The trace on the K-theory of group C^{∗}-algebras

Thomas Schick^{∗} FB Mathematik — Uni Münster Einsteinstr. 62 — 48149 Münster, Germany

Last compiled: November 17, 2000, last edited Nov 17, 2000 or later

Abstract

The canonical trace on the reduced C^{∗}-algebra of a discrete group gives rise to a homomorphism from the K-theory of this C^{∗}-algebra to the real numbers. This paper studies the range of this homomorphism. For torsion free groups, the Baum-Connes conjecture together with Atiyah’s L^{2}-index theorem implies that the range consists of the integers.

We give a direct and elementary proof that if G acts on a tree and admits a homomorphism α to another group H whose restriction α|_{G_{v}} to every stabilizer group of a vertex is injective, then

tr_{G}(K(C_{r}^{∗}G)) ⊂ tr_{H}(K(C_{r}^{∗}H)).

This follows from a general relative Fredholm module technique.

Examples are in particular HNN-extensions of H where the stable letter acts by conjugation with an element of H, or amalgamated free products G = H∗_{U} H of two copies of the same groups along a subgroup U. MSC: 19K (primary); 19K14, 19K35, 19K56 (secondary)

1 Introduction

Let G be a discrete group. All discrete groups considered in this paper are assumed to be countable. The trace tr_{G}: ℂG → ℂ: ∑_{g∈G} λ_{g}g ↦ λ_{1} (where 1 is the neutral element of G) extends to a trace on the reduced C^{∗}-algebra of G and therefore gives rise to a homomorphism

tr_{G}: K_{0}(C_{r}^{∗}G) → ℝ.

∗e-mail: thomas.schick@math.uni-muenster.de www: http://math.uni-muenster.de/u/schickt/ Fax: ++49 251/83 38370 Author’s work funded by Deutscher Akademischer Austauschdienst

If G is torsion free, we have the commutative diagram

K_{0}(BG) ⟶^{A} K_{0}(C_{r}^{∗}G)
↓^{ind}_{G}  ↓^{tr}_{G}
ℤ ⟶  ℝ,

where A is the Baum-Connes assembly map. The Baum-Connes conjecture says that A is an isomorphism. We denote by ind_{G} Atiyah’s L^{2}-index, which coincides with the ordinary index [1] and therefore takes values in the integers. Surjectivity of A of course implies that tr_{G} is also integer valued. We will denote this consequence of the Baum-Connes conjecture as the trace conjecture. It implies by a standard argument that there are no nontrivial projections in C_{r}^{∗}G.

The trace conjecture was verified directly for free groups using a special Fredholm module, which can be assigned to free groups, cf. e.g. [7]. Based on the ideas of this proof we get the following result.

1.1 Theorem. Let H, G be discrete countable groups and assume

tr_{H}(K(C_{r}^{∗}H)) ⊂ A ⊂ ℝ.

Let Ω and Δ be sets with commuting G-action from the left and H-action from the right such that Ω and Δ are free H-sets. Let Ω = Ω′ ∪ X and assume Δ and Ω′ are free G-sets and X consists of 1 ≤ r < ∞ H-orbits. Assume there is a bijective right H-map ϕ: Δ → Ω. Suppose that for every g ∈ G the set

R_{g} := {x ∈ Δ | ϕ(gx) ≠ gϕ(x)}

is contained in the union of finitely many H-orbits of Δ.

Then

tr_{G}(K(C_{r}^{∗}G)) ⊂ \frac{1}{r}A.

It remains now of course to give interesting examples of Theorem 1.1. For this, we use results of Dicks-Schick [5] and the following definition:

1.2 Definition. Let G be a group which acts on a tree. We say G is subdued by a group H if there is a homomorphism α: G → H whose restriction α|_{G_{v}} to the stabilizer group G_{v} := {g ∈ G | gv = v} is injective for each vertex v of the tree.

Remember that one can translate between groups acting on trees and fundamental groups of graphs in such a way that conjugacy classes of vertex stabilizers correspond to vertex groups.

1.3 Example. (1) If G = H ∗_{U} H, then G is the fundamental group of a graph consisting of two vertices joined by an edge. Hence there is an action of G on a tree such that the stabilizer group of every vertex is conjugate to one of the two copies of H. Using the obvious projection G → H which is the identity on both factors we see that G is subdued by H.

(2) Assume U < H and g ∈ H. Let ϕ: U → U^{g} be given by conjugation with g. Let G be the HNN-extension of H along ϕ. Then G is the fundamental group of a graph of groups with one vertex (the vertex group being H) and therefore acts on a tree such that each vertex stabilizer is conjugate to H. We define a homomorphism α: G → H such that the restriction of α to H is the identity and maps the stable letter t to g. This implies that G is subdued by H.

1.4 Theorem. Assume G is the fundamental group of a graph of groups which is subdued by another group H. Then

tr_{G} (K_{0}(C_{r}^{∗}[G])) ⊂ tr_{H} (K_{0}(C_{r}^{∗}[H])) .

In particular, this applies to the situations of Example 1.3.

If tr_{H}(K_{0}(C_{r}^{∗}[H])) ⊂ ℤ then this implies the trace conjecture for G. Baum and Connes conjecture [2, p.32] that for groups with torsion the range should be contained in ℚ. Our results support also this assertion (Baum and Connes make in fact a more precise and stronger conjecture, which however was disproved by Roy [17]).

1.5 Remark. In some cases, it is possible to derive the conclusions of Theorem 1.4 from elaborate K-theory calculations. One can use the exact sequence for the fundamental group of a graph of groups [16, Theorem 18]. In some cases elementary properties of the trace then imply Theorem 1.4.

One might hope to give a general treatment of the range of the traces in these cases as is done for certain HNN-extensions in [15] and [8]. However, even in the case of HNN-extensions, in general those results are difficult to interpret and it is not clear that [15] or [8] implies Example 1.3 (1).

Moreover, observe that Pimsner uses deep KK-theoretic methods to derive the exact sequence for a graph of groups. In contrast, our derivation is elementary.

1.6 Remark. To apply Theorem 1.4, essentially we have to know the trace conjecture for H. The obvious sufficient condition is that H fulfills the Baum-Connes conjecture, but it is also enough that H is a subgroup of such a group.

If H satisfies the Baum-Connes conjecture with coefficients, then the same is true for each group G(v) (since they are subgroups of H), hence by [14] G fulfills the Baum-Connes conjecture, and the statement of Theorem 1.4 follows also immediately from this fact.

However, these arguments do not apply to the Baum-Connes conjecture without coefficients. Lafforgue proves that cocompact discrete subgroups e.g. of Sl_{3}(ℝ) or Sp(n, 1) satisfy the Baum-Connes conjecture without coefficients [11]. However, it is unknown whether the Baum-Connes conjecture with coefficients is true for these groups. Therefore, the consequences of Theorem 1.4 are not included in the knowledge about the Baum-Connes conjecture for a non-cocompact subgroup H of Sl_{3}(ℝ) or Sp(n, 1) contained in a cocompact torsion-free subgroup.

In particular, for such an H every H ∗_{U} H fulfills the trace conjecture, but it is not clear whether it fulfills the Baum-Connes conjecture.

The method described in [7] was used by Linnell [12, 13] to prove the Atiyah conjecture about the integrality of L^{2}-Betti numbers for free groups, and starting with this for a lot of other groups. We investigate the Atiyah conjecture and obtain generalizations of Linnell’s results in [18, 5].

2 The trace conjecture for the K-theory of group C^{∗}-algebras

In this section we prove Theorem 1.1 and Theorem 1.4.

We first show how Theorem 1.4 follows from Theorem 1.1. The method for this is developed by Dicks and Schick in [5]. For the convenience of the reader we repeat the easy proof of the special case we are concerned with here.

Proof of Theorem 1.4. Assume G acts on the tree T with set of vertices V and set of edges E. We choose an arbitrary v_{0} ∈ V.

Let {∗} be a trivial G-set. Let ϕ̃: V → E ∨ {∗} denote the map which assigns to each v ∈ V the last edge in the T -geodesic from v_{0} to v, where this is taken to be ∗ if v = v_{0}. By Julg-Valette [9], ϕ̃ is bijective, and, for all v ∈ V, g ∈ G, we have ϕ̃(gv) = gϕ̃(v) if and only if v is not in the T -geodesic from v_{0} to g^{−1}v_{0}.

Define now Δ := V × H, Ω′ := E × H and Ω := (E ⨿ {∗}) × H. We define the G and the H-action on Δ and Ω′ be setting

g(x, u)h := (gx, α(g)uh) ∀g ∈ G, x ∈ T u, h ∈ H.

Since the restriction of α to each stabilizer group is injective, this is a free G- and of course also a free H-action, and they commute. Extend the action to Ω by g(∗, u)h = (∗, uh) for g ∈ G and u, h ∈ H.

We define ϕ: Δ = V × H → Ω = (E ⨿ {∗}) × H by

ϕ(v, u) = (ϕ̃(v), u).

Since ϕ̃ is bijective, the same is true for ϕ. Moreover, for fixed g ∈ G we have ϕ(g(v, u)) = (ϕ(gv), α(g)u) = (gϕ(v), α(g)u) = gϕ(v, u) exactly if ϕ(gv) = gϕ(v). Therefore, the set R_{g} of Theorem 1.1 is contained in the union of the finitely many H-orbits of Δ determined by the T -geodesic from v_{0} to g^{−1}v_{0}. Theorem 1.4 now follows from Theorem 1.1.

We are now going to prove Theorem 1.1. We use the language of Hilbert A-modules (for a C^{∗}-algebra A), compare e.g. [3, Section 13].

2.1 Definition. Let T be a free set of generators of the free H-set Δ. Then we can form the Hilbert C_{r}^{∗}H-module E := l^{2}(T) ⊗_{ℂ} C_{r}^{∗}H. Of course, E is nothing but the Hilbert sum of copies of C_{r}^{∗}H indexed by T (in the sense of Hilbert C_{r}^{∗}H-modules). Moreover, E ⊗_{C_{r}^{∗}H} l^{2}H ≅ l^{2}(Δ), and E is in a natural way a subset of l^{2}(Δ) because C_{r}^{∗}H ⊂ l^{2}H. The module E as a subset of l^{2}(Δ) does not depend on the choice of the basis T.

Let B(E) be the set of bounded adjointable Hilbert C_{r}^{∗}H-module homomorphisms. The map

B(E) → B(l^{2}(Δ)): A ↦ A_{Δ} := A ⊗ 1

is an injective algebra homomorphism by [3, p. 111], therefore an isometric injection of C^{∗}-algebras [10, 4.1.9]. Observe that the image of B(E) commutes with the right action of H on l^{2}(Δ) and therefore is contained in the corresponding von Neumann algebra with its canonical trace. We will prove in Lemma 2.2 that ℂG ⊂ B(E). It follows that the closure of ℂG in B(E) and B(l^{2}(Δ)) coincides. Since Δ is a free G-set, this closure is isomorphic to C_{r}^{∗}G.

2.2 Lemma. We have a natural inclusion ℂG ⊂ B(E) which extends the action of G on ℂΔ ⊂ E, in particular it is compatible with the injection ℂG ⊂ B(l^{2}Δ).

Proof. First observe that E is a closure of the algebraic tensor product of l^{2}(T) and C_{r}^{∗}H. Moreover, ℂ[T] is dense in l^{2}(T) and ℂH is dense in C_{r}^{∗}H. Therefore ℂΔ = ℂ[T] ⊗_{ℂ} ℂH is a dense subset of E.

Now fix g ∈ G. Since Δ = ⋃_{t∈T} tH, where the union is disjoint, for each t ∈ T we get unique elements t_{g,t} ∈ T, h_{g,t} ∈ H such that gt = t_{g,t}h_{g,t}. Since the actions of G and H commute, gt = gt′h implies t = t′. The map α_{g}: T → T: t ↦ t_{g,t} therefore is a bijection. Pick

x = ∑_{t∈T} tv_{t}, x′ = ∑_{t∈T} tv′_{t} ∈ ℂ[T] ⊗_{ℂ} ℂH, with v_{t}, v′_{t} ∈ ℂH.

By linearity we get

gx = ∑_{t∈T} t_{g,t}h_{g,t}v_{t} and gx′ = ∑_{t∈T} t_{g,t}h_{g,t}v′_{t}.

As is the convention in the theory of Hilbert A-modules, all of our inner products are linear in the second variable. Taking now the C_{r}^{∗}H-valued inner product we get (adjoint and products of elements in ℂH ⊂ C_{r}^{∗}H are taken in the sense of the C^{∗}-algebra)

\begin{aligned}⟨x, x′⟩_{C_{r}^{∗}H} & = ∑_{t∈T} v_{t}^{∗}v′_{t} = ∑_{t∈T}(h_{g,t}v_{t})^{∗}(h_{g,t}v′_{t}) \\ & = ∑_{t∈T}(h_{g,α_{g}^{−1}(t)}v_{α_{g}^{−1}(t)})^{∗}(h_{g,α_{g}^{−1}(t)}v′_{α_{g}^{−1}(t)}) = ⟨gx, gx′⟩_{C_{r}^{∗}H} \end{aligned}

(here we used that H acts unitarily, i.e. h_{g,t}^{∗} = h_{g,t}^{−1}). Hence G acts C_{r}^{∗}Hisometrically and this action extends to E. By linearity we get an ∗-algebra homomorphism ℂG → B(E) → B(l^{2}Δ). The composition is injective, therefore the same is true for the first map.

The above reasoning implies in the same way:

2.3 Lemma. For n ∈ ℕ we have canonical injections of ∗-algebras

M_{n}(ℂG) ⊂ M_{n}(C_{r}^{∗}G) ⊂ B(E^{n}) ⊂ B(l^{2}Δ^{n})^{H},

where B(l^{2}Δ^{n})^{H} denotes the operators which commute with the right action of H.

We have to compute the G-trace of operators in M_{n}(C_{r}^{∗}G), and we want to express this in terms of the H-trace of a suitable other operator. For this end, we repeat the following definition:

2.4 Definition. An operator A ∈ B(E) is of H-trace class if its image A_{l^{2}Δ} ∈ B(l^{2}(Δ))^{H} is of H-trace class in the sense of the von Neumann algebra (compare e.g. [19, 2.1]), i.e. if (with |A| = \sqrt{A^{∗}A})

∑_{t∈T}⟨t, |A_{l^{2}Δ}| t⟩_{l^{2}(Δ)} < ∞.

Then also ∑_{t∈T} ⟨t, A_{l^{2}Δ}t⟩_{l^{2}(Δ)} converges and we set

tr_{H}(A) := ∑_{t∈T}⟨t, A_{l^{2}Δ}t⟩_{l^{2}(Δ)}.

For A = (A_{ij}) ∈ B(E^{n}) = M_{n}(B(E)) we set

tr_{H}(A) = ∑_{i=1}^{n} tr_{H}(A_{ii}),

if |A_{l^{2}Δ}| of H-trace class (with the obvious definition for this).

2.5 Lemma. Let A, B, C ∈ B(E^{n}) and A be of H-trace class. The trace class operators form an ideal inside B(E^{n}) and we have tr_{H}(AB) = tr_{H}(BA). Moreover

|tr_{H}(A + B)| ≤ tr_{H}(|A|) + tr_{H}(|B|)
tr_{H}(|CAB|) ≤ ∥C∥ · ∥B∥ · tr_{H}(|A|).

Proof. These are standard properties of the (von Neumann) trace, compare [19, 2.3], [6, Théorème 8 and Corollaire 2 on p. 106].

Set S := ϕ(T). This is an H-basis for the free H-set Ω. Similarly to E we can build F := l^{2}(S) ⊗_{ℂ} C_{r}^{∗}H ⊂ l^{2}(Ω). If S′ := S ∩ Ω′ and S′′ := S ∩ X (i.e. S′ is an H-basis for Ω′ and S′′ is an H-basis for X) then with F′ := l^{2}(S′) ⊗_{ℂ} C_{r}^{∗}H and F′′ := l^{2}(S′′) ⊗_{ℂ} C_{r}^{∗}H we get a direct sum decomposition of Hilbert C_{r}^{∗}H-modules

F = F′ ⊕ F′′.

As in the case of E we get an canonical inclusion

ℂG ⊂ C_{r}^{∗}G ⊂ B(F′) ⊂ B(F) ⊂ B(l^{2}Ω)

(we extend the action of C_{r}^{∗}G to all of F by setting it zero on F′′). This composition is a non-unital ∗-algebra homomorphism.

Corresponding statements hold for matrices.

Denote the image of A ∈ M_{n}(C_{r}^{∗}G) in B(E^{n}) with A_{Δ} and in B(F^{n}) with A_{Ω}. We therefore have A_{Ω} = A_{Ω′} ⊕ 0 with the obvious notation.

The bijection ϕ: Δ → Ω induces a unitary map of Hilbert spaces ϕ: l^{2}(Δ)^{n} → l^{2}(Ω)^{n}. Since ϕ: Δ → Ω is H-equivariant, the same is true for the unitary map. Moreover, we get a Hilbert C_{r}^{∗}H-module unitary map ϕ: E^{n} → F^{n}. One key observations is now (this is an extension of the corresponding observation in the classical proof for the free group):

2.6 Lemma. Suppose A ∈ M_{n}(C_{r}^{∗}G) ⊂ B(E^{n}) is such that the Hilbert C_{r}^{∗}H-module morphism A_{Δ} − ϕ^{∗}A_{Ω}ϕ: E^{n} → E^{n} is of H-trace class. Then

tr_{G}(A) = \frac{1}{r} tr_{H}(A_{Δ} − ϕ^{∗}A_{Ω}ϕ).

Proof. Observe that ϕ is diagonal and traces are the sum over the diagonal entries. Therefore we may assume that n = 1. Since Δ is a free G-module, for every x ∈ Δ (which we identify with the element of l^{2}(Δ) which is 1 at x and zero everywhere else) we have

⟨x, A_{Δ}x⟩_{l^{2}(Δ)} = tr_{G}(A)

(simply identify Gx with G and the left and right hand side become identical). Similarly, since A_{Ω} = A_{Ω′} ⊕ 0 on l^{2}(Ω′)^{n} ⊕ l^{2}(X)^{n}

⟨ϕx, A_{Ω}ϕx⟩_{l^{2}(Ω)} = \begin{cases}tr_{G}(A); & if ϕ(x) ∈ Ω′ \\ 0; & if ϕ(x) ∈ X. \end{cases}

Moreover

tr_{H}(A_{Δ} − ϕ^{∗}A_{Ω}ϕ) = ∑_{t∈T}⟨t, A_{Δ}t⟩_{l^{2}(Δ)} − ⟨t, ϕ^{∗}A_{Ω}ϕ(t)⟩_{l^{2}(Δ)}

= ∑_{ϕ(t)∈X∩S′′} tr_{G}(A) = r tr_{G}(A)

since |X ∩ S′′| is the number of H-orbits in X, i.e. r. All other summands cancel each other out.

Because K(C_{r}^{∗}G) is generated by projections P ∈ M_{n}(C_{r}^{∗}G) ⊂ M_{n}(N G) and the trace we have to compute is exactly tr_{G}(P), we are tempted to apply Lemma 2.6 to such a P. A problem is that it is hard to check whether the trace class condition is fulfilled in general.

To circumvent these difficulties recall the following fact (compare e.g. [4, III.3, Proposition 3]):

2.7 Proposition. Let B be a C^{∗}-algebra and U ⊂ B a dense ∗-subalgebra that is closed under holomorphic functional calculus. Then the inclusion induces an isomorphism

K(U) ≅ K(B).

In particular, if B = C_{r}^{∗}G and U is closed under holomorphic functional calculus and contains ℂG, then the ranges of the canonical trace applied to K(C_{r}^{∗}G) and K(U) coincide.

As algebra U we will use the closure under holomorphic functional calculus of ℂG ⊂ B(E). This of course fulfills the conditions of Proposition 2.7. It remains to check:

2.8 Lemma. Let x ∈ M_{n}(U) ⊂ M_{n}(C_{r}^{∗}G) ⊂ B(E^{n}), where U is the closure under holomorphic functional calculus of ℂG in C_{r}^{∗}G. Then x − ϕ^{∗}xϕ: E → E is of H-trace class and C_{r}^{∗}H-compact.

Proof. Start with g ∈ G ⊂ ℂG. Since R_{g} as defined in Theorem 1.1 is contained in finitely many H-orbits, gt = ϕ^{∗}gϕt for all but a finitely many t ∈ T. In particular g − ϕ^{∗}gϕ is zero outside the C_{r}^{∗}H-submodule of E spanned by this finite number of elements of T, and is nonzero only on the complement, which is isomorphic to (C_{r}^{∗}H)^{N} for some N ∈ ℕ. Since id: C_{r}^{∗}H^{N} → C_{r}^{∗}H^{N} is of finite rank in the sense of Hilbert C_{r}^{∗}H-module morphisms, the same is true for g − ϕ^{∗}gϕ. Finite rank operators form a subspace, therefore the same is true if we replace g by v ∈ ℂG ⊂ B(E). Passage to finite matrices does preserve the finite rank property. Finite rank implies H-trace class and C_{r}^{∗}H-compactness. In particular, all operators in M_{n}(ℂG) give rise to C_{r}^{∗}H-compact operators, which also are of H-trace class. The map x ↦ x−ϕ^{∗}xϕ is norm continuous and the compact operators form a closed ideal, therefore x − ϕ^{∗}xϕ is compact even for arbitrary x ∈ M_{n}(C_{r}^{∗}G).

Assume A ∈ M_{n}(C_{r}^{∗}H) and 0 ≠ ξ ∉ Spec(A). Since the homomorphism A ↦ A_{Δ} is unital, we get

((ξ − A)^{−1})_{Δ} = (ξ − A_{Δ})^{−1}.

Similarly ((ξ − A)^{−1})_{Ω′} = (ξ − A_{Ω′})^{−1}. Consequently

((ξ − A)^{−1})_{Ω} = ((ξ − A)^{−1})_{Ω′} ⊕ 0 = (ξ − A_{Ω})^{−1} − ξ^{−1}P

where P: F → F is the projection onto F′′. Note that (ξ − A_{Ω})^{−1} acts by multiplication with ξ^{−1} on F′′. Here we need the assumption ξ ≠ 0. Since T′′ is finite, P is of finite rank as Hilbert C_{r}^{∗}H-module morphism.

Suppose now f is a function that is holomorphic in a neighborhood of Spec(A). Let Γ be a loop around Spec(A) and choose Γ so that it does not meet 0 ∈ ℂ. Then

f(A)_{Δ} = (∫_{Γ} f(ξ)(ξ − A)^{−1} dξ)_{Δ} = ∫_{Γ} f(ξ)(ξ − A_{Δ})^{−1} dξ
f(A)_{Ω} = ∫_{Γ} f(ξ) ((ξ − A_{Ω})^{−1} − ξ^{−1}P ) dξ.

For u, v ∈ B(E) and ξ ∉ Spec(u) ∪ Spec(v) we have

\begin{matrix}(ξ − u)^{−1} − (ξ − v)^{−1} = & (ξ − u)^{−1}(ξ − v − (ξ − u))(ξ − v)^{−1} \\ = & (ξ − u)^{−1}(u − v)(ξ − v)^{−1}. \end{matrix}

Therefore

f(A)_{Δ} − ϕ^{∗}f(A)_{Ω}ϕ = ϕ^{∗}P ϕ \underbrace{∫_{Γ} f(ξ)ξ^{−1} dξ}_{∈ℂ}

+ ∫_{Γ} \underbrace{f(ξ)(ξ − A_{Δ})^{−1}}_{=:f_{Δ}(ξ)} \underbrace{(A_{Δ} − ϕ^{∗}A_{Ω}ϕ)}_{=:A_{0}} \underbrace{(ξ − ϕ^{∗}A_{Ω}ϕ)^{−1}}_{=:f_{Ω}(ξ)} dξ.

As a consequence of Lemma 2.5 we have

tr_{H} |∫_{Γ} f_{Δ}(ξ)A_{0}f_{Ω}(ξ) dξ| ≤ ∫_{Γ} tr_{H}(|f_{Δ}(ξ)A_{0}f_{Ω}(ξ)|) dξ
≤ ∫ ∥f_{Δ}(ξ)∥ · ∥f_{Ω}(ξ)∥ · tr_{H}(|A_{0}|).

Since the operator valued functions f_{Δ} and f_{Ω} are norm-continuous, f(A)_{Δ}− ϕ^{∗}f(A)_{Ω})ϕ is of H-trace class if A_{0} is of H-trace class, in particular if A ∈ M^{n}(ℂG). This concludes the proof.

We determine now the range of the trace on the dense and holomorphically closed subalgebra U of C_{r}^{∗}G. Since the trace class condition is fulfilled, it only remains to calculate tr_{H}(P_{Δ} − ϕ^{∗}P_{Ω}ϕ) for a projection over U, and Theorem 1.1 follows from Lemma 2.6.

2.9 Lemma. Let E be the Hilbert C_{r}^{∗}H-module introduced in Definition 2.1 and P, Q ∈ B(E) be projections such that P − Q is of H-trace class and compact in the sense of Hilbert C_{r}^{∗}H-module morphisms. Then

w := ((P E ⊕ QE), 1, ( \begin{matrix}0 & P Q \\ QP & 0 \end{matrix} ))

is a Kasparov triple (in the sense of [3, 17.1.1]) representing an element in KK(ℂ, C_{r}^{∗}H) ≅ K_{0}(C_{r}^{∗}H) and

tr_{H}(P − Q) = ind_{H}(w) = tr_{H}([w]),

where tr_{H}(P − Q) is to be understood in the sense of Definition 2.4, whereas tr_{H}([w]) is the canonical trace defined on K_{0}(C_{r}^{∗}H).

Proof. Using Lemma 2.5 and the fact that P^{2} = P, Q^{2} = Q, (1 − P)P = 0, and tr_{H}(XY) = tr_{H}(Y X), we conclude

tr_{H}(P − Q) = tr_{H}(P^{2}(P − Q)) + tr_{H}((1 − P)(P − Q^{2}))
= tr_{H}(P (P − Q)P) − tr_{H}(Q(1 − P)Q)
= tr_{H}(P − P QP) − tr_{H}(Q − QP Q).

Let α_{P} : P E → P E be the orthogonal projection with image P E ∩ker(QP), and α_{Q}: QE → QE the orthogonal projection with image QE ∩ ker(P Q). Observe that α_{P} = (P − P QP)α_{P}. Therefore α_{P} is of H-trace class, since the same is true for P −P QP. In the same way we see that α_{Q} is of H-trace class. Set

\begin{aligned}T_{0} & := id_{P E} −P QP − α_{P} : P E → P E \\ T_{1} & := id_{QE} −QP Q − α_{Q}: QE → QE. \end{aligned}

Then

tr_{H}(P − Q) = tr_{H}(P − P QP) − tr_{H}(Q − QP Q)
= tr_{H}(T_{0}) − tr_{H}(T_{1}) + tr_{H}(α_{P}) − tr_{H}(α_{Q}).

(2.10)

Now QP: P E → QE is a bounded operator with adjoint P Q: QE → P E, and

tr_{H}(α_{P}) = dim_{H}(ker(QP: P E → QE))
tr_{H}(α_{Q}) = dim_{H}(ker(P Q: QE → P E))
= dim_{H}(coker(QP: P E → QE)).

(2.11)

For a complemented submodule X, one defines dim_{H}(X) := tr_{H}(pr_{X}), where pr_{X} is the orthogonal projection onto X. Since QP α_{P} = 0 and P Qα_{Q} = 0, and the latter implies 0 = (P Qα_{Q})^{∗} = α_{Q}QP, we have

QP T_{0} = QP (id_{P E} −P QP − α_{P})

= id_{QE} QP − QP Q^{2}P − α_{Q}QP = T_{1}QP.

Moreover, ker(QP: P E → QE) ⊂ ker(T_{0}), since QP (P x) = 0 implies α_{P} (P x) = P x, and in the same way we conclude ker(P Q) = ker((P Q)^{∗}) ⊂ ker(T_{1}) = ker(T_{1}^{∗}). It follows that QP “conjugates” T_{0} and T_{1}, and by [19, Proposition 2.6] (which goes back to a corresponding result in [1, p. 67]) that

tr_{H}(T_{0}) = tr_{H}(T_{1}).

Using Equation (2.10) and Equation (2.11) we arrive at

tr_{H}(P − Q) = ind_{H}(QP: P E → QE)

with the obvious definition of ind_{H}. This is exactly the H-index in the graded sense of the operator F := ( \begin{matrix}0 & P Q \\ QP & 0 \end{matrix} ) : P E ⊕ QE → P E ⊕ QE (where P E is the positive and QE the negative part of the graded Hilbert C_{r}^{∗}H-module P E ⊕ QE).

The only thing it remains to check is whether w fulfills all the axioms of Kasparov triples. Since the action of ℂ is unital and the operator is self adjoint, this amounts to check that 1 − F^{∗}F and 1 − F F^{∗} are compact in the sense of Hilbert C_{r}^{∗}H-module morphisms. Now F^{∗}F = F^{2} = F F^{∗} = ( \begin{matrix}P QP & 0 \\ 0 & QP Q \end{matrix} ). Since P − Q is compact, the same is true for P (P − Q)P = P − P QP: E → E. Then also the composition with the inclusion of P E into E and the projection P: E → P E is compact. This operator coincides with 1 − P QP: P E → P E. Similarly 1 − QP Q: QE → QE is compact. This concludes the proof.

To finish the proof of Theorem 1.1 observe that by Proposition 2.7 it suffices to compute tr_{G}(P) if P ∈ M_{n}(U) is a projection, where U is the holomorphic closure of ℂG ⊂ C_{r}^{∗}G. Since A → A_{Δ} and A → A_{Ω} are ∗-algebra homomorphisms, P_{Δ} and P_{Ω} are projections. Now Lemma 2.8 implies that we can apply Lemma 2.9 to P_{Δ} − ϕ^{∗}P_{Ω}ϕ. By assumption tr_{H}(K_{0}(C_{r}^{∗}H)) ⊂ A, therefore tr_{H}(P_{Δ} − ϕ^{∗}P_{Ω}ϕ) ∈ A. By Lemma 2.6 then tr_{G}(P) ∈ \frac{1}{r}A, and this concludes the proof of Theorem 1.1. 2.12 Remark. We use the language of Hilbert modules and Kasparov triples only for convenience. Observe that we don’t use much more than the definition: the single theorem we use is that our Kasparov triples indeed give rise to K-theory elements, and this is not very deep. By [3, 17.5.5] KK(ℂ, C_{r}^{∗}G) and K_{0}(C_{r}^{∗}G) are isomorphic, but to construct the map much less is needed. (Essentially we only have to perturb QP such that kernel and cokernel are finitely generated projective modules over C_{r}^{∗}G ⊗ 𝕂.)

3 Final remarks

We hope that Proposition 1.1 can be applied to more situations than the one described in Theorem 1.4. However, in [18] the situation where H is trivial (and consequently X is finite) is classified. It turns out that in this setting the assumptions of Theorem 1.1 can be fulfilled exactly if G is a finite extension of a free group. But then one has a transfer homomorphism for the K-theory of the reduced C^{∗}-algebras relating the trace for G to the trace of the free subgroup of finite index. One easily computes the range of the trace using this (and the known trace conjecture for the free group). The range is \frac{1}{d}ℤ where d is the smallest index of a free subgroup. Therefore it is not necessary to give details of the approach using Theorem 1.1 which gives the same result.

Acknowledgments. I am very much indebted to Warren Dicks. Without his help, I was able to apply Theorem 1.1 only in very basic situations. Moreover, I thank Nigel Higson and John Roe who pointed out that the trace of a difference of two projections on a Hilbert space is an index and therefore an integer and suggested that the same should work in a more general setting, inspiring the proof of Lemma 2.9. I also thank the referee for useful comments, in particular for pointing out Remark 1.5.

References

[1] Atiyah, M.: “Elliptic operators, discrete groups and von Neumann algebras”, Astérisque 32, 43–72 (1976)

[2] Baum, P. and Connes, A.: “Geometric K-theory for Lie groups and foliations”, Preprint IHES and Brown University (1982), to appear in L’enseignement mathématique

[3] Blackadar, B.: “K-theory for operator algebras”, vol. 5 of M.S.R.I. Monographs, Springer (1986)

[4] Connes, A.: “Non commutative geometry”, Academic Press (1994)

[5] Dicks, W. and Schick, T.: “The Atiyah conjecture for 1-relator groups and graphs of groups”, in preparation

[6] Dixmier, J.: “Les algèbre d’opérateurs dans l’espace Hilbertien (algèbres de von Neumann)”, Gauthier-Villars (1969)

[7] Effros, E.G.: “Why the circle is connected: an introduction to quantized topology”, Math. Intelligencer 11, 27–34 (1989)

[8] Exel, Y.: “Rotation numbers for Automorphisms of C^{∗}-algebras”, Pac. J. Math. 128, 31–89 (1987)

[9] Julg, P. and Valette, A.: “K-theoretic amenability for L_{2}(ℚ_{p}) and the action on the associated tree”, J. of Functional Analysis 58, 194–215 (1984)

[10] Kadison, R.V. and Ringrose, J.R.: “Fundamentals of the theory of operator algebras, volume I: Elementary theory”, Pure and Applied Mathematics, Academic Press (1983)

[11] Lafforgue, V.: “Une démonstration de la conjecture de Baum-Connes pour les groupes réductifs sur un corps p-adique et pour certains groupes discrts possédant la propriété (T)”, Preprint (1998)

[12] Linnell, P.: “Division rings and group von Neumann algebras”, Forum Math. 5, 561–576 (1993)

[13] Linnell, P.: “Analytic versions of the zero divisor conjecture”, in: Geometry and Cohomology in Group Theory, vol. 252 of London Math. Soc. Lecture Note Series, 209–248, Cambridge University Press (1998)

[14] Oyono-Oyono, Hervé: “La conjecture de Baum-Connes pour les groupes agissant sur les arbres”, C.R. Acad. Sci. Paris, séries 1 326, 799–804 (1998)

[15] Pimsner, M.: “Ranges of traces on K_{0} of reduced crossed products by free groups”, in: Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983), 374–408, Springer (1985)

[16] Pimsner, M.: “KK-groups of crossed products by groups acting on trees”, Invent. Math. 86, 603–634 (1986)

[17] Roy, R.: “The trace conjecture—a counterexample”, K-theory 17, 209-213 (1999)

[18] Schick, T.: “Integrality of L^{2}-Betti numbers”, Preprintreihe SFB 478 Münster, No. 73 (1999), Math. Ann.317, 727–750 (2000)

[19] Schick, T.: “L^{2}-index theorem for boundary manifolds”, preprint 1998, to appear in Pacific J. of Math.