we get:

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Lemma 1.16. — IfAis a filteredC^∗-algebra, thenK_∗^ε,r(A) =K₀^ε,r(A)⊕

K₁^ε,r(A)is aZ2-graded abelian group.

We have for any filteredC^∗-algebraAand any positive numbersr,r⁰,ε andε⁰ withε6ε⁰<1/4 andr6r⁰ natural group homomorphisms

• ι^ε,r₀ :K₀^ε,r(A)−→K0(A); [p, l]ε,r7→[κ0(p)]−[Il];

• ι^ε,r₁ :K₁^ε,r(A)−→K₁(A); [u]_ε,r7→[u];

• ι^ε,r∗ =ι^ε,r₀ ⊕ι^ε,r₁ ;

• ι^ε,ε₀ ⁰^,r,r⁰ :K₀^ε,r(A)−→K₀^ε⁰^,r⁰(A); [p, l]ε,r7→[p, l]ε⁰,r⁰;

• ι^ε,ε₁ ⁰^,r,r⁰ :K₁^ε,r(A)−→K₁^ε⁰^,r⁰(A); [u]ε,r7→[u]ε⁰,r⁰.

• ι^ε,ε_∗ ⁰^,r,r⁰ =ι^ε,ε₁ ⁰^,r,r⁰⊕ι^ε,ε₁ ⁰^,r,r⁰

If some of the indicesr, r⁰orε, ε⁰are equal, we shall not repeat it inι^ε,ε∗ ⁰^,r,r⁰. Remark 1.17. — Letp0 andp1be twoε-r-projections in a filteredC^∗- algebra such that κ₀(p₀) and κ₀(p₁) are homotopic projections. Then for anyεin (0,1/4), this homotopy can be approximated for somer⁰ by aε-r⁰- projection. Hence, using point (iii) of Remark 1.4, there exists a homotopy (q_t)_t∈[0,1]ofε-r⁰ projections inAsuch thatkp0−q₀k<3εandkp1−q₁k<

3ε. We can indeed assume thatr⁰ >rand thus by Lemma 1.7, we get that p₀andp₁are homotopic as 15ε-r⁰-projections. Proceeding in the same way for the odd case we eventually obtain:

there existsλ >1 such that for any filteredC^∗-algebraA, anyε∈(0,_4λ¹) and any positive numberr, the following holds:

Letxandx⁰be elements inK∗^ε,r(A) such thatι^ε,r∗ (x) =ι^ε,r∗ (x⁰) inK_∗(A), then there exists a positive numberr⁰ withr⁰ > rsuch thatι^ε,λε,r,r_∗ ⁰(x) = ι^ε,λε,r,r_∗ ⁰(x⁰) in K_∗^λε,r⁰(A).

Lemma 1.18. — Let p be a matrix in Mn(C) such that p = p^∗ and kp²−pk< εfor someεin(0,1/4). Then there is a continuous path(p_t)_t∈[0,1]

inMn(C)such that

• p₀=p;

• p1=Ik withk= dimκ0(p);

• p^∗_t =p_tandkp²_t−p_tk< εfor everyt in[0,1].

Proof. — The selfadjoint matrix p satisfies kp²−pk < ε if and only if the eigenvalues ofpsatisfy the inequality

−ε < λ²−λ < ε, i.e.

λ∈

1−√ 1 + 4ε 2 ,1−√

1−4ε 2

[√

1−4ε+ 1

2 ,

√1 + 4ε+ 1 2

.

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Letλ1, . . . , λk be the eigenvalues ofplying in

1−√ 1+4ε 2 ,¹⁻

√1−4ε 2

and let λk+1, . . . , λn be the eigenvalues ofplying in^√

1−4ε+1

2 ,

√1+4ε+1 2

. We set fort∈[0,1]

• λ_i,t=tλ_i fori= 1, . . . , k;

• λi,t=tλi+ 1−tfori=k+ 1, . . . , n.

Sinceλ7→λ²−λis decreasing on

1−√ 1+4ε 2 ,¹⁻

√1−4ε 2

and increasing on ^√_1−4ε+1

2 ,

√1+4ε+1 2

then,

−ε < λ²_i,t−λi,t< ε

for alltin [0,1] and i= 1, . . . , n. If we set pt=u·diag(λ1,t, . . . , λn,t)·u^∗ whereuis a unitary matrix ofMn(C) such thatp=u·diag(λ1, . . . , λn)·u^∗, then

• p₀=p;

• p1=κ0(p);

• p^∗_t =ptandkp²_t−ptk< εfor everyt in [0,1].

Since there is a homotopy of projections in Mn(C) between κ0(p) andIk

withk= dimκ₀(p), we get the result.

Let us equip Cwith the trivial filtration (i.eCr =C for every positive numberr). As a consequence of the previous lemma, we obtain:

Corollary 1.19. — For any positive numbers withε <1/4, then K₀^ε,r(C)→Z; [p, l]_ε,r7→dimκ₀(p)−l

is an isomorphism.

Lemma 1.20. — Letube a matrix inMn(C)such thatku^∗u−Ink< ε and kuu^∗ −Ink < ε for ε in (0,1/4). Then there is a continuous path (ut)_t∈[0,1] inMn(C)such that

• u₀=u;

• u1=In;

• ku^∗_tut−Ink< εandkutu^∗_t−Ink< εfor everyt in[0,1].

Proof. — Since uis invertible, u^∗uand uu^∗ have the same eigenvalues λ₁, . . . , λ_n, and thus ku^∗u−I_nk < ε and kuu^∗−I_nk < ε if and only if λi∈(1−ε,1 +ε) fori= 1, . . . , n. Let us set

• ht=w·diag(λ^−t/2₁ , . . . , λ^−t/2n )·w^∗ wherewis a unitary matrix of M_n(C) such thatu^∗u=w·diag(λ₁, . . . , λ_n)·w^∗;

• vt=u·htfor allt∈[0,1]. Thenv_t^∗vt=w·diag(λ^1−t₁ , . . . , λ^1−t_n )·w^∗.

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Since|λ^1−t_i −1|< εfor all allt ∈[0,1], we get that kv_t^∗v_t−I_nk< ε and kvtv^∗_t−Ink< εfor everytin [0,1]. The matrixv1is unitary and the result then follows from path-connectness of U_n(C).

As a consequence we obtain:

Corollary 1.21. — For any positive numbers r andε with ε <1/4, then we haveK₁^ε,r(C) ={0}.

1.4. Elementary properties of quantitativeK-theory

LetA1andA2be two unitalC^∗-algebras respectively filtered by (A1,r)r>0

and (A2,r)r>0and consider A1⊕A2 filtered by (A1,r⊕A2,r)r>0. Since we have identifications P^ε,r_∞(A1⊕A2) ∼= P^ε,r_∞(A1)×P^ε,r_∞(A2) and U^ε,r_∞(A1⊕ A₂)∼= U^ε,r_∞(A₁)×U^ε,r_∞(A₂) induced by the inclusions A₁ ,→A₁⊕A₂ and A2,→A1⊕A2, we see that we have isomorphismsK₀^ε,r(A1)⊕K₀^ε,r(A2)−→^∼ K₀^ε,r(A₁⊕A₂) andK₁^ε,r(A₁)⊕K₁^ε,r(A₂)−→^∼ K₁^ε,r(A₁⊕A₂).

Lemma 1.22. — LetAbe a filtered non unitalC^∗-algebra and letεand rbe positive numbers withε <1/4. We have a natural splitting

K₀^ε,r( ˜A)−→^∼⁼ K₀^ε,r(A)⊕Z.

Proof. — ViewingAas a subalgebra of ˜A, the group homomorphisms K₀^ε,r( ˜A) −→ K₀^ε,r(A)⊕Z

[p, l]ε,r 7→ ([p,dimκ0(ρA(p))]ε,r,dimκ0(ρA(p))−l) and

K₀^ε,r(A)⊕Z −→ K₀^ε,r( ˜A) ([p, l]ε,r, k−k⁰) 7→

p 0 0 Ik

, l+k⁰

ε,r

are inverse one of the other.

Let us set A⁺=A⊕Cequipped with the multiplication (a, x)·(b, y) = (ab+xb+ya, xy) foraandb inAandxandy inC. Notice that

• A⁺ is isomorphic toA⊕Cwith the algebra structure provided by the direct sum ifA is unital;

• A⁺= ˜AifAis not unital.

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Let us define alsoρ_A in the unital case by ρ_A :A⁺ →C; (a, x) 7→x. We know that in usual K-theory, we can equivalently define for A unital the Z2-graded group K_∗(A) asA⁺by

K0(A) = kerρA,∗:K0(A⁺)→K0(C)∼=Z and

K₁(A) =K₁(A⁺).

Let us check that this is also the case for ourZ2-graded groupsK_∗^ε,r(A). If theC^∗-algebraAis filtered by (Ar)r>0, thenA⁺is filtered by (Ar+C)r>0. Let us define for a unital filtered algebraA

K₀⁰^ε,r(A) ={[p, l]ε,r∈P^ε,r(A⁺)×N/∼ such that dimκ₀(ρ_A(p)) =l}

and

K₁⁰^ε,r(A) = U^ε,r(A⁺)/∼.

Proceeding as we did in the proof of Lemma 1.22, we obtain a natural splitting

K₀^ε,r(A⁺)−→^∼⁼ K₀⁰^ε,r(A)⊕Z.

But then, using the identificationA⁺∼=A⊕Cand in view of Lemmas 1.18 and 1.20, we get

Lemma 1.23. — TheZ2-graded groupsK_∗^ε,r(A)andK_∗⁰^ε,r(A)are nat- urally isomorphic.

This allows us to state functoriallity properties for quantitativeK-theory.

Ifφ:A→B is a homomorphism of unital filteredC^∗-algebras, then since φ preserve ε-r-projections and ε-r-unitaries, it obviously induces for any positive numberrand any ε∈(0,1/4) a group homomorphism

φ^ε,r_∗ :K_∗^ε,r(A)−→K_∗^ε,r(B).

In the non unital case, we can extend any homomorphismφ:A→B to a homomorphismφ⁺ :A⁺→B⁺ of unital filteredC^∗-algebras and then we use Lemmas 1.22 and 1.23 to defineφ^ε,r∗ :K∗^ε,r(A)−→K∗^ε,r(B).Hence, for any positive numberr and any ε ∈ (0,1/4), we get that K_∗^ε,r(•) is a co- variant additive functor from the category of filteredC^∗-algebras (together with filtered homomorphisms) to the category ofZ2-abelian groups.

Definition 1.24.

(i) Let A and B be filtered C^∗-algebras. Then two homomorphisms of filteredC^∗-algebrasψ₀:A→Bandψ₁:A→B are homotopic if there exists a path of homomorphisms of filtered C^∗-algebras ψ_t:A→Bfor06t61betweenψ₀andψ₁ and such thatt7→ψ_t is continuous for the pointwise norm convergence.

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(ii) A filtered C^∗-algebra A is said to be contractible if the identity map and the zero map ofAare homotopic.

Example 1.25. — IfAis a filteredC^∗-algebraA, then the cone ofA CA={f ∈C([0,1], A) such thatf(0) = 0}

is a contractible filteredC^∗-algebra.

We have then the following obvious result:

Lemma 1.26. — If φ : A → B and φ⁰ : A → B are two homotopic homomorphisms of filteredC^∗-algebras, thenφ^ε,r∗ =φ⁰^ε,r_∗ for every positive numbersεandrwith ε <1/4. In particular, ifAis a contractible filtered C^∗-algebra, then K∗^ε,r(A) ={0} for every positive numbers ε and r with ε <1/4.

LetAbe aC^∗-algebra filtered by (A_r)_r>0and let (B_k)_k∈_Nbe an increasing sequence ofC^∗-subalgebras ofAsuch that [

k∈N

B_k is dense inA. Assume thatS

r>0B_k∩A_ris dense inB_k for every integerk. Then for every integer k, theC^∗-algebraBk is filtered by (Bk∩Ar)r>0. IfAis unital, thenBk is unital for somek, and thus we will assume without loss of generality that Bk is unital for every integerk.

Proposition 1.27. — LetAbe a unitalC^∗-algebra filtered by(Ar)r>0

and let(B_k)_k∈_Nbe an increasing sequence ofC^∗-subalgebras ofAsuch that

• [

r>0

(Bk∩Ar)is dense inBk for every integerk,

• [

k∈N

(Bk∩Ar)is dense inAr for every positive numberr.

Then theZ2-graded groupsK_∗^ε,r(A)andlim

k K_∗^ε,r(B_k)are isomorphic.

Proof. — In particular, we see that [

k∈N

Bk is dense inA. Let us denote by

Υ∗,ε,r : lim

k K_∗^ε,r(Bk)→K_∗^ε,r(A)

the homomorphism of groups induced by the family of inclusionsB_k ,→A where k runs through integers. We give the proof in the even case, the odd case being analogous. Let p be an element of P^ε,r_n (A) and let δ = kp²−pk > 0 and choose α < ^ε−δ₁₂ . Since [

k∈N

(B_k∩A_r) is dense in A_r, there is an integerkand a selfadjoint elementqofM_n(B_k∩A_r) such that kp−qk < α. According to Lemma 1.18, q is a ε-r projection. Let q⁰ be

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another selfadjoint element ofM_n(B_k∩A_r) such thatkp−q⁰k< α. Then kq−q⁰k<2αand if we setqt= (1−t)q+tq⁰ fort∈[0,1], then

kq²_t−q_tk 6 kq_t²−q_tqk+kqtq−q²k+kq²−qk+kq−q_tk 6 kqt−qk(kqtk+kqk+ 1) + 4α+δ

6 12α+δ

< ε,

and thusqandq⁰are homotopic in P^ε,r_n (Bk). Therefore, forp∈P^ε,r_n (A) and qin someMn(Bk∩Ar) satisfyingkq−pk< ^kp²₁₂^−pk, we define Υ⁰_0,ε,r([p, l]ε,r) to be the image of [q, l]ε,r in lim

k K_∗^ε,r(Bk). Then Υ⁰_0,ε,r is a group homomorphism and is an inverse for Υ0,ε,r. We proceed similarly in the odd

case.

1.5. Morita equivalence

For any unital filtered algebra A, we get an identification between P^ε,r_n (Mk(A)) and P^ε,r_nk(A) and therefore between P^ε,r_∞(Mk(A)) and P^ε,r_∞(A).

This identification gives rise to a natural group isomorphism between K₀^ε,r(A) and K₀^ε,r(Mk(A)), and this isomorphism is induced by the inclusion ofC^∗-algebras

ιA:A ,→Mk(A);a7→diag(a,0).

Namely, if we sete_1,1 = diag(1,0, . . . ,0) ∈M_k(C), definition of the func- toriality yields

ι^ε,r_A,∗[p, l]_ε,r= [p⊗e_1,1+I_l⊗(I_k−e_1,1), l]_ε,r∈K₀^ε,r(M_k(A)) for anypin P^ε,r_n (A) and any integerl withl6n. We can verify that

(ι^ε,r_A,∗)⁻¹[q, l]_ε,r= [q, kl]_ε,r

for anyqin P^ε,r_n (Mk(A)) and any integerl withl6n, where on the right hand side of the equality, the matrixqofM_n(M_k(A)) is viewed as a matrix ofMnk(A).

In a similar way, we obtain in the odd case an identification between U^ε,r_∞(Mk(A)) and U^ε,r_∞(A) providing a natural group isomorphism between K₁^ε,r(A) andK₁^ε,r(Mk(A)). This isomorphism is also induced by the inclu- sionι_Aand we have

ι_A,∗[x]ε,r= [x⊗e1,1+In⊗(Ik−e1,1)]ε,r∈K₁^ε,r(Mk(A)) for anyxin U^ε,r_n (A).

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Let us deal now with the non-unital case. For usual K-theory, Morita equivalence for non-unitalC^∗-algebra can be deduced from the unital case by using the six-term exact sequence associated to the split extension 0→ A→A˜→C→0. But for quantitativeK-theory this splitting only gives rise (in term of Section 2.1) to a controlled isomorphism (see Corollary 4.9).

In order to really have a genuine isomorphism, we have to go through the tedious following computation. IfB is a non-unitalC^∗-algebra, let us identifyM_k( ˜B) withM_k(B)⊕M_k(C) equipped with the product

(b, λ)·(b⁰, λ⁰) = (bb⁰+λb⁰+bλ⁰, λλ⁰)

forb andb⁰ in M_k(B) andλandλ⁰ inM_k(C). Under this identification, if Ais not unital, let us check that the group homomorphism

Φ₁:K₁^ε,r( ˜A)→K₁^ε,r(M^_k(A)); [(x, λ)]_ε,r7→[(x⊗e_1,1, λ]_ε,r

induced by the inclusionι_A is invertible with inverse given by the composition

Ψ1:K₁^ε,r(M^k(A))→K₁^ε,r(Mk( ˜A))→^∼⁼ K₁^ε,r( ˜A),

where the first homomorphism of the composition is induced by the inclusion

M^k(A)→Mk( ˜A); (a, z)7→(a, zIk).

Let (x, λ) be an element of U^ε,r_n ( ˜A), withx∈M_n(A) andλ∈M_n(C). Then Ψ₁◦Φ₁[(x, λ)]_ε,r= [(x⊗e_1,1, λ⊗I_k)]_ε,r,

where we use the identificationM_nk(C)∼=M_n(C)⊗M_k(C) to see x⊗e_1,1 andλ⊗Ik respectively as matrices inMnk(A) andMnk(C). According to Lemma 1.20, as aε-r-unitary ofM_n(C),λis homotopic toI_n. Hence

[(x⊗e1,1, λ⊗Ik)]ε,r= [(x⊗e1,1, λ⊗e1,1+In⊗I_k−1)]

and from this we get that Ψ1◦Φ1 is induced inK-theory by the inclusion map ˜A ,→ M_k( ˜A);a 7→ diag(a,0) which is the identity homomorphism (according to the unital case).

Conversely, let (y, λ) be an element in U^ε,r_n (M^k(A)) with y∈Mn(Mk(A))∼=Mn(A)⊗Mk(C) andλ∈Mn(C). Then

Φ1◦Ψ1[(y, λ)]ε,r= [(y⊗e1,1, λ⊗Ik)]ε,r, where