The Baum-Connes conjecture, noncommutative Poincare duality and the boundary of the free Group

Every hyperbolic group acts continuously on its Gromov boundary. One can form the corresponding cross-product C*-algebra A. We show that there always exists a canonical Poincare duality map from the K-theory of A to the K-homology of A. We show that this map is an isomorphism when the group in question is the free group on two generators. There is a direct connection between our constructions and the Baum-Connes Conjecture, and we use the latter to deduce our result.


Introduction
The aim of this note is to point out a connection between the Baum-Connes conjecture with coefficients for the free group F 2 on two generators, and a Poincaré duality result for the 'noncommutative space' ∂F 2 /F 2 , where ∂F 2 is the Gromov boundary of F 2 , acted upon minimally by F 2 through homeomorphisms.
In order to formulate what Poincaré duality should mean for a noncommutative space such as ∂F 2 /F 2 , one passes to the C * -algebra cross product C(∂F 2 ) ⋊ F 2 and to K-theory and K-homology for C * -algebras.Poincaré duality for ∂F 2 /F 2 then means an isomorphism between the K-theory and K-homology of C(∂F 2 ) ⋊ F 2 , induced by cap product with a fixed K-homology class.
More generally one can speak of C * -algebras having Poincaré duality, or, as we call them in this paper, Poincaré duality algebras.It seems that such algebras are in some sense noncommutative analogs of spin c manifolds.For the commutative examples of such C *algebras are given precisely by the C * -algebras C(M), where M is a compact spin c manifold.Such a manifold has, corresponding to the spin c -structure, a canonical elliptic operator on it -the Dirac operator -and thus (see e.g.[9]) a canonical K-homology class.Cap product with this class induces the Poincaré duality isomorphism.
Various noncommutative examples of Poincaré duality C * -algebras have been produced by A. Connes, the first of which was the irrational rotation algebra A θ .Several other examples now exist, but all have the same character insofar as they are in some sense deformations of actual spin c -manifolds.Our example is somewhat different.The geometric data underlying ∂F 2 /F 2 is highly singular: the space ∂F 2 is not a homology manifold, and the group F 2 is not a Poincaré duality group.It turns out to be true, however, that in factoring the space by the action of the group, i.e. by forming the cross product C * -algebra C(∂F 2 ) ⋊ F 2 , the resulting noncommutative space satisfies Poincaré duality.
Part of our goal is thus to point out this example and also to place it in its proper context: that of hyperbolic groups acting on their Gromov boundaries.The second part is Date: September 10, 2001.to show as mentioned above, a connection between our constructions and the Baum-Connes conjecture for F 2 .
We begin by constructing -in the full generality of hyperbolic groups -the K-homology class cap product with which will induce our Poincaré duality isomorphism.It turns out that with Gromov hyperbolic groups Γ in general there is a certain duality between functions continuous on the Gromov boundary ∂Γ of Γ, and right translation operators on l 2 Γ.Using this duality, we produce an algebra homomorphism C(∂Γ) ⋊ Γ ⊗ C(∂Γ) ⋊ Γ → Q(l 2 Γ), where Q(l 2 Γ) = B(l 2 Γ)/K(l 2 Γ) denotes the Calkin algebra of l 2 Γ, and where Γ is an arbitrary hyperbolic group.Since C(∂Γ) ⋊ Γ is nuclear ( [3]), such an algebra homomorphism yields via the Stinespring construction a class ∆ ∈ KK 1 We next wish to prove that cap product with ∆ as above gives an isomorphism in the case of Γ = F 2 , the general case of hyperbolic groups being beyond the scope of this paper.To this end we observe that a sort of geodesic flow on the Cayley graph of F 2 may be used to construct a dual element to ∆, this time in the K-theory of C(∂F 2 ) ⋊ F 2 ⊗ C(∂F 2 ) ⋊ F 2 , playing the same role in this context as does the Thom class of the normal bundle of M in M × M in the commutative setting.We obtain a putative inverse map We then set about calculating the composition of these two maps.The connection with the Baum-Connes conjecture appears in that the composition turns out to be multiplication by the γ-element constructed by Julg and Vallette.
As mentioned, the construction of our fundamental class ∆ makes sense for a general hyperbolic group acting on its boundary, and in fact several of our other constructions have their counterparts for arbitrary hyperbolic groups; thus for instance it is possible by means of work of Gromov to make sense of 'geodesic flow' for an arbitrary hyperbolic group.Furthermore, although the statement 'γ = 1' for general hyperbolic groups is false due to the possible presence of Property T, it is nevertheless true by work of Tu ([12]) that γ ∂Γ⋊Γ = 1 C(∂Γ) , where γ ∂Γ⋊Γ is the γ-element for the amenable groupoid ∂Γ ⋊ Γ, which weaker statement is all we need.Nevertheless, the arguments for the general case, being substantially more involved, will be dealt with in a later paper.We have chosen to emphasise the free group case for two reasons: one, that the relationship to the Baum-Connes conjecture is extremely explicit, and two, that the geometry of our constructions is particularly visible.
Finally, we note that our arguments tend to suggest that the phenomenom of Poincaré duality for amenable groupoid algebras constructed from boundary actions of discrete groups is relatively common.Specifically, the author expects similar results for uniform lattices in semisimple lie groups acting on their Furstenberg boundaries, and for discrete, cocompact isometry groups of affine buildings acting on the boundaries of these buildings.Along these lines, we draw the attention of the reader to the work of Kaminker and Putnam on Cuntz-Krieger algebras (see [8]); indeed, our result (in the case of the free group of two generators) can be deduced from theirs.In fact, our work was partly motivated by the idea of finding a geometric explanation for theirs.

Geometric Preliminaries
In this section we work in the generality of a Gromov hyperbolic group Γ (see [5] or [4]).So let Γ be such.Thus, we have fixed a generating set S for Γ and the corresponding metric d(γ 1 , γ 2 ) = |γ −1 1 γ 2 |, where | • | denotes the word length of a group element with respect to S, and with this metric Γ is hyperbolic in the sense of Gromov as a metric space.Note that the metric is clearly invariant under left translation.
Recall that with the hypothesis of hyperbolicity, the group Γ viewed as a metric space can be compactified by addition of a boundary: thus there exists a compact metrizable space Γ = Γ ∪ ∂Γ such that Γ sits densely in Γ, and Γ is compact.The compactification is Γ-equivariant in the sense that the left translation action of Γ extends to an action by homeomorphisms on Γ.
There turns out to be an interesting duality between functions on Γ which extend continuously to the Gromov compactification Γ, and a certain class of operators on l 2 Γ, as follows.First we recall a definition.For what follows, let e x , e y , etc, denote the standard basis vectors in l 2 Γ corresponding to points x, y ∈ Γ.Also, if f is a function on Γ, we shall denote by M f the corresponding multiplication operator on l 2 Γ.Definition 1.An operator T ∈ B(l 2 Γ) is finite propagation if there exists R > 0 such that < T (e x ), e y >= 0 whenever d(x, y) ≥ R.
The duality we have alluded to is stated in the following: Lemma 2. If f is a function on Γ which extends continuously to Γ, then [M f , T ] is a compact operator for all finite propagation operators T on l 2 Γ.
For the proof, we shall need to use the following fact about the Gromov compactification of a hyperbolic group (see [5]).
Note that here and elsewhere in this paper, B r (x), for r > 0 and x ∈ Γ, denotes the ball of word-metric radius r centered at x. Lemma 3. If f is a continuous function on Γ, then for every R > 0, we have Let T be a finite propagation operator with propagation R, and f a bounded function on Γ which extends continuously to Γ.In Section 3 we will show how the above constructions can be organized to produce a K-homology class inducing a Poincaré duality isomorphism.

KK-theoretic preliminaries
In this section we recall some basic facts from KK-theory.For further details we refer the reader to [1], or to [9].
KK. KK can be understood categorically ( [6]): there is a category KK whose objects are separable, nuclear C * -algebras and whose morphisms A → B are the elements of KK(A, B).There is a functor from the category of C * -algebras to the category KK.If φ : A → B is an algebra homomorphism A → B, we denote its image under this functor as [φ].There is a composition, or intersection product operation We will sometimes use the notations φ * ([β]) and [φ] ⊗ B β interchangeably, as is warranted by clarity of notation.Similarly with φ * . If

and similarly a map KK(A, B) → KK(D ⊗ A, D ⊗ B).
Graded Commutativity.There are higher KK groups KK i (A, B) for all i ∈ Z, defined by KK i (A, B) = KK(A, B ⊗ C i ) where C i is the ith complex Clifford algebra, and one of the features of the theory is that the intersection product is graded commutative.If A 1 , . . ., A n are C * -algebras, let σ ij denote the map obtained by flipping the two factors.Then by graded commutativity we mean the following: Description of Even Cycles.We let B(E) denote bounded operators on a Hilbert module E, K(E) compact operators, and Q(E) the Calkin algebra B(E)/K(E).The quotient map B(E) → Q(E) will always be denoted by π.
Following Kasparov ([9]), if E is a Hilbert B-module and A acts on E by a homomorphism A → B(E), we will refer to E as a Hilbert (A, B)-bimodule.
Because all the algebras in this paper are ungraded -or alternatively, have trivial grading -we can make certain simplifications in the definitions of the KK groups (see [1]).With such ungraded A and B, cycles for KK(A, B) are given simply by pairs (E, F ) where E is an (A, B)-bimodule, F commutes modulo compact operators with the action of A, and a(F * F − 1) and a(F F * − 1) are compact for every a ∈ A.
Description of Odd Cycles.Cycles for KK 1 (A, B) are given by pairs (E, P ) for which P is an operator on the (A, B)-bimodule E satisfying the three conditions [a, P ], a(P 2 − P ), and a(P − P * ) are compact for all a ∈ A.
Let (E, P ) be an odd cycle.Then we obtain a homomorphism A → Q(E) by the formula a → π(P aP ).
Conversely, let τ : A → Q(E) be a homomorphism.Under the assumption of nuclearity of all algebras concerned, there exists a potentially larger Hilbert B-module Ẽ, a representation of A on Ẽ, an isometry U : E → Ẽ, and an operator P on Ẽ such that a(P 2 − P ), [a, P ], and a(P − P * ) are compact for all a ∈ A, and π(U * P aP U) = τ (a) for all a ∈ A (see [1]).The data ( Ẽ, P ) makes up an odd cycle.The process of constructing a Ẽ, U, and P , from an extension, we shall refer to as the Stinespring construction.
As a consequence, for A and B nuclear, we may regard KK 1 (A, B) as given by classes of maps τ : A → Q(E), where E is a right Hilbert B-module.This description of KK 1 -classes will be particularly appropriate to our purposes.
Bott Periodicity.Recall that x.We shall need to compute this map at the level of cycles in several simple cases.
Let ψ be the function ψ The significance of this simple lemma is that in the given setting it is not necessary to explicitly represent [τ ] as a KK-cycle (that is, perform the Stinespring construction) in order to calculate the Kasparov product of [ dR ] and [τ ].This is true also of the situation in the following lemma, which will be of direct use to us.
, where h ′ and h ′′ are homomorphisms.Suppose that the homomorphism h ′′ lifts to a homomorphism h′′ : ) is represented by the following cycle.The module is E with its original right A 2 -module structure and the left A 1 -module structure given by the homomorphism h′′ .The operator is given by U + 1 where U is any operator on E such that π(U) = h ′ (ψ).
The proof of both lemmas involves an application of the axioms for the intersection product, and is omitted (see [9]).
Equivariant KK.If Γ is a group acting on C * -algebras A and B, we have in addition to the group KK(A, B), an equivariant group KK Γ (A, B).We shall discuss this group briefly in connection with the γ-element and the work of Julg and Valette.Suffice it to say that the cycles for KK Γ (A, B) consist of the same cycles as for KK(A, B), but with the following extra conditions.(1) Γ acts as linear isometric maps on the Hilbert (A, B)-module E, in such a way that γ(aξb) = γ(a)γ(ξ)γ(b) for a ∈ A, b ∈ B and ξ ∈ E; (2) the operator F satisfies: γ(F ) − F is compact, for all γ ∈ Γ.
Regarding KK Γ as a category in its own right, with morphisms A → B the elements of KK Γ (A, B), and objects Γ-C * -algebras, there is a functor λ : KK Γ (A, B) → KK(A ⋊ Γ, B ⋊ Γ), called descent.The map λ : KK Γ (A, B) → KK(A ⋊ Γ, B ⋊ Γ) can be explicitly calculated on cycles; the formulas are given in [9].Since λ is a functor, it takes the unit 1 A ∈ KK Γ (A, A) to the unit 1 A⋊Γ ∈ KK(A ⋊ Γ, A ⋊ Γ), which fact we will make use of.

construction of the fundamental class
For this section, we shall return to the generality of a hyperbolic group Γ.Since Γ acts by homeomorphisms on ∂Γ, we can consider the cross product C * -algebra C(∂Γ) ⋊ Γ, which is our main object of interest in this paper.Note the cross product we are referring to is the reduced cross product; however, by the proof of the following lemma (whose proof can be found in [3]), the reduced and max cross products are in fact the same.
Our goal is to construct an element of the K-homology of the algebra ).This element will be presented as an extension; that is, as a map C(∂Γ) ⋊ Γ ⊗ C(∂Γ) ⋊ Γ → Q(H) for some Hilbert space H.By our remarks in the previous section and Lemma 9, such a map does produce a canonical class in KK 1 We construct two commuting maps λ, ρ : C(∂Γ) ⋊ Γ → Q(l 2 Γ).Let f ∈ C(∂Γ) and let f denote any extension of f to a continuous function on Γ.Let M f denote as above the multiplication operator on l 2 Γ corresponding to f , and let λ(f ) be the image in Q(l 2 Γ) of the operator M f .Let λ(γ) be the image in Q(l 2 Γ) of the unitary u γ corresponding to left translation by γ: u γ (e x ) = e γx , x ∈ Γ.It is easy to check that the assignments f → λ(f ), γ → λ(γ), define a covariant pair for the C * -dynamical system C(∂Γ), Γ , and so a homomorphism We can now define the 'cap-product map' interchanging the K-theory and K-homology of C(∂Γ) ⋊ Γ, which we are going to show is an isomorphism when Γ = F 2 .Specifically, define: Our main theorem is the following: Theorem 13.For Γ = F 2 and ∆ as in Defintion 11, the map ∩∆ is an isomorphism.

Connes' notion of Poincaré duality
In order to prove that the map ∩∆ of the previous section is an isomorphism, we shall use some ideas due to Connes.Theorem 14.Let A be a separable, nuclear C * -algebra, and ∆ a class in KK i (A ⊗ A, C).
Suppose we can find a class ∆ ∈ KK −i (C, A ⊗ A) such that the following equations hold: (2) Then the map ∩∆ : K j (A) → K j+i (A) defined previously, is an isomorphism with inverse (up to sign) the map If A is as above, with classes ∆ and ∆ satisfying Equations ( 1) and ( 2), we will call A a Poincaré duality algebra.

Proof. The hypotheses imply the two equations:
∆ and We show that as a consequence of these two equations, Expanding the product involved in (5), we obtain:

Consider the term (1
It is easy to check this is the same as Returning to the original product (5), we see the latter can be written Hence, putting back into the main product, we see that ( 5) can be written , where the last equality follows from equation ( 1).
Remark 15.We note that if we happen to have ∆ and ∆ as above, and (σ 12 ) * ( ∆) = (−1) i ∆, then the two equations ( 1) and ( 2) above would be the same, and it would suffice to show that one of them holds.This is the case in the commutative setting of a compact spin cmanifold, and will be the case for us, also, part of which we have already proven (Lemma 12).
We now set about proving Theorem 13 in the case of Γ = F 2 by verifying the equations ( 1) and (2) of Theorem 14 above, with, i.e.A = C(∂F 2 ) ⋊ F 2 and ∆ the fundamental class of Definition 11.We need first produce an element ∆ ∈ KK playing the role of the dual element in Theorem 14.We will then verify equation ( 1), the other being rendered superfluous as a consequence of Remark 15, which is applicable in this case.
It will turn out, rather surprisingly, that equation ( 1) can be shown to be equivalent to the equation , where γ ∂F 2 ⋊F 2 is the γ-element for the groupoid ∂F 2 ⋊ F 2 .Since this latter equation has been established by Julg and Valette, and also by J.L. Tu, we will by this device, i.e. by means of the Baum-Connes Conjecture, be done.

Construction of a dual element
In this section as for the rest of this note we specialize to the free group F 2 on two generators.We are going to define an element ∆ ∈ KK serving as an 'inverse' to ∆.
∆ shall be constructed by use of the fact that any two points of ∂F 2 may be connected by a unique geodesic.
By "geodesic" we shall mean an isometric map r : Z → F 2 .Topologize the collection of such r by means of the metric and denote the resulting metric space by GF 2 (we follow [4]).Both F 2 and Z act freely and properly on GF 2 , the former by translation (γr)(n) = γr(n), and the latter by flow (g n r)(k) = r(k − n).These actions commute.Note that GF 2 /F 2 is compact, whereas GF 2 /Z may be identified with the F 2 -space All these observations are easy to check.As a consequence of them, the C * -algebras C(GF 2 /F 2 ) ⋊ Z and C 0 (∂ 2 F 2 ) ⋊ F 2 are strongly Morita equivalent (see [10]).Let [E] denote the class of the strong Morita equivalence bimodule.It is an element of On the other hand, if u is the generator of We denote the class in It will be convenient for our later computations to define an auxilliary class [D], which will lie in Next, note that the cross product C 0 (∂ 2 F 2 ) ⋊ F 2 may be regarded as a subalgebra of C(∂F 2 ) ⋊ F 2 ⊗ C(∂F 2 ) ⋊ F 2 , via the composition of inclusions: Let i denote this composition.
Our class ∆ will be defined by: , where [ dR ] is as in Section 3, and [u − 1] and [E] are as above.
It will be convenient to calculate more explicitly the cycle corresponding to the class We will first describe an element , where χ is a projection.We will then set w = v − χ.Then, of course, w As the method of discovering such an explicit description (that is, of transfering Kclasses under strong Morita equivalences) is well known (see [2] in which a similar calculation is carried out in the context of A θ ) we give the outcome without further discussion.
As a function on Note that χ = v * v = vv * is the locally constant function on ∂ 2 F 2 given by χ(a, b) = 1 if some (therefore any) geodesic from a to b passes through e, and equals 0 else.
We can describe v in group-algebra notation as follows.Fix γ a generator.Then v(• , • , γ) is a function on ∂ 2 F 2 , and in particular is a function on ∂F 2 × ∂F 2 , whose representation as a tensor product of two functions on ∂F 2 is: where We can therefore represent v as Similarly we we represent the function χ by χ = χ γ ⊗ (1 − χ γ ), and it is easy to check that v * v = vv * = χ, as claimed.
Finally, we note the following: Proof.We have ∆ = i * ([D]), and so (σ by a direct calculation and we are done.
In the following sections we will show that in an appropriate sense ∆ provides an 'inverse' to the extension ∆.More precisely, we will show that the conditions of Theorem 14 are met by ∆ the fundamental class, and the element ∆ above.

The γ-element
Before proceeding to verify the equations of Theorem 14, we will need to recall the work of Julg and Vallette ([7]).
Up to now we have adopted the convention of writing even KK-cycles in the form (E, F ), where F is an operator on the module E. A different definition is possible, in which two modules are involved, and F is an operator between them.This was the set-up in the paper of [7].We will retain their notation temporarily.In a moment we will describe a means of geometrically describing their class in a way consistent with our conventions.
Consider the Cayley graph Σ for F 2 , which is a tree with edges Σ 1 and vertices Σ 0 .Note that we work with geometric edges, i.e. set theoretic pairs of vertices {x, x ′ }.If x is a vertex, let x ′ be the vertex one unit closer to e, the origin, and let s(x) be the edge {x, x ′ }.Define an operator b : Then it is clear that b is an isometry, is Fredholm, and has index 1.Next, note that F 2 acts unitarily on l 2 (Σ 0 ) and l 2 (Σ 1 ), and that, furthermore, γbγ −1 − b is a compact (in fact finite rank) operator, for all γ ∈ F 2 .
It follows that the pair Let γ denote its class.That γ = 1 in this group implies the Baum-Connes conjecture for F 2 .This fact (that γ = 1) was proved by Julg and Valette in [7].
We now set about describing a cycle equivalent to the above but which is in some sense simpler.To do this it will be notationally and conceptually simpler to work with fields.Thus, we note that E 0 and E 1 may be viewed as sections of the constant fields of Hilbert spaces {H 0 a | a ∈ ∂F 2 }, respectively {H 1 a | a ∈ ∂F 2 }, with H 0 a = l 2 (Σ 0 ) and H 1 a = l 2 (Σ 1 ) for all a ∈ ∂F 2 , and that the operator B may be regarded as the constant family of operators {b a | a ∈ ∂F 2 } with b a = b for all a ∈ ∂F 2 .What we are going to do is eliminate edges from the cycle at the expense of changing the constant field of operators to a nonconstant field.
To this end consider the field of unitary maps {U a : given by U a (e s ) = e x , where x is the vertex of s farthest from a.Note that the assignment a → U a , though not constant, is strongly continuous.For if a and b are two boundary points, then U a = U b except for edges lying on the geodesic (a, b).Consequently, if s is a fixed edge, and a and b are close enough, then U a (e s ) = U b (e s ), since if a and b are sufficiently close, s does not lie on (a, b).Now, consider the composition 2 , which we denote by W a .We see that for x = e, W a (e x ) = 0, and for x = e we have: where x ′ is the vertex one unit closer to e than x.
Since the assignment a → W a is continuous, we obtain a Hilbert C(∂F 2 )-module map E 0 → E 0 by defining for ξ ∈ C(∂F 2 ; l 2 F 2 ), (W ξ)(a) = W a (ξ(a)).Then, by unitary invariance of KK and the work of Julg and Vallette, we see: Since we have now altered the cycle of Julg and Valette up to equivalence so that only one Hilbert module is involved (it is now otherwise known as an 'evenly graded' Fredholm module), we may now return as promised to our conventions and write it simply (C(∂F 2 ; l 2 F 2 ), W ), consistent with the way we have been writing (even) KK-cycles up to now.
To summarize, we have: We shall next apply the descent map to the cycle described above, thus producing a cycle for KK(C(∂F which by functoriality of descent will be equivalent to the cycle corresponding to 1 C(∂F 2 )⋊F 2 .
A direct appeal to the definition of λ (see [9]) produces the cycle , the action of W on these functions is given by the formula ( W ξ)(γ) = W (ξ(γ)).We have: This concludes our preparatory work.We will now show that the class of the cycle given in the above lemma is the same as the class of the Kasparov product of the elements ∆ and ∆, concluding thus as a consequence of the work of Julg and Valette that equation (1) holds.

untwisting
We are interested in calculating the cycle corresponding to the Kasparov product ∆ .In this section we will do something we call -following an analogous procedure in [8] -'untwisting.'A simple but fundamental property of hyperbolic groups -and in particular of the free group -will be used: specifically, if two points a and b on ∂F 2 are sufficiently far apart then any geodesic connecting them passes quite close to the identity e of the group.This follows immediately from the definition of the topology on the compactified space F 2 .More precisely: Lemma 22.Let Ñ be a neighbourhood of the diagonal {(a, a) | a ∈ ∂F 2 } in ∂F 2 × F 2 .Then there exists R > 0 such that if (a, b) ∈ (∂F 2 × F 2 )\ Ñ, then the (unique) geodesic from a to b passes through B R (e).
Note 23.To simplify notation in this section, we shall denote by A the cross product C(∂F 2 ) ⋊ F 2 , and by B the algebra C 0 (∂ 2 F 2 ) ⋊ F 2 .
We remind the reader that the statement: "x ∈ [e, y]," for x, y ∈ F 2 may be equivalently read: "the reduced expression of y contains x as an initial subword," or more shortly, "y begins with x." With this in mind, consider the first case above.If g −1 a begins with γ but g −1 does not, it follows there is cancellation between g −1 and a; more precisely, a must begin with g, followed by γ. (Since g −1 does not begin with γ, g does not end in γ −1 , and hence gγ is in fact reduced.)We have: gγ ∈ [e, a), and g does not end in γ −1 0 else .

Now consider the operator
. This operates by ( f h h) ⊗ e g → ( F ′ γ ( • , gγ −1 )f h h) ⊗ e gγ −1 for f h h an arbitrary element of the cross product A. From our above work, we see that F ′ γ ( • , gγ −1 ) = 0 unless g ends in γ.On the other hand, if g does end in γ, gγ −1 does not end in γ −1 .Hence we see that the above operator sends ( f h h) ⊗ e g → ( χ g f h h) ⊗ e gγ −1 g ends in γ 0 else .
We see finally, that V = γ∈S F ′ γ • (1 ⊗ v γ ), which is a lift of τ (v), acts on A ⊗ l 2 F 2 by where the prime notation is as in the discussion just prior to Lemma 20.In particular, V as an operator on A ⊗ l 2 F 2 , where the latter is regarded as functions F 2 → C(∂F 2 ) ⊗ l 2 F 2 , has the form where V is the operator Otherwise expressed, let ξ be an element of C(∂F 2 ; l 2 F 2 ) of the form ξ(a) = ξ g (a)e g , where each ξ g is a scalar-valued function on ∂F 2 .Then (V ξ)(a) = g∈[e,a) ξ g (a) ⊗ e g ′ .Now apply the same calculations to the element τ (χ).We obtain the operator (projection) P on A ⊗ l 2 F 2 given by P = F ′ γ ∈ A ⊗ B(l 2 F 2 ) ⊂ B(A ⊗ l 2 F 2 ).
y) T xy e y where T xy denotes as usual < T (e x ), e y >.Therefore < [M f , T ](e x ), e y >= 0 if d(x, y) ≥ R, and equals f (x) − f (y) T xy else.The result follows immediately from Lemma 3. Let γ ∈ Γ, and ρ(γ) denote the unitary l 2 Γ → l 2 Γ induced from right translation by γ, ρ(γ)e x = e xγ −1 .The relevance of the above remarks to us lies in the following observation:
suitably interpreted in terms of the Clifford gradings.The class [ dR ] allows us to identify, for any C * -algebras A and B, the groups KK 1 (C 0 (R) ⊗ A, B), and KK(A, B), by the map KK 1 + .We begin by stating the simplest version of what we shall need.Lemma 7. Let τ be a homomorphism C 0 (R) → Q(H) to the Calkin algebra of some Hilbert space H. Let [τ ] denote the class in KK 1 (C 0 (R), C) corresponding to τ .Then the class [ dR ] ⊗ C 0 (R) [τ ] ∈ KK(C, C) is represented by the cycle (H, U + 1), where U is any operator on H such that π(U) = τ (ψ).