FROM ACTIONS OF N

FROM ACTIONS OF N

IULIAN POPESCU and IONEL POPESCU

We define a semidirect product groupoid of a system of partially defined local homeomorphims T = (T1, . . . , Tr). We prove that this construction gives rise to amenable groupoids. The algebra associated is a Cuntz-like algebra. We use this construction for higher rank graph algebras in order to give a topological interpretation for the duality inE-theory betweenC^∗(Λ) andC^∗(Λ^op).

AMS 2000 Subject Classification: 46L80, 37A60, 54H99.

Key words: Toeplitz algebras, grupoids,K-homology.

INTRODUCTION

Toeplitz algebras have been used to define extensions of C^∗-algebras.

The beginning of this paper is in [20] where a Toeplitz algebra was the main tool in constructing a K-homology class for higher rank graph algebras. Let (Λ, σ) be a higher rank graph with shape (see [17]), Λ^∗ the set of morphism of nonzero shape and Λ = Ω∪Λ^∗ where Ω is a symbol (the vacuum morphism) which does not belong to Λ^∗. We define left and right creations on the Fock space F_Λ=F=l²(Λ):

Lλδµ=

δλµ ifs(λ) =t(µ) 0 otherwise, R_λδ_µ=

δ_µλ ifs(µ) =t(λ)

0 otherwise, L_λΩ =R_λΩ =δ_λ.

The left sided Toeplitz algebra isE_l=C^∗(Lλ;λ∈Λ^∗) the right sided Toeplitz algebra is E_r = C^∗(R_λ;λ ∈ Λ^∗) and the two sided Toeplitz algebra is E = C^∗(L_λ, R_λ;λ∈Λ^∗). These algebras are well known in the rank one case since they give rise to short exact sequences. For example, one can use E to define

0→ K → E →OE⊗OE^op →0,

where E is an ordinary oriented graph with certain conditions which give the uniqueness of the algebras OE andOE^op,E^opis the graph obtain by reversing the arrows. In the higher rank case the algebraEhas a more complicated ideal

REV. ROUMAINE MATH. PURES APPL.,55(2010),3, 205–240

structure. It is this fact which gives a longer exact sequence. It would give a KK-class if the sequence was semisplit [20, 24]. This would follow immediately from the nuclearity of E. We tried to give a more conceptual construction of this algebra as a groupoid algebra of an amenable groupoid. We have not been able to do it except in the rank one case. However, the construction appearing in [22] can be generalized to give a groupoid description of one-sided Toeplitz algebras of a higher rank graph. We believe that this construction can be generalized to actions of other semigroups. For example actions of cones in Rⁿ should give algebras generated by Wiener-Hopf operators (see [18, 19]).

The organization of this paper is as follows. In the first section we give the main construction of this paper, the semidirect product of an action of N^r by partial local homeomorphism which we also call a multiply generated dynamical system. Such an action is obtained by choosing r commuting par- tially defined local homeomorphisms. The main examples come from shifts on finite, semifinite and infinite paths of a higher rank graph. There are two groupoids associated to such an action, the groupoid of germs of the pseu- dogroup generated by these local homeomorphism and the semidirect product groupoid. We think that these groupoids have been around in the study of Toeplitz algebras of higher rank graphs as well as in the general theory of crossed products by partial actions (see for example [11, 12]). The semidirect product can be defined only if we have a condition on the domains of the partial maps. Otherwise it may not even give rise to an algebraic groupoid.

This condition is fullfiled for one sided Toeplitz algebras associated to higher rank graphs. Following the lines in [22], we prove that these two groupoids are isomorphic if and only if the dynamical system is essentially free. Next we prove the amenability of the semidirect product. This is done by decom- posing the action in subaction and then applying a result on the amenability of extensions.

In the second section we give a groupoid approach for the two-sided Toeplitz extension of a rank-one graph. The description is ad-hoc since we do not know an elegant construction like a semidirect product. The definition of the unit space is indicated by the diagonal algebraE ∩l^∞(Λ) which is generated by the projections L_λL^∗_λ and R_λR^∗_λ. These groupoids are not Hausdorff and this fact is best illustrated by the graph with one vertex and one edge (whose C^∗-algebra is C(T)).

In the third section we remind several facts about duality in a bivariant theory. We give the definition of Spanier-Whitehead duality and a condition which is often taken as definition in literature. We remind then a way to con- struct exact sequences starting from an algebra and a tuple of ideals. We prove that irrational rotation algebras are dual. Here we give an operator approach and a long exact sequence approach. We start with simple observations of the

duality of circles along the lines of [15]. Ideas appearing in this example are inspiring for more general cases, even for higher rank graph algebras. The example of rotation algebras is included in the last section using a twist of the Cuntz-Krieger relation for a higher rank graph.

In the last section we improve the results in [20] and give a groupoid approach to the duality of higher rank graph algebras. We make some simple observations inspired by the example of rotation algebras to improve the result to any locally finite higher rank graph with finite set of objects, regardless the uniqueness of the generating relation of our graph algebras as for examples the graph N^r. We obtain a duality for the universal C^∗-algebras C^∗(Λ) and C^∗(Λ^op). TheK-theory fundamental class is given byrpartial unitaries which can be best described in the groupoid picture as two-sided shifts. Even if we do not have a conceptual groupoid approach to the two-sided Toeplitz algebra, we can define it as a groupoid of germs. The main problem in understanding better this groupoid is the unit spaceX which is given by the spectrum of the diagonal algebra E ∩l^∞(Λ). The isometriesL^∗_λR_µ with σ(λ) =σ(µ) =e_j are not given by something like shift equivalence. However, the partial isometries can be viewed as partial homeomorphisms using Gelfand duality . The space X has three properties inherited from the graph Λ: the shape σ extends to a map from X to N^r, the multiplication λµν extends to a multiplication λxµ, and the unique factorizationα=λα⁰µwhereσ(α)≥σ(λ) +σ(µ) extends to a unique factorizationx=λx⁰µwhereσ(x)≥σ(λ) +σ(µ). These properties are enough to define a MGDS given by shifts on the spaceX×Λ^∞ together with an equivariant map to the MGDS N^r×Λ^∞ given again by shifts. This map induces a map of the semidirect product groupoids which in turn induces a morphism between the algebras T^⊗r⊗C^∗(Λ) andE ⊗C^∗(Λ). This morphism is the crucial step in the proof of the duality.

Finally, we want to draw the attention to the results in [9, 10]. Our groupoid approach may give a hint to what a higher rank hyperbolic group would be.

1. PRELIMINARIES

In this paper we shall use locally compactr-discrete groupoids (possibly non-Hausdorff) ([23]). The canonical Haar systemλ^u is the discrete measure.

A 2-cocycle α in Z²(G,T) is a map α defined on the set G² of composable pairs toTsuch thatα(x, yz)α(y, z) =α(xy, z)α(x, y) for any (x, y),(y, z)∈G² ([21]). The convolution algebra C_c(G, σ) is given by the operation

f∗g(x) = Z

f(xy)g(y⁻¹)α(xy, y⁻¹)dλ^d(x)(y), f^∗(x) =f(x⁻¹)α(x, x⁻¹).

We shall use mainly the reduced algebraCr(G, α) which is a certain completion of C_c(G, α). Any open invariant set of G⁰ gives rise to an ideal of C_r(G, α).

We use the cocycle σ in order to include in our study rotation algebras and some twisted higher rank graph algebras.

Two partial maps on a set X,L: dom(L)→ran(L) andR : dom(R)→ ran(R) can be composed if one defines

dom(LR) ={x∈dom(R);R(x)∈dom(L)}

and LR(x) = L(R(x) for any x ∈ dom(LR). We say that two partial maps L and R on a set X commute if dom(LR) = dom(RL) and LR = RL on dom(LR). We setL⁰ = id for any partial map onX. IfLis injective, we denote by L⁻¹ the partial mapL⁻¹ : ran(L)→ dom(L). A local homeomorphism is a mapφ:X→Y with the property that each pointx has a neighborhoodU such that φ|_U :U →φ(U) is a homeomorphism.

A partial homeomorphism on X is a local homeomorphism S from an open set dom(S) ofXonto an open set ran(S) ofX. A setGof partial homeo- morphisms of X which is closed under composition, inversion and containing the identity is called a pseudogroup ([2]). For any setS of partial homeomor- phisms on X, there exists the smallest pseudogroup [S] generated by S. The semi-direct product groupoid XoG of a pseudogroup G is the set of triples (x, S, y) where S∈ G, y ∈dom(S), x=S(y) with the obvious operations

(x, S, y)(y, T, z) = (x, ST, y), (x, S, y)⁻¹= (y, S⁻¹, x).

The topology is given by the product topology of X and G where G has the discrete topology. The groupoid of germs is a quotion of the semidirect product groupoid by the equivalence relation (x1, S1, y1) ∼ (x2, S2, y2) if and only if y₁ =y₂ and S₁ =S₂ on a neighborhood of y₁. The topology is the quotient topology. These two groupoids are r-discrete, that is the range and source maps are local homeomorphisms.

In the monoid N^r we write n = (n₁, . . . , n_r) with n_j ∈ N and e_k the coordinates (0, . . . ,1, . . . ,0).

2. CUNTZ-LIKE ALGEBRAS ASSOCIATED

WITH MULTIPLY GENERATED DYNAMICAL SYSTEMS Definition 2.1 (Conform [22]). A multiply generated dynamical system (MGDS) is a pair (X, T) whereX is a topological space andT = (T1, . . . , Tr) a system of r commuting partial homeomorphisms on X.

Examples 2.2. (i) ([12], Definition 5.1) Form∈N^r let N^r_m be the higher rank graph {(n, n⁰) ∈ N^r×N^r :n≤ n⁰ ≤m} where s(n, k) =k,t(n, k) =n, σ(n, k) =k−nand the composition is given by (n, k)(k, p) = (n, p). Let Λ be

a finitely aligned rank r graph and

XΛ={x:N^r_m →Λ;m∈N^r_m}

the space of finite, semifinite and infinite paths. The shapeσ can be extended to XΛ,σ(x) =m wherex is defined on N^r_m. For r = 2 an element inXΛ can be seen graphically as one of the following:

oo

oo oo oooo oo

A basis of a topology onX_Λis given by the sets{x:x(0, k) =λ, k ≤σ(x)≤m}

where k ∈ N^rm, λ ∈ Λ and m ∈ N^rm. Then the partial homeomorphisms T_k with dom(T_k) = {x ∈ X_Λ : σ(x) ≥ k}, T_k(x) : N^r_σ(x)−k → Λ, T_k(n, n⁰) = x(n+k, n⁰+k) give a MGDS.

(ii) In the example above the restriction to boundary paths∂X gives a subsystem. If Λ has no sources then∂X = Λ^∞={x:σ(x) = (∞. . .∞)} (see [12], Definition 5.10 and [17]).

(iii) Let X be the free monoid on r letters a1, . . . , ar, that is the set of words a_i1. . . a_ik. Put on X the discrete topology and define T = (L, R) the translations on the left and on the right dom(L) = dom(R) ={a_i1. . . a_ik: k≥ 1}(the set of nonvoid words),L(ai1ai2. . . ai_k) =ai2. . . ai_k,R(ai1. . . aik−1ai_k) = a_i1. . . a_ik−1. For instance, if r = 1 then X = N and dom(L) = dom(R) = N\ {0},L(k) =R(k) =k−1. It is clear that L andR commute.

We denote byG(X, T) the full pseudogroup generated by the restrictions of T_j|_U where U is an open subset of X on which T_j is injective. Because of the commutation conditions on T, we can defineTⁿ =T₁ⁿ¹T₂ⁿ²· · ·T_k^nr for n ∈ N^k. One can see a MGDS as an action of the semigroup N^r on X by partial local homeomorphisms. We write sometimesT_ninstead ofTⁿwhen we want to viewT as a semigroup. We need a technical condition on the domains of Tⁿ (domain condition)

(DC) dom(Tⁿ)∩dom(T^m)⊂dom(T^n∨m),

wheren∨mis the componentwise maximum. The following lemma is basically Lemma 2.4 from [22] for MGDS with (DC).

Lemma 2.3. Let (X, T) be a MGDS with (DC).

(i) A partial homeomorphism S belongs to G(X, T) if and only if it is locally of the form (T_|U^m)⁻¹T_|Vⁿ where m, n ∈ N^r, U is an open set on which T^m is injective and V is an open set on which Tⁿ is injective.

(ii) Let a ∈ X. Suppose that (T_|U^m)⁻¹T_|Vⁿ and (T_|W^p )⁻¹T_|Y^q are two par- tial homeomorphisms as in (i) having a in their domains and (T_|U^m)⁻¹T_|Vⁿa= (T_|W^p )⁻¹T_|Y^q a. If m−n = p−q, then (T_|U^m)⁻¹T_|Vⁿ and (T_|W^p )⁻¹T_|Y^q have the same germ at a.

Proof. (i) It is clear that (T_|U^m)⁻¹T_|Vⁿ ∈ G(X, T) and, since G(X, T) is full, it is still true for a partial homeomorphism locally of this form. The inverse of (T_|U^m)⁻¹T_|Vⁿ is (T_|Vⁿ)⁻¹T_|U^m which belongs to G(X, T). It remains to show that the product of (T_|U^m)⁻¹T_|Vⁿ and (T_|W^p )⁻¹T_|Y^q is locally of the same form. When r = 1 this is the alternative n ≥p or n≤ p given in the proof of Lemma 2.4 of [22]. In our setting this alternative does not work since N^r is not totally ordered. For this reason we need the condition (DC). Let x ∈ Y such that (T_|W^p )⁻¹T_|Y^q x = y ∈ V. We can suppose that this happens in a neighborhood of x and so we assume that it is true on Y. Then with z = (T_|U^m)⁻¹T_|Vⁿy we have T^qx =T^py and Tⁿy=T^mz. From condition (DC) we havey∈dom(T^p∨n) andT^q+p∨n−px=T^p∨ny =T^m+p∨n−nzfor anyx∈Y so that (T_|U^m)⁻¹T_|Vⁿ(T_|W^p )⁻¹T_|Y^q is locally of the form (T_|Z^m+p∨n−n)⁻¹T_|Z^q+p∨n−p0 . (ii) Taking neighborhoods of a and (T_|U^m)⁻¹T_|Vⁿa, we may assume that W =U and Y =V,T^p(U) =T^q(V) and T^m∨p is injective on U. Forx ∈V, (T_|U^m)⁻¹T_|Vⁿx = y ∈ U, (T_|U^p )⁻¹T_|V^q x = y⁰ ∈ U we have T^my = Tⁿx and T^py⁰ = T^qx. As U ⊂dom(T^m)∩dom(T^p) ⊂dom(T^m∨p) we have T^m∨py = T^n+m∨p−mx = T^q+m∨p−px = T^m∨py⁰ and therefore y = y⁰ so (T_|U^m)⁻¹T_|Vⁿ = (T_|U^p )⁻¹T_|V^q .

Having defined a pseudogroup, we denote by Germ(X, T) the groupoid of germs of G(X, T).

Following [22], Definition 2.5 we consider another groupoid, the semidi- rect product groupoid (simply replacing Zwith Z^r):

Definition 2.4. Let (X, T) be a MGDS with (DC). Its semidirect grou- poid is

G(X, T) ={(x, m−n, y);m, n∈N^r, x∈dom(T^m), y∈dom(Tⁿ), T^mx=Tⁿy}

with the groupoid structure induced by the product structure of the trivial groupoid X×X and of the group Z^r. The topology is defined by the basic open sets

U(U;m, n;V) ={(x, m−n, y) : (x, y)∈U×V, T^mx=Tⁿy},

whereU (respectively V) is an open subset of the domain ofT^m (respectively Tⁿ) on whichT^m (respectively Tⁿ) is injective.

The family of given subsets is indeed a basis for a topology since U(U;m, n;V)∩ U(U⁰;m⁰, n⁰;V⁰)⊃ U(U∩U⁰;m∨m⁰, n∨n⁰;V ∩V⁰).

Thus, γ = (x, z, y) and η = (x⁰, z⁰, y⁰) in G(X, T) are composable if and only if y = x⁰ and then γη = (x, z +z⁰, y⁰). The range and domain are r(x, z, y) = x and d(x, z, y) = y. This is a groupoid indeed since y ∈ dom(T^n∨m0) soT^m+n∨m0−nx=T^{n0+n∨m0−m0}y⁰. In the absence of the condition (DC),G(X, T) may not be a groupoid. Consider, for example, (X, L, R) as in Example 2.2(iii). The condition (DC) is not satisfied since dom(L)∩dom(R) is the set of nonvoid words, while dom(LR) is the set of words with length greater or equal to 2. Let|·|be the word length onXand Ω the empty word. Forx, y∈ X we have γ = (x,(|x|,−|y|), y)∈G(X, T) andη = (y,(|y|,0),Ω) ∈G(X, T) since T^(|x|,0)x =L^|x|x = Ω =R^|y|y=T^(0,|y|) and T^(|y|,0)y =L^|y|y= Ω =T⁰Ω butγη= (x,(|x|+|y|,−|y|),Ω)∈/G(X, T) since x /∈dom(Tⁿ) for anyn∈N² with n1>|x|.

We assume again that (X, T) have (DC). According to (ii) of the previous lemma, there is a map π fromG(X, T) onto Germ(X, T) which sends (x, m− n, y) into the germ [x,(T_|U^m)⁻¹T_|Vⁿ, y] where U is an open neighborhood of x on which T^m is injective andV is an open neighborhood ofy on whichTⁿ is injective. This map is continuous and is a groupoid homomorphism. It is an isomorphism when (X, T) is essentially free.

Definition 2.5 (Conform [22], Definition 2.6). We shall say that a MGDS (X, T) is essentially free if for every pair of distinct m, n ∈ N^r, there is no open set on which Tⁿ and T^m agree.

Lemma2.6 (Conform [22], Lemma 2.7). Let (X, T) be an essentially free MGDS with (DC). Then

(i) If (T_|U^m)⁻¹T_|Vⁿ and (T_|W^p )⁻¹T_|Y^q , where m, n, p, q ∈ N and U, V, W, Y are open sets such thatT_|U^m, T_|Vⁿ, T_|W^p , T_|Y^q are injective and have the same germ at a, then m−n=p−q.

(ii) The map c: Germ(X, T)→Z^r such that c[(T_|U^m)⁻¹T_|Vⁿx,(T_|U^m)⁻¹T_|Vⁿ, x] =m−nis a continuous homomorphism.

Proof. (i) By assumption, we have T^my = Tⁿx and T^py = T^qx for x and y in neighborhoods of a respectively b = (T_|U^m)⁻¹T_|Vⁿa. Then on these neighborhoods we have T^n+m∨p−mx = T^m∨py = T^q+m∨p−px. The essential freeness implies that n+m∨p−m=q+m∨p−pson−m=q−p.

(ii) We have seen in the proof of the previous lemma that the product of (T_|U^m)⁻¹T_|Vⁿ and (T_|W^p )⁻¹T_|Y^q is locally of the form (T_|Z^m+p∨n−n)⁻¹T_|Z^q+p∨n−p0

and that the inverse of (T_|U^m)⁻¹T_|Vⁿ is (T_|Vⁿ)⁻¹T_|U^m. This shows that c is a homomorphism and by construction it is locally constant.

Proposition 2.7. Let (X, T) be a MGDS with (DC). Then (X, T) is essentially free if and only if the above surjection π:G(X, T)→Germ(X, T) is an isomorphism.

Proof. Word with word as in [22], Proposition 2.8.

This homomorphism induces a morphism of algebras even if (X, T) is not essentially free. We construct it in

Lemma 2.8. Let π : G1 → G2 a surjective morphism between two r- discrete groupoids with the property that if s(π(x)) = r(π(y)) then s(x) = r(y) Then the correspondence Cc(G1) 3 f → π(f˜ ) ∈ Cc(G2), π(f˜ )(x) = P

π(x⁰)=xf(x⁰), is a *-homomorphism between the topological algebras C_c(G₁) and Cc(G2). Composing it with the regular representation of Cc(G2) we ob- tain a bounded representation of C_c(G₁) therefore a morphism π˜ :C^∗(G₁) → C_r^∗(G₂).

Proof. The condition in the statement is (π ×π)⁻¹(G²₂) = G²₁. This means that composable pairs lifts only to composable pairs. In particular, the restrictionπ⁰ ofπ to the unit space must be a bijection. Moreover,π satisfies this condition if π⁰ is a bijection which identifies the equivalence relations as- sociated toG₁andG₂. In particular ifπ(u) =r(x) then{x⁰;π(x⁰) =x} ⊂G^u₁. Indeed, if π(x⁰) = π(y⁰) =x then π(y⁰⁻¹) and π(x⁰) are composable, so y⁰⁻¹ and x⁰ are composable, hence r(x⁰) = r(y⁰). This says that π(G^u₁) = G^π₂^0(u). Iff ∈C_c(G₁) then {x⁰ :π(x⁰) =x} ⊂supp(f)∩G^u₁ which is finite. Therefore, the sum which defines ˜π(f) is finite. We have supp(˜π(f)) ⊂ π(supp(f) so

˜

π(f)∈C_c(G₂).

We compute

˜

π(f ? g)(x) = X

π(x⁰)=x

f ? g(x⁰) = X

π(y⁰z⁰)=x

f(y⁰)g(z⁰),

˜

π(f)?π(g)(x) =˜ X

yz=x

˜

π(f)(y)˜π(g)(z).

Since π is surjective we have

˜

π(f)?π(g)(x) =˜ X

π(y⁰)π(z⁰)=x

f(y⁰)g(z⁰).

The first sum runs over {(y⁰, z⁰) : π(y⁰)π(z⁰) = x} and the second over sum runs over {(y⁰, z⁰) :π(y⁰z⁰) =x}. These two sets are equal by the assumption on π so ˜π is an algebraic morphism.

If fn ∈ Cc(G1), supp(fn) ⊂ K, fn →^u f on K, then ˜π(fn) →^u π(f˜ ) since the cardinal of the set {x⁰ :π(x⁰) = x} is bounded when x runs over a compact set.

Finally, one has

k˜π(f)k_I = sup

u∈G⁰₂

X

r(x)=u

|˜π(f)(x)|=

= sup

u∈G⁰₂

X

r(x)=u

X

π(x⁰)=x

f(x⁰)

≤ sup

u∈G⁰₂

X

r(x)=u, π(x⁰)=x

|f(x⁰)|.

Sinceπ(G^u₁) =G^π₂^0(u) and π⁰ is a bijection, the last sum is sup

u∈G⁰₁

X

r(x)=u

|f(x)|=kfk_I ≥ k˜π(f)k_I.

Therefore, kfk_I ≥ kπ(f˜ )k_B(L2(G2)) since the regular representation is bounded.

Proposition2.9. Lemma2.8holds forG₁ =G(X, T),G₂ = Germ(X, T) and π the morphism of Lemma 2.3.

Proof. π is continuous and surjective. π(x, z, y) andπ(x⁰, z⁰, y⁰) are com- posable if and only if y =x⁰ which is the condition for (x, z, y) and (x⁰, z⁰, y⁰) to be composable.

We assume from now on that X is Hausdorff, second countable and locally compact. Then G(X, T) becomes a Hausdorff, locally compact ´etale groupoid. In the next theorem we prove the amenability of G(X, T). For the proof we need some notations. The motivation comes from Example 2.2. For 1 ≤ j ≤ r let Xj = T

n∈Ndom(T_jⁿ) and for a subset J ⊂ {1, . . . , r} (J may be void) we denote by X_J =T

j∈JX_j ∩T

j /∈J(X\X_j). Clearly, X_J∩X_J⁰ = Φ for J 6=J⁰ and for x ∈ X we have x ∈ XJx where Jx = {j :x ∈Xj} so the subsets X_J provide a partition ofX. When r = 1 we get the partition in the end of the proof of Proposition 2.9 (i) in [22]. For x∈X let

σ(x) = sup{n∈N^r :x∈dom(Tⁿ)} ∈N^r.

For the rank one case σ(x) may be considered as the exit time of T, the first time when x escapes the domain of T. In general σ(x)_j can be thought of as the exit time ofTj so we call σ(x) the exit time of T. The crucial assumption (DC) ensures the existence of this supremum. Then X_J ={x∈ X :σ(x)_j =

∞ forj ∈J and σ(x)_j finite for j /∈J}. Forn∈N^r andJ ={j:n_j =∞}we have σ⁻¹(n) = T

j∈JX_j∩T

j /∈Jdom(T^nj)\dom(T^nj+ej)⊂ X_J which is Borel

and analytic so that a preimage byσ is analytic. Note that from the condition (DC) we have

\

j /∈J

dom(Tⁿ)\dom(T^n+ej) = \

j /∈J

dom(T_j^nj)\dom(T_j^nj+1).

Denote byZ^Jthe subgroup ofZ^rgiven by the inclusionZ^|J|3z7→P

j∈Jzjej ∈ Z^r and, for z ∈ Z^r, let zJ =P

j∈Jzjej ∈ Z^J. We use a similar notation for N^J andn_J.

Lemma 2.10. (i) Forx, y∈X and n∈N^r we have σ(Tⁿx) =σ(x)−n.

(ii) For (x, z, y) ∈ G(X, T)|_XJ we have zJ^c = σ(x)J^c −σ(y)J^c, where J^c ={1, . . . , r}\J.

Proof. (i) By definition of σ we have σ(Tⁿx) = sup{m ∈ N^r : x ∈ dom(T^m+n)} = sup{m−n ∈ N^r : m ≥ n, x ∈ dom(T^m)} = sup{m ∈ N^r : m≥n, x∈dom(T^m)} −nσ(x)−n.

(ii) By definition ofG(X, T) there existn, m∈N^rsuch thatTⁿx=T^my so by (i) we haveσ(Tⁿx) =σ(x)−n=σ(T^my) =σ(y)−m. We can substract the finiteJ^c coordinates to getz_J^c =n_J^c−m_J^c =σ(x)_J^c−σ(y)_J^c.

The following lemma is very likely folklore

Lemma 2.11. Let G be a Borel groupoid, (X_j)j∈J, J a finite set, a par- tition of G⁰ by invariant Borel sets. Then G is measurewise amenable if and only if G|_Xj is measurewise amenable.

Proof. This follows directly from the definition of amenability ([1], Defi- nition 3.2.8). Precisely, we denote bytthe disjoint union of Borel set with the obvious Borel structure. We know that G⁰ =tX_j, G= tG|_Xj so L^∞(G) =

⊕L^∞(G|_Xj) and L^∞(G⁰) = ⊕L^∞(X_j). Therefore a mean m : L^∞(G) → L^∞(G⁰) is invariant if and only if the restrictions mj :L^∞(G|_Xj)→ L^∞(Xj) are invariant. Conversely, we can piece m_j together to define m = ⊕m_j an invariant mean L^∞(G)→L^∞(G⁰).

Theorem2.12 (Conform [22], Proposition 2.9). Let (X, T) be a MGDS with (DC). Then

i)G(X, T) is amenable;

ii) the full and reduced C^∗-algebras coincide;

iii) the C^∗-algebra C^∗(X, T) =C^∗(G(X, T)) is nuclear.

Proof. (i) We will check measurewise amenability (according to [1], 3.3.7, it is equivalent to topological amenability for ´etale groupoids). Each XJ is an invariant Borel set for G(X, T). By Lemma 2.11 it is enough to prove that the reduction of G(X, T) on XJ is amenable. By the definition of XJ,

the homomorphism cJ : G(X, T)|_XJ → Z^J defined by cJ(x, z, y) = zJ is strongly surjective in the sense given in [1], Definition 5.3.7. We shall show that R_J =c⁻¹_J (0) is an amenable equivalence relation. Once this proven, we can now apply a result on the amenability of an extension ([1], 5.3.14) to conclude that G(X, T)|_XJ is amenable, henceG(X, T) amenable.

We have

RJ ={(x, m−n, y) :x, y∈XJ,∃n, m∈N^r, withmJ =nJ, x∈dom(Tⁿ), y∈dom(T^m), T^m(x) =Tⁿ(y)}.

If (x, z, y) ∈R_J we havez_J = 0 and z_J^c =σ(x)_J^c−σ(y)_J^c (conform Lemma 2.10(ii)) so z depends only on x, y and RJ ⊂XJ ×XJ. We leavez out when we have such an element. We show that R_J is an equivalence relation. If (x, y),(y, z) ∈ R_J then we can find n, n⁰, m, m⁰ ∈ N^r such that n_J = n⁰_J, Tⁿx=Tⁿ⁰y,mJ =m⁰_J,T^my =T^m0z. As we have seen before, we can assume n_J^c = σ(x)_J^c, n⁰_Jc =σ(y)_J^c,m_J^c =σ(y)_J^c, m⁰_Jc =σ(z)_J^c. (DC) gives again T^{nJ∨mJ+σ(x)J c}x = T^{nJ∨mJ+σ(y)J c}y = T^{nJ∨mJ+σ(z)J c}z and then (x, z) ∈ RJ

which proves that R_J is an equivalence relation.

We shall show that R_J is an inductive limit of amenable equivalence relations. For N ∈N^J let

R^N_J ={(x, y)∈R_J :∃n, m∈N^r such thatn_J =m_J ≤N, Tⁿ(x) =T^m(y)}.

By Lemma 2.10(ii), we can choose n, m such that n_J^c =σ(x)_J^c and m_J^c = σ(y)_J^c soR^N_J can be described as

R^N_J ={(x, y)∈R_J :∃n∈N^J, n≤N, such thatT^{n+σ(x)J c}(x) =T^{n+σ(y)J c}(y)}.

R^N_J is an equivalence relation onX_J since we have seen before that T^{nJ∨pJ+σ(x)J c}x=T^{nJ∨pJ+σ(z)J c}z

for (x, y, m, n),(y, z, p, q) defining two elements inR^N_J as above. RJ =S

N∈N^J ·

·R_J^N, (R^N_J)⁰ = XJ = (RJ)⁰ and R^N_J = R^N_J⁺¹. We shall show that R^N_J is a proper equivalence relation. This ensures that R_J is the inductive limit of R^N_J in the sense of [1], Chapter 5.3.f. Then Proposition 5.3.37 of [1] gives the amenability of R_J.

Since R^N_J is a discrete equivalence relation we have to show that the space XJ/R^N_J is analytic (conform [1], Example 2.1.4(2)). According to [3], Corollary 4.12, this follows from the countable separability of X_J/R^N_J. Since σ is a Borel map, we can find a sequence Yi of Borel subsets of XJ such that Y_i ⊂ {x∈X_J :σ(x)_J^c =k} for somek∈N^J

c and the restriction of Tⁿ to Y_i is injective. We can suppose furthermore that (Yi)i is separating forXJ and is closed under finite intersections. The saturation of a Borel set A ⊂ X_J is [A] =S

n∈N^J, n≤N(T^{n+σ(·)J c})⁻¹T^{n+σ(·)J c}(A) which is a Borel set. We use here

the notation T^{n+σJ c(·)} for the Borel map x7→ T^{n+σJ c(x)}(x). Let (x, y)∈/ R_J^N, x∈Y_i. We prove that we can choose Y_j such thaty /∈[Y_j] soπ(Y_j) separates π(x) and π(y) where π is the projection from X_J to X_J/R^N_J. If y /∈ Y_i we are done. If y ∈ [Yi], there exists y⁰ ∈ Yi such that (y, y⁰) ∈ R^N_J. Since the family (Y_i)_i separates X_J and is closed under finite intersections, we can find another set Yj ⊂Yi which separates x and y⁰, that is x ∈Yj and y⁰ ∈/ Yj. If y /∈ [Y_j] we are back to the case y /∈ [Y_i]. If y ∈ [Y_j] then (y, y⁰⁰) ∈ R^N_J for some y⁰⁰ ∈ Yj ⊂ Yi so (y⁰, y⁰⁰) ∈ R^N_J. This means that there exists n ∈ N^J, n ≤ N such that T^{n+σ(y0)J c}(y⁰) = T^{n+σ(y00)J c}(y⁰⁰). Since y⁰, y⁰⁰ ∈ Y_i we have σ(y⁰)J^c =σ(y⁰⁰)J^c. T^{n+σ(·)J c} is injective onYisoy⁰ =y⁰⁰. This contradicts the choice of Y_j (y⁰ ∈/Y_j). Therefore,y /∈[Y_j].

Corollary 2.13. There is a morphism

˜

π:Cr(G(X, T))→Cr(Germ(X, T)) induced by the canonical morphism π :G(X, T)→Germ(X, T).

Proof. According to Proposition 2.9 there is a morphism C_r^∗(G(X, T)) =C^∗(G(X, T))→C_r^∗(Germ(X, T)).

3. GRUPOIDS OF TWO-SIDED TOEPLITZ ALGEBRAS OF A DIRECTED GRAPH

Let us start with the classical Toeplitz extension. The two-sided Toeplitz extension is given by pulling back the Toeplitz extension of C(T) with the diagonal morphism. If S is a proper isometry this Toeplitz algebra is T = C^∗((S, z1),(S, z2))⊂C^∗(S)⊕C(T)⊗C(T). If we think of C(T) as C^∗(Z) the diagonal morphism is given by the morphismZ×Z3(x, y)7→x+y∈Z. C^∗(S) is a Cuntz-like algebra of the groupoid given by (N, T) where T : N\ {0} → N, T(n) = n−1. We replace now the points (∞, z,∞) by (∞,(z1, z2),∞) wherez₁, z₂∈Zto get another groupoid, G. Two pairs (x, z, y) and (x⁰, z⁰, y⁰) are composable if y = x⁰ and then (x, z, y)(y, z⁰, y⁰) = (x, z +z⁰, y⁰) where z+z⁰ is the addition inZ orZ² accordingly. As an algebraic object, Gis the reduction of G(N, T)×G(N, T) to the diagonal of N×N. Its reduction to N (an invariant subset) isN×Nand to{∞}isZ×Z. The basis of the topology is given by {(n, n−m, m)}, n, m∈Nand

Bz1,z2 ={(n+z1+z2, z1+z2, n) :n∈N} ∪ {(∞,(z1, z2),∞)}.

This is not a Hausdorff topology since if z1+z2 =z₁⁰ +z₂⁰ then the sequence (n, z₁+z₂, n+z₁+z₂) converges both to (∞,(z₁, z₂),∞) and (∞,(z₁⁰, z₂⁰),∞).

However, the sets in the above base are Hausdorff and compact, G^u, u ∈ N

is a discrete set in the relative topology from G and N, the unit space, is Hausdorff. G is therefore a non-Hausdorff r-discrete groupoid. The algebra Cc(G) is generated by 1{(m,n)} with m, n ∈ N and 1_B(z1,z2). It is easy to check that the map which sends 1_B(1,0) to(S, z1) and 1_B(0,1) to (S, z2) can be extended to an isomorphism from C^∗(G) to T.

We define next a groupoid which gives rise to the algebraC^∗(L_λ⊕s_λ, R_λ⊕ t_λ)⊂ E ⊕C^∗(Λ)⊗C^∗(Λ^op) where Λ is a rank one graph. Let X= Λ∪(Λ^∞× (Λ^op)^∞) be the space of finite words and pairs (x, y) of infinite words given by the graphs Λ respectively Λ^op. The sets {λ}, λ ∈ Λ and {λxµ : x ∈ Λ} ∪ {(λx, µy)(x, y) ∈Λ^∞×(Λ^op)^∞} provide a basis for a Hausdorff locally compact topology on X. X is a compactification of Λ provided that Λ is connected (there is at least one finite paths between any two objects). We assume this hereafter. A sequence in λn∈Λ converges inX if and only if it is eventually constant or converges in Λ^∞ and (Λ^op)^∞}. Equivalently, it means that (λ_n, λ_n) converges in X_Λ×X_Λ^op. This shows again that the unit space is a kind of diagonal. Note that for x∈Λ, λxµ isµ(λx) if we think ofλx as an element in Λ^op. We define a groupoid G

{(λ, σ(λ)−σ(µ), µ) :λ, µ∈Λ} ∪ {((λx, µy),(σ(λ)−σ(λ⁰),

σ(µ)−σ(µ⁰)),(λ⁰x, µ⁰y)) : (λ, µ),(λ⁰, µ⁰)∈Λ×Λ^op, (x, y)∈Λ^∞×(Λ^op)^∞}, where two pairs (x, z, y) and (x⁰, z⁰, y⁰) are composable if y = x⁰ and then (x, z, y)(y, z⁰, y⁰) = (x, z+z⁰, y⁰) where z+z⁰ is the addition inZorZ² accor- dingly. We see that the piece of information σ(λ)−σ(µ) in (λ, σ(λ)−σ(µ), µ) is redundant, that is G reduced to Λ is the principal groupoid Λ×Λ. We freely avoid it whenever convenient. As an algebraic objectG can be thought of as the reduction ofG(X_Λ, T_Λ)×G(X_Λ^op, T_Λ^op) to X. ButGdoes not have the induced topology which is rather odd. A basis for the topology is given by the sets {(λ, µ)},λ, µ∈Λ and

B(λ, µ, λ⁰, µ⁰) ={(λxµ, λ⁰xµ⁰) :x∈Λ} ∪ {((λx, µy),(σ(λ)−σ(λ⁰), σ(µ)−σ(µ⁰)),(λ⁰x, µ⁰y)); (λ, µ),(λ⁰, µ⁰)∈Λ×Λ^op, (x, y)∈Λ^∞×(Λ^op)^∞}.

This groupoid may not be Hausdorff, the non-Hausdorffness appearing from periodic infinite paths. To see it, we construct a sequence which converges to two distinct points. Start with λ ∈ Λ^∗, o(λ) = t(λ). Then the sequence (λⁿ, λⁿ)_n converges to ((λ^∞, λ^∞),(σ(λ^k),−σ(λ^k),(λ^∞, λ^∞)) for any k ∈ N. Nevertheless, the topology induced on the sets B(λ, µ, λ⁰, µ⁰) is Hausdorff.

Indeed, it is clear that the points (λ, µ) are separated being open sets. Let (x, z, y) 6= (x⁰, z, y⁰) ∈ B(λ, µ, λ⁰, µ⁰) such that x ∈ Λ^∞×(Λ^op)^∞. Here z = (σ(λ)−σ(λ⁰), σ(µ)−σ(µ⁰)). Let us suppose thatx= (x1, x2)6=x⁰ = (x⁰₁, x⁰₂), the case y6=y⁰ being similar. Assumex₁ 6=x⁰₁ the casex₂ 6=x⁰₂ being similar.

Then there exist γ 6=γ⁰∈Λ,a, b∈Λ^∞ such thatγ ≥λ,γ⁰≥λ⁰,σ(γ) =σ(γ⁰)

and x1 =λa, x⁰₁ =γ⁰b. The sets B(γ, δ, µ, µ⁰) and B(γ⁰, δ⁰, µ, µ⁰) are disjoint for anyδ, δ⁰ ∈Λ,δ, δ⁰ ≥λ⁰,σ(δ) =σ(δ⁰) =σ(λ⁰) +σ(γ)−σ(λ). Therefore,Gis a non-Hausdorff locally compact groupoid. The correspondence 1_{B(λ,Ω,Ω,Ω)} → L_λ⊕s_λ extends to an isomorphism from C^∗(G) to C^∗(L_λ⊕s_λ, R_λ⊕t_λ).

4. DUALITY IN BIVARIANT THEORIES

K-theory and K-homology are particular cases of the Kasparov groups.

The product inKK-theory provides the analogue of the cup and cap products in algebraic topology. Therefore KK-theory and other bivariant theories are the framework for notions of duality. KK-theory is suitable when geometric classes are available Ext-theory was used in [15, 9, 10]. The novelty in [20] is to use Ext^r but a technical condition of semisplitness leads toE-theory. We give first a general notion of duality analogous with the Spanier-Whitehead duality in algebraic topology. Then we give a condition which implies this duality. It appeared for the first time in [16], Chapter 4, Theorem 6, but was highlighted by Connes in [7], Chapter VI.4.β. After that we recall the construction of long exact sequences starting from an algebra and a tuple of ideals.

LetF be any of the bivariant theoriesKK-theory,E-theory, Ext-theory or KExt-theory. Forx∈F(A1⊗. . .⊗A_n, B1⊗. . .⊗B_m) we define the flip maps σ_ij(x) and σ^ij(x) the induced maps on F-groups by the flip homomorphisms ofC^∗-algebrasA1⊗. . .⊗A_i⊗. . .⊗A_j⊗A_n→A1⊗. . .⊗A_j⊗. . .⊗Ai⊗. . .⊗A_n respectivelyB₁⊗. . .⊗B_i⊗. . .⊗B_j⊗. . .⊗B_n→B₁⊗. . .⊗B_j⊗. . .⊗B_i⊗. . .⊗B_n. Definition 4.1 (Spanier-Whitehead duality, [15], Definition 2.2). LetA, B be separable C^∗-algebras, ∆∈F^r(A⊗B,C), δ∈F^r(C, A⊗B). We say that A andB are Spanier-Whiteheadr-dual inF-theory if the maps ∆_i:K_i(A)→ K^i+r(B), ∆i(x) = x⊗_A∆ and σ¹²(∆)i : Ki(B) → K^i+r(A), σ¹²(∆)i(x) = x⊗_B∆ are isomorphisms with inversesδ_i:K^i+r(B)→K_i(A), δ_i(y) =δ⊗_By respectively σ12(δ)i :K^i+r(A)→Ki(B),σ12(δ)i(y) =σ12(δ)⊗_Ay.

The duality classes, ∆ and δ, are the K-homology and the K-theory classes.

Theorem 4.2 ([10], Theorem 11). Let A, B,∆, δ be C^∗-algebras and classes as above such that

δ⊗_Bσ¹²(∆) = [1A], σ12(δ)⊗_A∆ = [(−1)ⁱ1B].

Then the maps ∆i andδi defined above are isomorphisms.

The proof can be found in [10]. We shall call such algebras simplyr-dual.

The sign in the second condition comes from the change of sign under flip maps σ. The sign is not given in [16] and [7]. We do not know of any example of two dual C^∗-algebras such that the assumptions of Theorem 4.2 are not fulfilled.

If B is the same as A we say that A is a Poincar´e duality algebra. The dual is not unique since ifAhas a dualB and AisKK-equivalent toA⁰ thenA⁰ is also dual with B.

Bivariant theories work well when the algebra in the first argument is separable. For KK-theory this is forced by the Kasparov technical theorem.

Since in the definition of duality both algebras appear in the first argument we are forced to work with separable algebras. For a separable algebra the K-theory group is countable. For this reason not every algebra has a dual.

For example M₂^∞, the CAR algebra, does not have a dual. If there was a dual B for M₂^∞ we would have K_i(B) = K¹(M₂^∞) where i is 0 or 1. But K¹(M2^∞) isZ(2)/Z([14]), whereZ(2) is the group of 2-adic numbers, which is uncountable. Therefore, B cannot be separable.

In [20] theK-homology class was given as an E-theory class associated to a long exact sequences. We recall the construction of long exact sequences starting from an algebra and a tuple of ideals.

LetEbe an arbitraryC^∗-algebra andJ₁, . . . , J_rberideals in it. We shall describe next a procedure of defining an r-fold exact sequence starting with B = T_r

i=1Ji and ending with A = E.

J1+. . .+Jr. The motivation comes from the Zekri’s Yonneda product ([24]) ₁ ⊗_{C 2} of two 1-fold extensions 1 ∈Ext(A1, B1) and2 ∈Ext(A2, B2). Let

0→B₁→E₁ →A₁ →0 0→B₂→E₂ →A₂ →0

be the extensions 1 and 2. Forgetting for the moment the troubles caused by tensor products, the product ₁⊗_C2 is computed by tensoring₁ withB₂ on the right, ₂ withA₁ on the left and then splicing:

0→B1⊗B2→E1⊗B2 →A1⊗E2 →A1⊗A2 →0.

Define E =E1⊗E2,J1 =B1⊗E2,J2 =E1⊗B2. We have B₁⊗B₂=J₁∩J₂ =J₁J₂, E₁⊗B₂ =J₂,

A1⊗E2 =E/J₁, A1⊗A2 =E/J₁+J₂ so the exact sequence is

0→J1∩J2 →J2→ E/J₁ → E/J₁+J₂→0.

For a nonempty subset S ⊆ {1, . . . , r} let us define J^S = \

j∈S

Jj and JS=X

j∈S

Jj.

We shall use the abbreviation{k, k+ 1, . . . , l}=k, l. Define E₀=J1∩. . .∩Jr =J_1,r,E₁ =J2∩. . .∩Jr=J_2,r, E_k=Jk+1∩. . .∩Jr

.

(J₁+. . .+Jk−1)∩J_k+1∩. . .∩J_r=

=J^k+1,r.

J_1,k−1∩J^k+1,r, for any k∈ {2, . . . r−1} and

E_r=E.

J1+. . .+Jr−1 =E. J^1,r−1, E_r+1 =E.

J1+. . .+Jr =E. J^1,r.

E₀ ⊆ E₁ so we can define i0 : E₀ ,→ E₁. Since E₁ ⊆ J3∩. . .∩Jr, there is a mapi₁ :E₁ → E₂ given by the inclusion composed with the quotient map. For k∈ {2, . . . , r−2} we have

J^k+1,r =J_k+1∩. . .∩Jr⊆J_k+2∩. . .∩Jr =J^k+2,r and

J_1,k−1∩J^k+1,r ⊆J_1,k∩J^k+2,r

so that we can again define a map i_k : E_k → E_k+1 as the bottom line of the diagram

J^{k+1,r  //}

πk

J^k+2,r

πk+1

E_k ^{ik //}E_k+1

where the vertical arrows are projections. Using the isomorphism J.

J∩I ' J +I.

I we write E_r−1Jr

.

J_1,r−2∩J_rJ1+· · ·+Jr−2+Jr

.

J₁+· · ·+Jr−2.

Therefore, there is also a natural homomorphism ir−1 :E_r−1 → E_r since J1+

· · ·+Jr−2+J_r⊆ E andJ₁+· · ·+J_r−2 ⊆J₁+· · ·+Jr−1. Finally,i_r :E_r → E_r+1 is defined since J1+· · ·+Jr−1⊆J1+· · ·+Jr.

Theorem 4.3 ([20], Proposition 3.1). The r-fold sequence 0 ^//B ^{i0 //}E₁ ^{i1 //}· · · ^{ir−1 //}E_r ^{ir //}A ^//0 is exact.