The relation between the Baum-Connes Conjecture and the Trace Conjecture

Wolfgang Lück^{∗} Fachbereich Mathematik und Informatik Westfälische Wilhelms-Universität Münster Einsteinstr. 62 48149 Münster Germany

February 15, 2001

Abstract

We prove a version of the L^{2}-index Theorem of Atiyah, which uses the universal center-valued trace instead of the standard trace. We construct for G-equivariant K-homology an equivariant Chern character, which is an isomorphism and lives over the ring ℤ ⊂ Λ^{G} ⊂ ℚ obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the Baum-Connes Conjecture implies the modified Trace Conjecture, which says that the image of the standard trace K_{0}(C_{r}^{∗}(G)) → ℝ takes values in Λ^{G}. The original Trace Conjecture predicted that its image lies in the additive subgroup of ℝ generated by the inverses of all the orders of the finite subgroups of G, and has been disproved by Roy [13]. Key words: Baum-Connes Conjecture, Trace Conjecture, equivariant Chern character, L^{2}-index theorem. Mathematics subject classification 2000: 19L47, 19K56, 55N91

0. Introduction and statements of results

Throughout this paper let G be a discrete group. The Baum-Connes Conjecture for G says that the assembly map

asmb^{G} : K_{0}^{G}(\underline{E}G) → K_{0}(C_{r}^{∗}(G))

from the equivariant K-homology of the classifying space for proper G-actions \underline{E}G to the topological K-theory of the reduced C^{∗}-algebra C_{r}^{∗}(G) is bijective [3, page 8], [5, Conjecture 3.1]. In connection with this conjecture Baum and Connes [3, page 21] also made the sometimes so called Trace Conjecture. It says that the image of the composition

K_{0}(C_{r}^{∗}(G)) ⟶^{i} K_{0}(N (G)) ⟶^{tr_{N (G)}} ℝ

∗email: lueck@math.uni-muenster.de, www: http://www.math.uni-muenster.de/u/lueck, FAX: 49 251 8338370

is the additive subgroup of ℚ generated by all numbers \frac{1}{|H|}, where H ⊂ G runs though all finite subgroups of G. Here N (G) is the group von Neumann algebra, i the change of rings homomorphism associated to the canonical inclusion C_{r}^{∗}(G) → N (G) and tr_{N (G)} is the map induced by the standard von Neumann trace tr_{N (G)} : N (G) → ℂ. Roy has construced a counterexample to the Trace Conjecture in this form in [13] based on her article [14]. She constructs a group Γ, whose finite subgroups are all of order 1 or 3, together with an element in K_{0}^{G}(\underline{E}G), whose image under tr_{N (Γ)} ∘i ∘ asmb is −\frac{1105}{9}. The point is that 3 · \frac{1105}{9} is not an integer. Notice that Roy’s example does not imply that the Baum-Connes Conjecture does not hold for Γ. Since the group Γ contains a torsionfree subgroup of index 9 and the Trace Conjecture for torsionfree groups does follow from the Baum-Connes Conjecture, the Baum-Connes Conjecture predicts that the image of tr_{N (Γ)} ∘i : K_{0}(C_{r}^{∗}(Γ)) → ℝ is contained in {r ∈ ℝ | 9 · r ∈ ℤ}. So one could hope that the following version of the Trace Conjecture is still true. Denote by

Λ^{G} := ℤ [\frac{1}{|Fin(G)|}]  (0.1)

the ring ℤ ⊂ Λ^{G} ⊂ ℚ obtained from ℤ by inverting all the orders |H| of finite subgroups of G. For Roy’s group Γ this is \{m · 3^{−n} | m, n ∈ ℤ, n ≥ 0\} and obviously contains −\frac{1105}{9}.

Conjecture 0.2 (Modified Trace Conjecture for a group G) The image of the composition

K_{0}(C_{r}^{∗}(G)) ⟶^{i} K_{0}(N (G)) ⟶^{tr_{N (G)}} ℝ

is contained in Λ^{G}.

The motivation for this paper is to prove

Theorem 0.3 The image of the composition

Λ^{G} ⊗_{ℤ} K_{0}^{G}(\underline{E}G) ⟶^{id ⊗ asmb^{G}} Λ^{G} ⊗_{ℤ} K_{0}(C_{r}^{∗}(G)) ⟶^{i} Λ^{G} ⊗_{ℤ} K_{0}(N (G)) ⟶^{tr_{N (G)}} ℝ

is Λ^{G}.

In particular the modified Trace Conjecture 0.2 holds for G, if the assembly map asmb^{G} : K_{0}^{G}(\underline{E}G) → K_{0}(C_{r}^{∗}(G)) appearing in the Baum-Connes Conjecture is surjective.

In order to prove Theorem 0.3 (actually a generalization of it in Theorem 0.8), we will prove a slight generalization of Atiyah’s L^{2}-Index Theorem and construct an equivariant Chern character for equivariant K-homology of proper G-CW -complexes, which is bijective and defined after applying Λ^{G} ⊗_{ℤ} −.

Let M be a closed Riemannian manifold and D^{∗} = (D^{∗}, d^{∗}) be an elliptic complex of differential operators of order 1 on M. Denote by index(D^{∗}) ∈ ℤ its index. Let \overline{M} → M be a G-covering. Then one can lift D^{∗} to an elliptic G-equivariant complex \overline{D}^{∗}. Using the trace tr_{N (G)} : N (G) → ℂ Atiyah [1] defines its L^{2}-index index_{N (G)}(\overline{D}^{∗}) ∈ ℝ and shows

index(D^{∗}) = index_{N (G)}(\overline{D}^{∗}).

The L^{2}-index theorem of Atiyah implies that the composition

K_{0}^{G}(EG) ⟶^{asmb^{G}} K_{0}(C_{r}^{∗}(G)) ⟶^{i} K_{0}(N (G)) ⟶^{tr_{N (G)}} ℝ

agrees with the composition

K_{0}^{G}(EG) ⟶^{ind_{G→{1}}} K_{0}(BG) ⟶^{K_{0}(pr)} K_{0}(∗) ⟶^{asmb^{{1}}} K_{0}(C_{r}^{∗}({1})) ⟶^{dim_{ℂ}} ℤ ↪ ℝ.

Since for a torsionfree group G the spaces EG and \underline{E}G agree, the Baum-Connes Conjecture for

a torsionfree group G does imply that the image of K_{0}(C_{r}^{∗}(G)) ⟶^{i} K_{0}(N (G)) ⟶^{tr_{N (G)}} ℝ is ℤ [3, Corollary 1 on page 21]. Instead of using the the standard von Neumann trace tr_{N (G)} : N (G) → ℂ, one can use the universal center-valued trace tr_{N (G)}^{u} : N (G) → Z(N (G)) to define an index

index_{N (G)}^{u}(\overline{D}^{∗}) ∈ Z(N (G)),

which takes values in the center Z(N (G)) of the group von Neumann algebra N (G). Thus we get additional information, namely, for any element g ∈ G, whose conjugacy class (g) is finite, we get a complex number. However, it turns out that the value at classes (g) with g ≠ 1 is zero and that the value at (1) is the index of D^{∗}. Namely, we will show in Section 1

Theorem 0.4 Under the conditions above we get in Z(N (G))

index_{N (G)}^{u}(\overline{D}^{∗}) = index(D^{∗}) · 1_{N (G)}.

As an illustration we discuss the special case, where G is finite, M is an oriented closed 4kdimensional manifold with free orientation preserving G-action and D^{∗} is the signature operator. Then Theorem 0.4 reduces to the well-known statement that the equivariant signature

sign^{G}(M) := [H_{2k}(M)^{+}] − [H_{2k}(M)^{−}] ∈ Rep_{ℂ}(G)

is equal to sign(G\M) · [ℂG] for sign(G\M) ∈ ℤ the (ordinary) signature of G\M. We mention that this implies sign(M) = |G| · sign(G\M). Theorem 0.4 is a special case of Theorem 5.4 but we will need it in the proof of Theorem 5.4 and therefore will have to prove it first.

The second ingredient is a variation of the equivariant Chern character of [11] for equivariant Khomology of proper G-CW -complexes. Recall that proper means that all isotropy groups are finite. The construction in [11] works for equivariant homology theories with a Mackey structure on the coefficient system in general, but requires to invert all primes. The construction we will give here works after applying Λ^{G}⊗_{ℤ}? and has a different source.

Denote for a proper G-CW -complex X by F(X) the set of all subgroups H ⊂ G, for which X^{H} ≠ ∅, and by

Λ^{G}(X) := ℤ [\frac{1}{F(X)}]  (0.5)

the ring ℤ ⊂ Λ^{G}(X) ⊂ Λ^{G} obtained from ℤ by inverting the orders of all subgroups H ∈ F(X). Denote by

J^{G} resp. J^{G}(X)  (0.6)

the set of conjugacy classes (C) of finite cyclic subgroups C ⊂ G resp. the subset J^{G}(X) ⊂ J^{G} of conjugacy classes (C) of finite cyclic subgroups C ⊂ G, for which X^{C} is non-empty. Obviously Λ^{G} = Λ^{G}(\underline{E}G) and J^{G} = J^{G}(\underline{E}G) since \underline{E}G is characterized up to G-homotopy by the property that \underline{E}G^{H} is contractible (and hence non-empty) for finite H ⊂ G and empty for infinite H ⊂ G. Let C ⊂ G be a finite cyclic subgroup. Let C_{G}C be the centralizer and N_{G}C be the normalizer of C ⊂ G. Let W_{G}C be the quotient N_{G}C/C_{G}C. We will construct an idempotent Θ_{C} ∈ Λ^{C} ⊗_{ℤ} Rep_{ℚ}(C) which acts on Λ^{C} ⊗_{ℤ} Rep_{ℂ}(C). We will see in Lemma 3.4 (b) that the cokernel of

⊕_{D⊂C,D≠C} ind_{D}^{C} : ⊕_{D⊂C,D≠C} ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(D) → ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C)

is isomorphic to the image of the idempotent endomorphism

θ_{C} : ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C) → ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C).

After introducing and proving some preliminary results about modules over a category and representation theory of finite groups in Sections 2 and 3, we will prove in Section 4

Theorem 0.7 Let X be a proper G-CW -complex. Put Λ = Λ^{G}(X) and J = J^{G}(X). Then there is for p = 0, 1 a natural isomorphism called equivariant Chern character

ch_{p}^{G}(X) : ⊕_{(C)∈J} Λ ⊗_{ℤ} K_{p}(C_{G}C\X^{C}) ⊗_{Λ[W_{G}C]} im (θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C))

⟶^{≅} Λ ⊗_{ℤ} K_{p}^{G}(X).

Notice that the equivariant Chern character of Theorem 0.7 reduces to the obvious isomorphism K_{0}(G\X) ⊗_{ℤ} Rep_{ℂ}({1}) ⟶^{≅} K_{0}^{G}(X), if G acts freely on X. In the special case, where G is finite, X is the one-point-space {∗} and p = 0, the equivariant Chern character reduces to an isomorphism

⊕_{(C)∈J^{G}} ℤ [\frac{1}{|G|}] ⊗_{ℤ[\frac{1}{|G|}][W_{G}C]} im (θ_{C} : ℤ [\frac{1}{|G|}] ⊗_{ℤ} Rep_{ℂ}(C) → ℤ [\frac{1}{|G|}] ⊗_{ℤ} Rep_{ℂ}(C))

⟶^{≅} ℤ [\frac{1}{|G|}] ⊗_{ℤ} Rep_{ℂ}(G).

This is a strong version of the well-known theorem of Artin that the map induced by induction

⊕_{(C)∈J^{G}} ℚ ⊗_{ℤ} Rep_{ℂ}(C) → ℚ ⊗_{ℤ} Rep_{ℂ}(G)

is surjective for any finite group G. Theorem 0.7 gives a computation of Λ^{G} ⊗ K_{0}^{G}(\underline{E}G), namely

⊕_{(C)∈J^{G}} Λ^{G} ⊗_{ℤ} K_{p}(B(C_{G}C)) ⊗_{Λ^{G}[W_{G}C]} im (θ_{C} : Λ^{G} ⊗_{ℤ} Rep_{ℂ}(C) → Λ^{G} ⊗_{ℤ} Rep_{ℂ}(C))

⟶^{≅} Λ^{G} ⊗_{ℤ} K_{p}^{G}(\underline{E}G).

Another construction of an equivariant Chern character using completely different methods can be found in [4]. However, it works only after applying ℂ ⊗_{ℤ} − and therefore cannot be used for our purposes here.

In Theorem 5.4 we will identify the composition of the Chern character of Theorem 0.7 with the map

Λ^{G} ⊗_{ℤ} K_{0}^{G}(\underline{E}G) ⟶^{id ⊗ asmb^{G}} Λ^{G} ⊗_{ℤ} K_{0}(C_{r}^{∗}(G)) ⟶^{id ⊗i} Λ^{G} ⊗_{ℤ} K_{0}(N (G))

with an easier to understand and to calculate homomorphism, whose image is obvious from its definition. This will immediately imply

Theorem 0.8 Let Λ^{G} resp. J^{G} be the ring resp. set introduced in (0.1) resp. (0.6). Then the image of the composition

Λ^{G} ⊗_{ℤ} K_{0}^{G}(\underline{E}G) ⟶^{id ⊗_{ℤ} asmb^{G}} Λ^{G} ⊗_{ℤ} K_{0}(C_{r}^{∗}(G)) ⟶^{id ⊗_{ℤ}i} Λ^{G} ⊗_{ℤ} K_{0}(N (G))

is the image of the map given by induction

⊕_{(C)∈J^{G}} id ⊗ ind_{C}^{G} : ⊕_{(C)∈J^{G}} Λ^{G} ⊗_{ℤ} Rep_{ℂ}(C) → Λ^{G} ⊗_{ℤ} K_{0}(N (G)).

Now Theorem 0.3 follows from Theorem 0.8.

The change of rings and K-theory map l : K_{0}(ℂG) → K_{0}(C_{r}^{∗}(G)) from the algebraic K_{0}-group of the complex group ring ℂG to the topological K_{0}-group of C_{r}^{∗}(G) is in general far from being surjective. There is some evidence that it is injective after applying Λ⊗_{ℤ}? (see [11, Theorem 0.1]). Theorem 0.8 gives some evidence for the conjecture that the image of Λ^{G} ⊗_{ℤ} K_{0}(C_{r}^{∗}(G)) ⟶^{id ⊗_{ℤ}i} Λ^{G} ⊗_{ℤ} K_{0}(N (G)) agrees with the image of the composition Λ^{G}⊗_{ℤ}K_{0}(ℂG) ⟶^{l} Λ^{G}⊗_{ℤ}K_{0}(C_{r}^{∗}(G)) ⟶^{id ⊗_{ℤ}i} Λ^{G}⊗_{ℤ}K_{0}(N (G)). The paper is organized as follows

1. The L^{2}-index theorem
2. Modules over a category
3. Some representation theory for finite groups
4. The construction of the Chern character
5. The Baum-Connes Conjecture and the Trace Conjecture

References

The author wants to thank the Max-Planck-Institute for Mathematics in Bonn for the hospitality during his stay in January and February 2001, when parts of the paper were written.

1. The L^{2}-index theorem

In this section we prove a slight generalization of the L^{2}-index theorem of Atiyah [1]. Let \overline{M} be a Riemannian manifold (without boundary) together with a cocompact free proper action of G by isometries. In other words, M = G\\overline{M} is a closed Riemannian manifold, the projection p : \overline{M} → M is a G-covering and \overline{M} is equipped with the Riemannian metric induced by the one of M. Let D^{∗} = (D^{∗}, d^{∗}) be an elliptic complex of differential operators d^{p} : D^{p} → D^{p+1} of order 1 acting on the space of sections D^{p} = C^{∞}(E^{p}) of vector bundles E^{p} → M. Define \overline{E}^{p} by p^{∗}E^{p} and \overline{D}^{p} by L^{2}C^{∞}(\overline{E}_{p}). Then G-acts on \overline{E}^{p} and \overline{D}^{p}. Since differential operators are local operators, there is a unique lift of each operator d^{p} to a G-equivariant differential operator d̂^{p} : C^{∞}(\overline{E}^{p}) → C^{∞}(\overline{E}^{p+1}). We obtain an elliptic G-complex (C^{∞}(\overline{E}^{∗}), d̂^{∗}). Let \overline{d}^{p} : \overline{D}^{p} → \overline{D}^{p+1} be the minimal closure of d̂^{p} which is the same as its maximal closure [1, Proposition 3.1].

Since D^{∗} is elliptic, each cohomology module H^{p}(D^{∗}) := ker(d^{p})/ im(d^{p−1}) is a finitely generated ℂ-module. Hence we can define the index of the elliptic complex D^{∗} by

index(D^{∗}) := ∑_{p≥0} dim_{ℂ}(H^{p}(D^{∗})) ∈ ℤ.  (1.1)

Next we want to define an analogous invariant for the lifted complex \overline{D}^{∗}. The group von Neumann algebra N (G) of G is the ∗-algebra B(l^{2}(G))^{G} of all bounded G-equivariant operators l^{2}(G) → l^{2}(G), where we equip l^{2}(G) with the obvious left G-action. Let

tr_{N (G)} : N (G) → ℂ  (1.2)

be the standard von Neumann trace, which sends f ∈ N (G) = B(l^{2}(G))^{G} to ⟨f(e), e⟩_{l^{2}(G)}, where e denotes the element in l^{2}(G) given by the unit element in G ⊂ l^{2}(G). Denote by Z(N (G)) the center of N (G). There is the universal center-valued trace [7, Theorem 7.1.12 on page 462, Proposition 7.4.5 on page 483, Theorem 8.2.8 on page 517, Proposition 8.3.10 on page 525, Theorem 8.4.3 on page 532]

tr_{N (G)}^{u} : N (G) → Z(N (G))  (1.3)

which is uniquely determined by the following two properties:

(a) tr^{u} is a trace with values in the center, i.e. tr^{u} is ℂ-linear, for a ∈ N (G) with a ≥ 0 we have tr^{u}(a) ≥ 0 and tr^{u}(ab) = tr^{u}(ba) for all a, b ∈ N (G);

(b) tr^{u}(a) = a for all a ∈ Z(N (G)).

The map tr^{u} has the following further properties:

(c) tr^{u} is faithful;

(d) tr^{u} is normal. Equivalently, tr^{u} is continuous with respect to the ultraweak topology on N (G);

(e) || tr^{u}(a)|| ≤ ||a|| for a ∈ N (G);

(f) tr^{u}(ab) = a tr^{u}(b) for all a ∈ Z(N (G)) and b ∈ N (G);

(g) Let p and q be projections in N (G). Then p and q are equivalent, i.e. p = vv^{∗} and q = v^{∗}v, if and only if tr^{u}(p) = tr^{u}(q);

(h) Any linear functional f : N (G) → ℂ, which is continuous with respect to the norm topology on N (G) and which is central, i.e. f(ab) = f(ba) for all a, b ∈ N (G), factorizes as

N (G) ⟶^{tr^{u}} Z(N (G)) ⟶^{f|_{Z(N (G))}} ℂ.

In particular tr_{N (G)} ∘ tr_{N (G)}^{u} = tr_{N (G)}.

A Hilbert N (G)-module V is a Hilbert space V together with a G-action by isometries such that there exists a Hilbert space H and a G-equivariant projection p : H ⊗ l^{2}(G) → H ⊗ l^{2}(G) with the property that V and im(p) are isometrically G-linearly isomorphic. Here H ⊗ l^{2}(G) is the tensor product of Hilbert spaces and G acts trivially on H and on l^{2}(G) by the obvious left multiplication. Notice that p is not part of the structure, only its existence is required. We call V finitely generated if H can be choosen to be finite-dimensional.

Our main examples of Hilbert N (G)-modules are the Hilbert spaces \overline{D}^{p} which are isometrically G-isomorphic to L^{2}(C^{∞}(E^{p})) ⊗ l^{2}(G). This can be seen using a fundamental domain F for the Gaction on \overline{M} which is from a measure theory point of view the same as M. A morphism f : V → W of Hilbert N (G)-modules is a densely defined closed G-equivariant operator. The differentials \overline{d}^{p} are morphisms of Hilbert N (G)-modules.

Let f : V → V be a morphism of Hilbert N (G)-modules which is positive. Choose a G-projection p : H ⊗ l^{2}(G) → H ⊗ l^{2}(G) and an isometric invertible G-equivariant operator u : im(p) → V. Let \{b_{i} | i ∈ I\} be a Hilbert basis for H. Let \overline{f} be the composition

H ⊗ l^{2}(G) ⟶^{p} im(p) ⟶^{u} V ⟶^{f} V ⟶^{u^{−1}} im(p) ↪ H ⊗ l^{2}(G).

Define the von Neumann trace of f : V → V by

tr_{N (G)}(f) := ∑_{i∈I}⟨\overline{f}(b_{i} ⊗ e), b_{i} ⊗ e⟩_{H⊗l^{2}(G)}  ∈ [0, ∞].  (1.4)

This is indeed independent of the choice of p, u and the Hilbert basis {b_{i} | i ∈ I}. If V is finitely generated, then tr_{N (G)}(f) < ∞ is always true. Define the von Neumann dimension of a Hilbert N (G)-module V by

dim_{N (G)}(V) := tr_{N (G)}(id : V → V)  ∈ [0, ∞].  (1.5)

If V is a finitely generated Hilbert N (G)-module, we define the universal center-valued von Neumann dimension

dim_{N (G)}^{u}(V) := tr_{N (G)}^{u}(id : V → V) ∈ Z(N (G))  (1.6)

analogously to dim_{N (G)}(V) replacing tr_{N (G)} by tr_{N (G)}^{u}. Given a finitely generated Hilbert N (G)- module V, we have tr_{N (G)}(dim_{N (G)}^{u}(V)) = dim_{N (G)}(V).

Define the L^{2}-cohomology H_{(2)}^{p}(\overline{D}^{∗}) to be ker(\overline{d}^{p})/ clos(im(\overline{d}^{p−1})), where clos(im(\overline{d}^{p−1})) is the

closure of the image of \overline{d}^{p−1}. Define the p-th Laplacian by \overline{Δ}_{p} = (\overline{d}^{p})^{∗}\overline{d}^{p} + \overline{d}^{p−1}(\overline{d}^{p−1})^{∗}. By the L^{2}- Hodge-deRham Theorem we get a G-equivariant isometric isomorphism ker(\overline{Δ}_{p}) ⟶^{≅} H_{(2)}^{p}(\overline{D}^{∗}). Thus

H_{(2)}^{p}(\overline{D}^{∗}) inherits the structure of a Hilbert N (G)-module. Moreover, it turns out to be a finitely generated Hilbert N (G)-module. This can be deduced from the results of [12], where an index already over C_{r}^{∗}(G) is defined and the problem of getting finitely generated modules over C_{r}^{∗}(G) is treated. Namely, one can deduce from [12] after passing to the group von Neumann algebra, that there are finitely generated Hilbert N (G)-modules U_{1}, U_{2}, V_{1} and V_{2} and Hilbert N (G)-modules W_{1} and W_{2} together with a morphism v : V_{1} → V_{2} and isomorphisms of Hilbert N (G)-modules w : W_{1} ⟶^{≅} W_{2},

u_{1} : \overline{D}^{p} ⊕ U_{1} ⟶^{≅} V_{1} ⊕ W_{1} and u_{2} : \overline{D}^{p} ⊕ U_{2} ⟶^{≅} V_{2} ⊕ W_{2} such that u_{2} ∘ (Δ_{p} ⊕ 0) = (v ⊕ w) ∘ u_{1}.

Obviously the kernel of v and hence the kernel of \overline{Δ}_{p} are finitely generated Hilbert N (G)-modules.

Define the center-valued L^{2}-index and the L^{2}-index

index_{N (G)}^{u}(\overline{D}^{∗}) := ∑_{p≥0} dim_{N (G)}^{u}(H_{(2)}^{p}(\overline{D}^{∗})) ∈ Z(N (G));  (1.7)
index_{N (G)}(\overline{D}^{∗}) := ∑_{p≥0} dim_{N (G)}(H_{(2)}^{p}(\overline{D}^{∗})) ∈ ℝ.  (1.8)

The rest of this section is devoted to the proof of Theorem 0.4

Notation 1.9 Denote by con(G)_{cf} the set of conjugacy classes (g) of elements g ∈ G such that the set (g) is finite, or, equivalently, the centralizer C_{g}(g) = {g′ ∈ G | g′g = gg′} has finite index in G. For c ∈ con(G)_{cf} let N_{c} be the element ∑_{g∈c} g ∈ ℂG. In the sequel L_{c} resp. L_{g} denotes left multiplication with N_{c} resp. g for c ∈ con(G)_{cf} resp. g ∈ G.

Notice for the sequel that N_{c} ∈ Z(N (G)) and L_{c} is G-equivariant and commutes with all Goperators.

Lemma 1.10 Consider a ∈ Z(N (G)). Then we have a = 0 if and only if tr_{N (G)}(N_{c}a) = 0 holds for any c ∈ con_{cf}(G).

\underline{\underline{Proof}} : Consider a ∈ N (G) = B(l^{2}(G))^{G} which belongs to Z(N (G)). Write a(e) = ∑_{g∈G} λ_{g} · g ∈ l^{2}(G). Since aR_{g} = R_{g}a holds for g ∈ G and R_{g} : l^{2}(G) → l^{2}(G) given by right multiplication with g ∈ G, we get λ_{g} = λ_{hgh^{−}1} for g, h ∈ G. This implies that λ_{g} = 0 if the conjugacy class (g) is infinite. On easily checks for an element g with finite (g)

λ_{g} = tr_{N (G)}(N_{(g^{−1})}a).

Lemma 1.11 We get under the conditions above.

tr_{N (G)} (index_{N (G)}^{u}(\overline{D}^{∗})) = index(D^{∗}).

\underline{\underline{Proof}} : The L^{2}-index theorem of Atiyah [1, (1.1)] says

index_{N (G)}(\overline{D}^{∗}) = index(D^{∗}).

We have

tr_{N (G)} (index_{N (G)}^{u}(\overline{D}^{∗})) = tr_{N (G)} (∑_{p≥0}(−1)^{p} dim_{N (G)}^{u}(H_{(2)}^{p}(\overline{D}^{∗})))
= ∑_{p≥0}(−1)^{p} tr_{N (G)} (dim_{N (G)}^{u}(H_{(2)}^{p}(\overline{D}^{∗})))
= ∑_{p≥0}(−1)^{p} dim_{N (G)} (H_{(2)}^{p}(\overline{D}^{∗}))
= index_{N (G)}(\overline{D}^{∗}).

Next we want to prove

Lemma 1.12 Consider an element c ∈ con(G)_{cf} with c ≠ (1). Then

tr_{N (G)} (N_{c} · index_{N (G)}^{u}(\overline{D}^{∗})) = 0.

\underline{\underline{Proof}} : In the sequel we denote by \overline{pr}_{p} : \overline{D}^{p} → \overline{D}^{p} the projection onto the kernel of the p-th Laplacian \overline{Δ}_{p} = (\overline{d}^{p})^{∗}\overline{d}^{p} + \overline{d}^{p−1}(\overline{d}^{p−1})^{∗}. By the L^{2}-Hodge-deRham Theorem we get a G-equivariant isometric isomorphism im(\overline{pr}_{p}) ⟶^{≅} H_{(2)}^{p}(\overline{D}^{∗}). This implies

tr_{N (G)} (N_{c} · index_{N (G)}^{u}(\overline{D}^{∗}))
= ∑_{p≥0}(−1)^{p} · tr_{N (G)} (N_{c} · tr_{N (G)}^{u} (id : H_{(2)}^{p}(\overline{D}^{∗}) → H_{(2)}^{p}(\overline{D}^{∗})))
= ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} : H_{(2)}^{p}(\overline{D}^{∗}) → H_{(2)}^{p}(\overline{D}^{∗}))
= ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ \overline{pr}_{p} : \overline{D}^{p} → \overline{D}^{p}) .  (1.13)

The operator e^{−t\overline{Δ}_{p}} : \overline{D}^{p} → \overline{D}^{p} is a bounded G-equivariant operator and has a smooth kernel e^{−t\overline{Δ}_{p}}(\overline{x}, \overline{y}) : \overline{E}p ; \overline{x} → \overline{E}p ; \overline{y} for \overline{x}, \overline{y} ∈ \overline{M}. Thus e^{−t\overline{Δ}_{p}}(ω) applied to a section ω is given at \overline{y} ∈ \overline{M} by ∫\overline{M} e^{−t\overline{Δ}_{p}}(\overline{x}, \overline{y})(ω(\overline{x}))dvol\overline{x}. The operator L_{c} ∘e^{−t\overline{Δ}_{p}} is also a bounded G-equivariant operator and has a smooth kernel (L_{c} ∘ e^{−t\overline{Δ}_{p}}) (\overline{x}, \overline{y}) satisfying

(L_{c} ∘ e^{−t\overline{Δ}_{p}}) (\overline{x}, \overline{y}) = ∑_{g∈c} L_{g} ∘ e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{y}).

If F is a fundamental domain for the G-action, then [1, Proposition 4.6].

tr_{N (G)}(L_{c} ∘ e^{−t\overline{Δ}_{p}}) = ∫_{F} tr_{ℂ} ((L_{c} ∘ e^{−t\overline{Δ}_{p}}) (\overline{x}, \overline{x})) dvol\overline{x};
= ∑_{g∈c} ∫_{F} tr_{ℂ} (L_{g} ∘ e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{x})) dvol\overline{x} .  (1.14)

where tr_{ℂ} is the trace of an endomorphism of a finite-dimensional complex vector space. Since M is compact, we can find ϵ > 0 such that the distance of \overline{x} and g\overline{x} is bounded from below by ϵ for all \overline{x} ∈ \overline{M} and g ∈ c. We have

lim_{t→0} sup \{||e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{x})|| | \overline{x} ∈ F\} = 0,  (1.15)

where ||e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{x})|| is the operator norm of the linear map e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{x}) of finite-dimensional Hilbert spaces. This follows from the finite propagation speed method of [6]. There only the standard Laplacian on 0-forms is treated, but the proof presented there carries over to the Laplacian \overline{Δ}_{p} associated to the lift \overline{D}^{∗} to the G-covering \overline{M} of an elliptic complex D^{∗} of differential operators of order 1 on a closed Riemannian manifold M in any dimension p. The point is that \overline{M} has bounded

geometry, \overline{Δ}_{p} is essentially selfadjoint and positive so that \sqrt{\overline{Δ}_{p}} makes sense, and \frac{∂^{2}}{∂t^{2}} + \overline{Δ}_{p} is strictly hyperbolic. Now one applies the results of [6, Section 1] and uses the estimate in [9, page 475], where the special case of D^{∗} being the deRham complex is treated.

Since

|tr_{ℂ} (L_{g} ∘ e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{x}))| ≤ dim_{ℂ}(E^{p}) · ||e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{x})||

and F is compact, we conclude from (1.14) and (1.15)

lim_{t→0} tr_{N (G)}(L_{c} ∘ e^{−t\overline{Δ}_{p}}) = 0.  (1.16)

Since the trace tr_{N (G)} is ultraweakly continuous and lim_{t→∞} e^{−t\overline{Δ}_{p}} = \overline{pr}_{p} in the weak topology, we get

lim_{t→∞} tr_{N (G)}(L_{c} ∘ e^{−t\overline{Δ}_{p}}) = tr_{N (G)}(L_{c} ∘ \overline{pr}_{p}).  (1.17)

We conclude from (1.13) and (1.17)

tr_{N (G)} (N_{c} · index_{N (G)}^{u}(\overline{D}^{∗})) = lim_{t→∞} ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ e^{−t\overline{Δ}_{p}}) .  (1.18)

We have

\frac{d}{dt} ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ e^{−t\overline{Δ}_{p}} : \overline{D}^{p} → \overline{D}^{p})

= ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ \frac{d}{dt}e^{−t\overline{Δ}_{p}})
= ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (−\overline{Δ}_{p}) ∘ e^{−t\overline{Δ}_{p}})
= − ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ \overline{d}^{p−1} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{−t\overline{Δ}_{p}})

− ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p})^{∗} ∘ \overline{d}^{p} ∘ e^{−t\overline{Δ}_{p}})

= − ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ \overline{d}^{p−1} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}})

− ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p})^{∗} ∘ e^{−t\overline{Δ}_{p+1}} ∘ \overline{d}^{p})

= − ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}} ∘ \overline{d}^{p−1} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}})

+ ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{−t\overline{Δ}_{p}} ∘ \overline{d}^{p−1})

= − ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}} ∘ \overline{d}^{p−1})

+ ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{−t\overline{Δ}_{p}} ∘ \overline{d}^{p−1})

= − ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{−t\overline{Δ}_{p}} ∘ \overline{d}^{p−1})

+ ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{−t\overline{Δ}_{p}} ∘ \overline{d}^{p−1})

= 0.  (1.19)

Here are some justifications for the calculation above. Recall that L_{c} is a bounded G-operator and commutes with any G-equivariant operator. We can commute tr_{N (G)} and \frac{d}{dt} since tr_{N (G)} is ultraweakly continuous. We conclude e^{−t\overline{Δ}_{p+1}} ∘ \overline{d}^{p} = \overline{d}^{p} ∘ e^{−t\overline{Δ}_{p}} from the fact that \overline{Δ}_{p+1} ∘ \overline{d}^{p} = \overline{d}^{p} ∘ \overline{Δ}_{p} holds on C^{∞}(\overline{E}^{p−1}). We have used at several places the typical trace relation tr_{N (G)}(AB) = tr_{N (G)}(BA) which is in each case justified by [1, section 4]. In order to be able to apply this trace relation we have splitted e^{−t\overline{Δ}_{p}} into e^{− \frac{t}{2} \overline{Δ}_{p}} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}} in the calculation above.

Hence ∑_{p≥0}(−1)^{p}·tr_{N (G)} (L_{c} ∘ e^{−t\overline{Δ}_{p}} : \overline{D}^{p} → \overline{D}^{p}) is independent of t and we conclude from (1.18)

tr_{N (G)} (N_{c} · index_{N (G)}^{u}(\overline{D}^{∗})) = lim_{t→0} ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ e^{−t\overline{Δ}_{p}} : \overline{D}^{p} → \overline{D}^{p}) . (1.20)

Now Lemma 1.12 follows from (1.16) (1.20).

Finally Theorem 0.4 follows from Lemma 1.10, Lemma 1.11 and Lemma 1.12.

2. Modules over a category

In this section we recall some facts about modules over the category Sub = Sub(G; F(X)) for a proper G-CW -complex X as far as needed here. For more information about modules over a category we refer to [10].

Let Sub := Sub(G; F(X)) be the following category. Objects are the elements of the set F(X) of subgroups H ⊂ G, for which X^{H} ≠ ∅. For two finite subgroups H and K in F(X) denote by conhom_{G}(H, K) the set of group homomorphisms f : H → K, for which there exists an element g ∈ G with gHg^{−1} ⊂ K such that f is given by conjugation with g, i.e. f = c(g) : H → K, h ↦ ghg^{−1}. Notice that c(g) = c(g′) holds for two elements g, g′ ∈ G with gHg^{−1} ⊂ K and g′H(g′)^{−1} ⊂ K if and only if g^{−1}g′ lies in the centralizer C_{G}H = {g ∈ G | gh = hg for all h ∈ H} of H in G. The group of inner automorphisms of K acts on conhom_{G}(H, K) from the left by composition. Define the set of morphisms mor_{Sub} (H, K) by Inn(K)\ conhom_{G}(H, K). Let N_{G}H be the normalizer {g ∈ G | gHg^{−1} = H} of H. Define H · C_{G}H = {h · g | h ∈ H, g ∈ C_{G}H}. This is a normal subgroup of N_{G}H and we define W_{G}H := N_{G}H/(H ·C_{G}H). One easily checks that W_{G}H is a finite group and that there is an isomorphism from W_{G}H to aut_{Sub} (H) which sends g(H · C_{G}H) ∈ W_{G}H to the automorphism of H represented by c(g) : H → H. Notice that there is a morphism from H to K if and only if H is subconjugated to K. There is an isomorphism from H to K if and only if H and K are conjugated. The category Sub is a so called EI-category, i.e. any endomorphism in Sub is an isomorphism.

Let R be a commutative associative ring with unit. A covariant resp. contravariant RSub-module M is a covariant resp. contravariant functor from Sub to the category of R-modules. Morphisms are natural transformations. The structure of an abelian category on the category of R-modules carries over to the category of RSub-modules. In particular the notion of a projective RSub-module is defined. Given a contravariant RSub-module M and a covariant RSub-module N, one can define a R-module, their tensor product over Sub

M ⊗_{RSub} N = ⊕_{H∈F(X)} M(H) ⊗_{R} N(H)/ ∼,

where ∼ is the typical tensor relation mf ⊗ n = m ⊗ fn, i.e. for each morphism f : H → K in Sub, m ∈ M(K) and n ∈ N(H) we introduce the relation M(f)(m) ⊗ n − m ⊗ N(f)(n) = 0.

Given a left R[W_{G}H]-module N for H ∈ F(X), define a covariant RSub-module E_{H}M by

(E_{H}M)(K) := R mor_{Sub} (H, K) ⊗_{R[W_{G}H]} N  for K ⊂ G, |K| < ∞,  (2.1)

where R mor_{Sub} (H, K) is the free R-module generated by the set mor_{Sub} (H, K). Given a covariant RSub-module M and H ∈ F(X), define M(H)_{s} to be the left R-submodule of M(H), which is spanned by the images of all R-maps M(f) : M(K) → M(H), where f runs through all morphisms f : K → H in Sub, which have H as target and are not isomorphisms. Obviously M(H)_{s} is an R[W_{G}H]-submodule of M(H). Define a left R[W_{G}H]-module S_{H}M by

S_{H}M := M(H)/M(H)_{s}.  (2.2)

Both functors E_{H} and S_{H} respect direct sums and the property finitely generated and the property projective. Given a left R[W_{G}H]-module M, S_{K} ∘ E_{H}M is M, if H = K and is 0, if H and K are not conjugated in G.

Let M be a covariant RSub-module. We want to check whether it is projective or not. A necessary (but not sufficient) condition is that S_{H}M is a projective R[W_{G}H]-module. Assume that S_{H}M is R[W_{G}H]-projective for all objects H in Sub. We can choose a R[W_{G}H]-splitting σ_{H} : S_{H}M → M(H) of the canonical projection M(H) → S_{H}M = M(H)/M(H)_{s}. For a finite subgroup H ⊂ G define the morphism of covariant RSub-modules

i_{H}M : E_{H}(M(H)) → M

by (i_{H}M)(K)((f : H → K) ⊗_{R[W_{G}H]} m) = M(f)(m). We obtain after a choice of representatives H ∈ (H) for any conjugacy class (H) of subgroups H ∈ F(X) a morphism of covariant RSub-modules

T : ⊕_{(H),H∈F(X)}E_{H}S_{H}M ⟶^{⊕_{(H),H∈F(X)}E_{H}(σ_{H})} ⊕_{(H),H∈F(X)}E_{H}(M(H)) ⟶^{⊕_{(H),H∈F(X)}i_{H}M} M. (2.3)

We get as a special case of [11, Theorem 2.11]

Theorem 2.4 The morphism T is always surjective. It is bijective if and only if M is a projective RSub-module.

3. Some representation theory for finite groups

Denote for a finite group H by Rep_{ℚ}(H) resp. Rep_{ℂ}(H) the ring of finite dimensional Hrepresentations over the field ℚ resp. ℂ. Recall for the sequel that these are finitely generated free abelian groups. Given an inclusion of finite groups H ⊂ G, we denote by ind_{H}^{G} : Rep_{ℚ}(H) → Rep_{ℚ}(G) and res_{G}^{H} : Rep_{ℚ}(G) → Rep_{ℚ}(H) the induction and restriction homomorphism and similar for R ⊗_{ℤ} Rep_{ℚ}, Rep_{ℂ} and R ⊗_{ℤ} Rep_{ℂ} for a commutative ring R with ℤ ⊂ R. Let con_{ℚ}(H) be the set of ℚ-conjugacy classes of elements in H, where h and h′ are called ℚ-conjugated if the cyclic subgroups ⟨h⟩ and ⟨h′⟩ are conjugated in G. Let con(G) be the set of conjugacy classes of elements in G. Denote by class_{ℚ}(H) resp. class_{ℂ}(H) the rational resp. complex vector space of functions con_{ℚ}(H) → ℚ resp. con(G) → ℂ. Character theory yields isomorphisms [15, page 68 and Theorem 29 on page 102]

χ_{ℚ} : ℚ ⊗_{ℤ} Rep_{ℚ}(H) ⟶^{≅} class_{ℚ}(H);
χ_{ℂ} : ℂ ⊗_{ℤ} Rep_{ℂ}(H) ⟶^{≅} class_{ℂ}(H).

For a finite cyclic group C denote by θ_{C} ∈ ℚ ⊗_{ℤ} Rep_{ℚ}(C) the element whose character χ_{ℚ}(θ_{C}) sends c ∈ C to 1, if c generates C, and to 0 otherwise.

Let C ⊂ H be a cyclic subgroup of the finite group H. Then we get for h ∈ H

\frac{1}{[H : C]} · χ_{ℚ} (ind_{C}^{H} θ_{C}) (h) = \frac{1}{[H : C]} · \frac{1}{|C|} · ∑_{l∈H,l^{−1}hl∈C} χ_{ℚ} (θ_{C}) (l^{−1}hl) = \frac{1}{|H|} · ∑_{l∈H,⟨l^{−1}hl⟩=C} 1.

Denote by [ℚ] ∈ Rep_{ℚ}(H) the class of the trivial H-representation ℚ. Notice that χ_{ℚ}([ℚ]) is the constant function with values 1. We get in ℚ ⊗_{ℤ} Rep_{ℚ}(H)

1 ⊗_{ℤ} [ℚ] = ∑_{C⊂H,C cyclic} \frac{1}{[H : C]} · ind_{C}^{H} θ_{C},  (3.1)

since for any l ∈ H and h ∈ H there is precisely one cyclic subgroup C ⊂ H with C = ⟨l^{−1}hl⟩ and χ_{ℚ} is bijective. In particular we get for a finite cyclic group C in ℚ ⊗_{ℤ} Rep_{ℚ}(C)

Θ_{C} = 1 ⊗_{ℤ} [ℚ] − ∑_{D⊂C,D≠C} \frac{1}{[C : D]} · ind_{D}^{C} θ_{D}.  (3.2)

Now one easily checks by induction over the order of the finite cyclic subgroup C that the element Θ_{C} satisfies

Θ_{C} ∈ ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℚ}(C).  (3.3)

Obviously Θ_{C} is an idempotent in ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℚ}(C). By the obvious change of rings homo-

morphism, ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C) becomes a ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℚ}(C)-module. Hence multiplication with θ_{C} defines an idempotent endomorphism

θ_{C} : ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C) → ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C).

It is natural with respect to group automorphisms of C, since Θ_{C} is invariant under group automorphisms of C.

Lemma 3.4 (a) For a finite group H the map

⊕_{C⊂H,C cyclic} ind_{C}^{H} : ⊕_{C⊂H,C cyclic} ℤ [\frac{1}{|H|}] ⊗_{ℤ} Rep_{ℂ}(C) → ℤ [\frac{1}{|H|}] ⊗_{ℤ} Rep_{ℂ}(H)

is surjective;

(b) Let C be a finite cyclic group. Then the image resp. cokernel of

⊕_{D⊂C,D≠C} ind_{D}^{C} : ⊕_{D⊂C,D≠C} ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(D) → ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C)

is equal resp. isomorphic to the kernel resp. image of the idempotent endomorphism

θ_{C} : ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C) → ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C).

\underline{\underline{Proof}} : (a) follows from the following calculation for x ∈ ℤ [\frac{1}{|H|}] ⊗_{ℤ} Rep_{ℂ}(H) based on (3.1)

x = (1 ⊗_{ℤ} [ℚ]) · x = ( ∑_{C⊂H,C cyclic} \frac{1}{[H : C]} · ind_{C}^{H} θ_{C} ) · x = ∑_{C⊂H,C cyclic} \frac{1}{[H : C]} · ind_{C}^{H}(θ_{C} · res_{H}^{C} x).

(b) follows from the following two calculations based on (3.2) for x ∈ ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(H)

x − θ_{C} · x = (1 ⊗ [ℚ] − θ_{C}) · x
= ( ∑_{D⊂C,D≠C} \frac{1}{[C : D]} · ind_{D}^{C} θ_{D}) · x

= ∑_{D⊂C,D≠C} \frac{1}{[C : D]} · ind_{D}^{C}(θ_{D} · res_{C}^{D} x)

and for D ⊂ C, D ≠ C and y ∈ ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(D)

θ_{C} · ind_{D}^{C} y = ind_{D}^{C}(res_{C}^{D} θ_{C} · y) = ind_{D}^{C}(0 · y) = 0.

This finishes the proof of Lemma 3.4.

The next result is analogous to [11, Lemma 7.4] but we have to go through its proof again because here we want to invert only the orders of finite subgroups of G, whereas in [11] we have considered everything over ℚ.

Theorem 3.5 Let G be a group and Λ = Λ^{G}(X) as defined in (0.5). Consider the covariant ΛSubmodule Λ ⊗_{ℤ} Rep_{ℂ}(?) which sends a finite subgroup group H ⊂ G to Λ ⊗_{ℤ} Rep_{ℂ}(H). Then

(a) S_{H}Λ ⊗_{ℤ} Rep_{ℂ}(?) is trivial if the finite subgroup H ⊂ G is not cyclic.

For a finite cyclic subgroup C ⊂ G, the Λ[W_{G}C]-module S_{C}Λ ⊗_{ℤ} Rep_{ℂ}(?) is isomorphic to the image of the idempotent Λ[W_{G}C]-homomorphism

θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C).

The isomorphism is given by the composition of the obvious inclusion im(θ_{C}) → Rep_{ℂ}(C) with the obvious projection Rep_{ℂ}(C) → S_{C}Λ ⊗_{ℤ} Rep_{ℂ}(?);

(b) Λ ⊗_{ℤ} Rep_{ℂ}(?) is a projective ΛSub-module;

(c) Let M be a contravariant ΛSub-module. There is a natural isomorphism of Λ-modules

⊕_{(C),C cyclic,C∈F(X)} M(C) ⊗_{Λ[W_{G}C]} im (θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C))
≅ M ⊗_{ΛSub} Λ ⊗_{ℤ} Rep_{ℂ}(?);

(d) Λ ⊗_{ℤ} Rep_{ℂ}(H) is a flat ΛSub-module, i.e. for an exact sequence 0 → M_{0} → M_{1} → M_{2} → 0 of contravariant ΛSub-modules the induced sequence of R-modules 0 → M_{0} ⊗_{ΛSub} Λ⊗_{ℤ} Rep_{ℂ}(?) → M_{1} ⊗_{ΛSub} Λ ⊗_{ℤ} Rep_{ℂ}(?) → M_{2} ⊗_{ΛSub} Λ ⊗_{ℤ} Rep_{ℂ}(?) → 0 is exact.

\underline{\underline{Proof}} : (a) Consider H ∈ F(X). Since Rep_{ℂ}(H) is finitely generated free abelian and |W_{G}H| is invertible in Λ, the Λ[W_{G}H]- module Λ ⊗_{ℤ} Rep_{ℂ}(H) is projective. We conclude from Lemma 3.4 (a) that S_{H}Λ ⊗_{ℤ} Rep_{ℂ}(?) is trivial if H is not cyclic. If H = C for a finite cyclic subgroup C ⊂ G, the assertion follows from Lemma 3.4 (b).

(b) Notice that S_{H}Λ ⊗_{ℤ} Rep_{ℂ}(?) is a projective Λ[W_{G}H]-module for all H ∈ F(X) by assertion (a). Because of Theorem 2.4 it suffices to show for the morphism T defined in (2.3) that T (K) is injective for any given element K ∈ F(X).

Consider an element u in the kernel of T (K). Put J(H) = mor_{Sub} (H, K)/(W_{G}H) for H ∈ F(X) and put I = {(H) | H ∈ F(X)}. Choose for any (H) ∈ I a representative H ∈ (H). Then fix for any element \overline{f} ∈ J(H) a representative f : H → K in mor_{Sub} (H, K). For the remainder of the proof of assertion (b) we abbreviate L(?) := Λ ⊗_{ℤ} Rep_{ℂ}(?). We can find elements x_{H,f} ∈ S_{H}L for (H) ∈ I and \overline{f} ∈ J(H) such that only finitely many of the x_{H,f}-s are different from zero and u can be written as

u = ∑_{(H)∈I} ∑_{\overline{f}∈J(H)}(f : H → K) ⊗_{Λ[W_{G}H]} x_{H,f}.

We want to show that all elements x_{H,f} are zero. Suppose that this is not the case. Let (H_{0}) be maximal among those elements (H) ∈ I for which there is \overline{f} ∈ J(H) with x_{H,f} ≠ 0, i.e. if for (H) ∈ I the element x_{H,f} is different from zero for some morphism f : H → K in Sub and there is a morphism H_{0} → H in Sub, then (H_{0}) = (H). In the sequel we choose for any of the morphisms f : H → K in Sub a group homomorphism denoted in the same way f : H → K representing it. Recall that f : H → K is given by conjugation with an appropriate element g ∈ G. Fix f_{0} : H_{0} → K with x_{H_{0}.f_{0}} ≠ 0. We claim that the composition

A : ⊕_{(H)∈I}E_{H} ∘ S_{H}(L(K)) ⟶^{T (K)} L(K) ⟶^{res_{K}^{im(f_{0})}} L(im(f_{0})) ⟶^{ind_{f{−1 \atop 0}:im(f_{0})→H_{0}}} L(H_{0}) ⟶^{pr_{H_{0}}} S_{H_{0}}L

maps u to m·x_{H_{0},f_{0}} for some integer m > 0 which is invertible in Λ. This would lead to a contradiction because of T (K)(u) = 0 and x_{H_{0},f_{0}} ≠ 0.

Consider (H) ∈ I and \overline{f} ∈ J(H). It suffices to show that A ((f : H → K) ⊗_{Λ[W_{G}H]} x_{H,f}) is [K ∩ N_{G} im(f_{0}) : im(f_{0})] · x_{H,f} if (H) = (H_{0}) and \overline{f} = \overline{f_{0}}, and is zero otherwise. One easily checks that A((f : H → K) ⊗_{Λ[W_{G}H]} x_{H,f}) is the image of x_{H,f} under the composition

a(H, f) : S_{H}L ⟶^{σ_{H}} L(H) ⟶^{ind_{f:H→im(f)}} L(im(f)) ⟶^{ind_{im(f)}^{K}} L(K) ⟶^{res_{K}^{im(f_{0})}} L(im(f_{0}))

⟶^{ind_{f{−1 \atop 0}:im(f_{0})→H_{0}}} L(H_{0}) ⟶^{pr_{H_{0}}} S_{H_{0}}L.

The Double Coset formula implies

res_{K}^{im(f_{0})} ∘ ind_{im(f)}^{K} = ∑_{k∈im(f_{0})\K/ im(f)} ind_{c(k):im(f)∩k^{−1} im(f_{0})k→im(f_{0})} ∘ res_{im(f)}^{im(f)∩k^{−1} im(f_{0})k} .

The composition pr_{H_{0}} ∘ ind_{f_{0}^{−1}:im(f_{0})→H_{0}} ∘ ind_{c(k):im(f)∩k^{−1} im(f_{0})k→im(f_{0})} is trivial, if c(k) : im(f) ∩ k^{−1} im(f_{0})k → im(f_{0}) is not an isomorphism. Suppose that c(k) : im(f) ∩ k^{−1} im(f_{0})k → im(f_{0}) is an isomorphism. Then k^{−1} im(f_{0})k ⊂ im(f). Since H_{0} has been choosen maximal among the H for which x_{H,f} ≠ 0 for some morphism f : H → K, this implies x_{H,f} = 0 or that k^{−1} im(f_{0})k = im(f). Suppose k^{−1} im(f_{0})k = im(f). Then (H) = (H_{0}) which implies H = H_{0}. Moreover, the homomorphisms in Sub represented by f_{0} and f agree. Hence the group homomorphisms f_{0} and f agree themselves and we get k ∈ N_{G} im(f_{0}) ∩ K. This implies that a(H, f) = [K ∩ N_{G} im(f_{0}) : im(f_{0})] · id if (H) = (H_{0}) and \overline{f} = \overline{f_{0}}, and that otherwise a(H, f) = 0 or x_{H,f} = 0 holds. Hence the map T is injective.

(c) follows from assertion (a) and the bijectivity of the isomorphism T defined in (2.3) because there is a natural isomorphism

M ⊗_{ΛSub} E_{H}S_{H}Λ ⊗_{ℤ} Rep_{ℂ}(?) ⟶^{≅} M(H) ⊗_{Λ[W_{G}H]} S_{H}Λ ⊗_{ℤ} Rep_{ℂ}(?).  (3.6)

Now (d) follows from (c) and the fact that the Λ[W_{G}H]-module S_{H}Λ ⊗_{ℤ} Rep_{ℂ}(?) ≅ im(Θ_{C}) is projective. This finishes the proof of Theorem 3.5.

4. The construction of the Chern character

In this section we want to prove Theorem 0.7. There are similarities with the construction in [11]. The main difference is that here we want to give a construction, where we only have to invert the orders of elements in F(X), whereas in [11] we have worked over the rationals. In [11] we have used the Hurewicz homomorphism from stable homotopy to singular homology, which is only an isomorphism after inverting all primes. We will use the multiplicative structure of K_{∗}^{G} instead and work with a different source for the equivariant Chern character, which allows us to invert only the orders of finite subgroups of G.

In the sequel we denote by K_{p}^{G}(X) the equivariant K-homology of a proper G-CW -complex X. It is defined by colim_{Y ⊂X} KK_{G}^{p}(C_{0}(Y), ℂ), where Y runs over all cocompact G-subcomplexes of X and KK_{G}^{p}(C_{0}(Y), ℂ) denotes equivariant KK-theory of the G-C^{∗}-algebra C_{0}(X) of continuous functions X → ℂ, which vanish at infinity, and the C^{∗}-algebra ℂ with the trivial G-action. Given a homomorphism ϕ : H → G of groups and a proper H-CW -complex, then ind_{ϕ} X := G ×_{ϕ} X is a proper G-CW -complex and there is an induction homomorphism

ind_{ϕ} : K_{0}^{H}(X) → K_{0}^{G}(ind_{ϕ} X).

If the kernel of ϕ acts freely on X, then ind_{ϕ} is bijective. In particular we get for a proper G-CW - complex X a homomorphism

K_{p}^{G}(X) ⟶^{ind_{G→{1}}} K_{p}(G\X),

which is bijective if G acts freely on X. There is an external product

μ : K_{p}^{G}(X) × K_{q}^{G′}(X′) → K_{p+q}^{G×G′}(X × X′)

for groups G and G′, a proper G-CW -complex X and a proper G′-CW -complex X′. External products and induction are compatible. For more information about equivariant K-homology and KK-theory we refer to [8] and in particular for the induction homomorphisms to [16].

Let X be a proper G-CW -complex. We have introduced the ring Λ = Λ^{G}(X) in (0.5). We want to construct for H ∈ F(X) and p = 0, 1 a Λ-homomorphism

\underline{ch}_{p}^{G}(X)(H) : Λ ⊗_{ℤ} K_{p}(C_{G}H\X^{H}) ⊗_{ℤ} Rep_{ℂ}(H) → Λ ⊗_{ℤ} K_{p}^{G}(X),  (4.1)

where K_{p}(C_{G}H\X^{H}) is the (non-equivariant) K-homology of the CW -complex C_{G}H\X^{H}. The map will be defined by the following composition

Λ ⊗_{ℤ} K_{p}(C_{G}H\X^{H}) ⊗_{ℤ} Rep_{ℂ}(H)
id ⊗_{ℤ}K_{p}(pr_{1};R)⊗_{ℤ}id↑≅
Λ ⊗_{ℤ} K_{p}(EG ×_{C_{G}H} X^{H}) ⊗_{ℤ} Rep_{ℂ}(H)
id ⊗_{ℤ} ind_{C_{G}H→{1}} ⊗j↑≅
Λ ⊗_{ℤ} K_{p}^{C_{G}H}(EG × X^{H}) ⊗_{ℤ} K_{0}^{H}(∗)
μ↓
Λ ⊗_{ℤ} K_{p}^{C_{G}H×H}(EG × X^{H})
ind^{m_{H}} ↓≅
Λ ⊗_{ℤ} K_{p}^{G}(ind_{m_{H}} EG × X^{H})
id ⊗_{ℤ}K_{p}^{G}(ind_{m_{H}} pr_{2})↓
Λ ⊗_{ℤ} K_{p}^{G}(ind_{m_{H}} X^{H})
id ⊗_{ℤ}K_{p}^{G}(v_{H})↓
K_{p}^{G}(X)

Some explanations are in order. We have a left C_{G}H-action on EG × X^{H} by g(e, x) = (ge, gx) for g ∈ C_{G}H, e ∈ EG and x ∈ X^{H}. It extends to a C_{G}H × H-action by letting the factor H acting trivially. The map pr_{1} : EG ×_{C_{G}H} X^{H} → C_{G}H\X^{H} is the canonical projection. It induces an isomorphism

Λ ⊗_{ℤ} K_{p}(pr_{1}; R) : Λ ⊗_{ℤ} K_{p}(EG ×_{C_{G}H} X^{H}) ⟶^{≅} Λ ⊗_{ℤ} K_{p}(C_{G}H\X^{H})

since each isotropy group of the C_{G}H-space X^{H} is finite and for any finite group L the projection induces an isomorphism Λ ⊗_{ℤ} H_{p}(BL) ⟶^{≅} Λ ⊗_{ℤ} H_{p}(∗) and hence by the Atiyah-Hirzebruch spectral sequence an isomorphism Λ ⊗_{ℤ} K_{p}(BL) ⟶^{≅} Λ ⊗_{ℤ} K_{p}(∗) for all p. The isomorphism j : K_{0}^{H}(∗) ⟶^{≅} Rep_{ℂ}(H) is the canonical isomorphism. The group homomorphism m_{H} : C_{G}H × H → G sends (g, h) to gh. We denote by pr_{2} : EG × X^{H} → X^{H} the canonical projection. The G-map v_{H} : ind_{m_{H}} X^{H} = G ×_{m_{H}} X^{H} → X sends (g, x) to gx.

Notice that we obtain a contravariant Sub-module K_{0}(C_{G}?\X^{?}) by assigning to a finite subgroup H ⊂ G the Λ-module Λ ⊗_{ℤ} K_{p}(C_{G}H\X^{H}). We have already introduced the covariant Λ-module Λ ⊗_{ℤ} Rep_{ℂ}(?). Analogously to [11] one checks that the various maps \underline{ch}_{p}^{G}(X)(H) defined above induce a map of Λ-modules

ch_{p}^{G}(X) : Λ ⊗_{ℤ} K_{p}(C_{G}?\X^{?}) ⊗_{ΛSub} Λ ⊗_{ℤ} Rep_{ℂ}(?) → Λ ⊗_{ℤ} K_{p}^{G}(X).  (4.2)

Notice that for L ∈ F(X) and X = G/L the ΛSub-module K_{0}(C_{G}?\(G/L)^{?}) is isomorphic to the ΛSub-module Λ mor_{Sub} (?, L), which sends a finite subgroup H ⊂ G to the free Λ-module with base mor_{Sub} (H, K). By the Yoneda Lemma one obtaines a canonical isomorphism

Λ ⊗_{ℤ} K_{p}(C_{G}?\(G/L)^{?}) ⊗_{ΛSub} Λ ⊗_{ℤ} Rep_{ℂ}(?) ⟶^{≅} Rep_{ℂ}(L).

One easily checks that under this identification ch_{0}^{G}(G/L) becomes the canonical identification of Rep_{ℂ}(L) with K_{0}^{G}(G/L). Notice that K_{1}(C_{G}?\(G/L)^{?}) and K_{1}^{G}(G/L) are both trivial. Hence ch_{p}^{G}(G/L) is bijective for all L ∈ F(X) and p = 0, 1. Because of Theorem 3.5 (d) the source of ch_{∗}^{G} is an equivariant homology theory on proper G-CW -complexes Y with F(Y) ⊂ F(X). One easily checks that ch_{∗}^{G} is compatible with the Mayer-Vietoris sequences. By induction over the number of equivariant cells and the Five-Lemma ch_{p}^{G}(Y) is bijective for any finite proper G-CW -complex Y with F(Y) ⊂ F(X). Notice that K_{p}^{G}(Y) is the colimit colim_{Z⊂Y} K_{p}^{G}(Z), where Z runs through all finite G-CW -subcomplexes Z of Y. The analogous statement holds for the source of ch_{∗}^{G}. Hence ch_{p}^{G}(Y) is bijective for all proper G-CW -complexes Y with F(Y) ⊂ F(X) and p = 0, 1. Now Theorem 0.7 follows from Theorem 3.5 (c).

5. The Baum-Connes Conjecture and the Trace Conjecture

In the sequel we denote for a proper G-CW -complex X by

asmb^{G} : K_{0}^{G}(X) → K_{0}(C_{r}^{∗}(G))  (5.1)

the assembly map which essentially assigns to an element in K_{0}^{G}(X) represented by an equivariant Kasparov cycle its index. Given a homomorphism ϕ : H → G of groups with finite kernel, there is an induction homomorphism ind_{ϕ} : K_{p}(C_{r}^{∗}(H)) → K_{p}(C_{r}^{∗}(G)) such that the following diagram commutes [16, Theorem 1]

K_{0}^{H}(X) ⟶^{asmb^{H}} K_{0}(C_{r}^{∗}(H))
ind^{ϕ}↓  ^{ind}^{ϕ}↓
K_{0}^{G}(ind_{ϕ} X) ⟶^{asmb^{G}} K_{0}(C_{r}^{∗}(G))

These induction homomorphisms, the assembly maps and the change of rings homomorphisms associated to the passage from C_{r}^{∗}(G) to N (G) are compatible with the external products

μ : K_{p}^{G}(X) × K_{q}^{G′}(X′) → K_{p+q}^{G×G′}(X × X′);
μ : K_{p}(C_{r}^{∗}(G)) × K_{q}(C_{r}^{∗}(G′)) → K_{p+q}(C_{r}^{∗}(G × G′));
μ : K_{p}(N (G)) × K_{q}(N (G′)) → K_{p+q}(N (G × G′))

for groups G and G′, a proper G-CW -complex X and a proper G′-CW -complex X′. We will use in the sequel the elementary fact that for any G-map f : X → Y of proper G-CW -complexes the

composition K_{0}^{G}(X) ⟶^{K_{0}^{G}(f)} K_{0}^{G}(Y) ⟶^{asmb^{G}} K_{0}(C_{r}^{∗}(G)) is asmb^{G} : K_{0}^{G}(X) → K_{0}(C_{r}^{∗}(G)). In the sequel the letter i denotes change of rings homomorphism for the canonical map C_{r}^{∗}(G) → N (G).

Let X be a proper G-CW -complex. We have introduced J = J^{G}(X) in (0.6). Define the homomorphism

ξ_{1} : ⊕_{(C)∈J} Λ ⊗_{ℤ} K_{0}(C_{G}C\X^{C}) ⊗_{Λ[W_{G}C]} im (θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C))
→ K_{0}(N (G)) (5.2)

by the composition of the equivariant Chern character of Theorem 0.7

ch_{0}^{G}(X) : ⊕_{(C)∈J} Λ ⊗_{ℤ} K_{0}(C_{G}C\X^{C}) ⊗_{Λ[W_{G}C]} im (θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C))

⟶^{≅} Λ ⊗_{ℤ} K_{0}^{G}(X),

the assembly map

id ⊗ asmb^{G} : Λ ⊗_{ℤ} K_{0}^{G}(X) → Λ ⊗_{ℤ} K_{0}(C_{r}^{∗}(G))

and the change of rings homomorphism

id ⊗i : Λ ⊗_{ℤ} K_{0}(C_{r}^{∗}(G)) → Λ ⊗_{ℤ} K_{0}(N (G)).

This is the homomorphism which we want to understand. In particular we are interested in its image. We will identify it with a second easier to compute homomorphism

ξ_{2} : ⊕_{(C)∈J} Λ ⊗_{ℤ} K_{0}(C_{G}C\X^{C}) ⊗_{Λ[W_{G}C]} im (θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C))
→ K_{0}(N (G)), (5.3)

which is defined as follows. Let l : im(Θ_{C}) → Rep_{ℂ}(C) be the inclusion. Let K_{0}(pr) : K_{0}(C_{G}C\X^{C}) → K_{0}(∗) be induced by the projection from C_{G}C\X^{C} to the one-point space ∗. We obtain a map

(i ∘ asmb^{{1}} ∘K_{0}(pr)) ⊗ l : K_{0}(C_{G}C\X^{C}) ⊗ im(θ_{C}) → K_{0}(N ({1})) ⊗ Rep_{ℂ}(C).

Define

α : K_{0}(N ({1})) ⊗ Rep_{ℂ}(C) → Rep_{ℂ}(C) [U] ⊗ [W] ↦ dim_{ℂ}(U) · [W].

Notice that α is essentially given by the external product and K_{0}(N (H)) = Rep_{ℂ}(H) holds by definition for any finite group H. Induction yields a map

ind_{C}^{G} : K_{0}(N (C)) → K_{0}(N (G)).

The composition of these three maps above induces for any finite cyclic subgroup C ⊂ G a homomorphism

ξ_{2}(C) : Λ ⊗_{ℤ} K_{0}(C_{G}C\X^{C}) ⊗_{Λ[W_{G}C]} im (θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C)) → K_{0}(N (G)).

Define ξ_{2} to be the direct sum ⊕_{(C)∈J} ξ_{2}(C) after the choice of a representative C ∈ (C) for each (C) ∈ J.

Theorem 5.4 Let X be a proper G-CW -complex. Then the maps ξ_{1} of (5.2) and ξ_{2} of (5.3) agree.

\underline{\underline{Proof}} : In the sequel maps denoted by the letter μ will be given by external products and pr denotes the projection from a space to the one-point space ∗. Fix a cyclic subgroup C ∈ F(X). Notice that the homomorphism m_{C} : C_{G}C × C → G (g, c) ↦ gc has a finite kernel so that induction is defined also on the level of the reduced group C^{∗}-algebra and the group von Neumann algebra. Denote by ν : Λ⊗_{ℤ}K_{0}^{C_{G}C×C}(EG×X^{C}) → K_{0}^{G}(X) the composition of the maps id ⊗K_{0}^{G}(v_{C}), id ⊗K_{0}^{G}(ind_{m_{C}} pr_{2}) and ind_{m_{C}} appearing in the definition of \underline{ch}_{0}(X)(C). Then the following diagram commutes

Λ ⊗_{ℤ} K_{0}^{C_{G}C}(EG × X^{C}) ⊗_{ℤ} K^{C}(∗) ⟶^{id ⊗i∘asmb^{C_{G}C} ⊗j} Λ ⊗_{ℤ} K_{0}(N (C_{G}C)) ⊗_{ℤ} Rep_{ℂ}(C)
μ↓  ^{μ}↓
Λ ⊗_{ℤ} K_{0}^{C_{G}C×C}(EG × X^{C})  ⟶^{id ⊗i∘asmb^{C_{G}C×C}}  Λ ⊗_{ℤ} K_{0}(N (C_{G}C × C))
ν↓  ^{ind}^{m_{C}} ↓
Λ ⊗_{ℤ} K_{0}^{G}(X)  ⟶^{id ⊗i∘asmb^{G}}  Λ ⊗_{ℤ} K_{0}(N (G))

For any group G the map induced by the center-valued von Neumann dimension

dim_{N (G)}^{u} : K_{0}(N (G)) → Z(N (G))

is injective. Given a CW -complex Z and an element η ∈ K_{0}(Z), there is a closed manifold M with a map f : M → BG and an elliptic complex D^{∗} of differential operators of order 1 over M such that K_{0}(f) : K_{0}(M) → K_{0}(Z) maps the class [D^{∗}] ∈ K_{0}(M) to η [2]. In the case Z = BG the composition

K_{0}(M) ⟶^{K_{0}(f)} K_{0}(BG) ⟶^{(ind_{G→{1}})^{−1}} K_{0}^{G}(EG) ⟶^{asmb^{G}} K_{0}(C_{r}^{∗}(G)) ⟶^{i} K_{0}(N (G)) ⟶^{dim_{N (G)}^{u}} Z(N (G))

resp. the composition

K_{0}(M) ⟶^{K_{0}(pr)} K_{0}(∗) ⟶^{asmb^{{1}}} K_{0}(C^{∗}({1}) ⟶^{i} K_{0}(N ({1})) ⟶^{ind_{{1}}^{G}} K_{0}(N (G)) ⟶^{dim_{N (G)}^{u}} Z(N (G))

maps [D^{∗}] to the element index_{N (G)}^{u}(\overline{D}^{∗}) resp. index(D^{∗}) · 1_{N (G)}, where index_{N (G)}^{u}(\overline{D}^{∗}) resp. index(D^{∗}) has been defined in (1.7) resp. (1.1). We conclude from Theorem 0.4 and the injectivity of the map dim_{N (G)}^{u} of (5.5) that the following diagram commutes

K_{0}^{G}(EG) ⟶^{i∘asmb^{G}} K_{0}(N (G))
K_{0}(pr)∘ind_{G→{1}}^{−1}↓  ^{ind}_{{1}}^{G}↑
K_{0}(∗) ⟶^{i∘asmb^{{1}}} K_{0}(N ({1}))

Since there is a C_{G}C-map EG × X^{C} → EC_{G}C, we conclude from the diagram above applied to the case G = C_{G}C that the following diagramm commutes

Λ ⊗_{ℤ} K_{0}^{C_{G}C}(EG × X^{C}) ⊗_{ℤ} K^{C}(∗) ⟶^{id ⊗i∘asmb^{C_{G}C} ⊗j} Λ ⊗_{ℤ} K_{0}(N (C_{G}C)) ⊗_{ℤ} Rep_{ℂ}(C)
id ⊗(K_{0}(pr)∘ind_{C_{G}C→{1}})⊗j↓  ↑^{id ⊗ ind_{{1}}^{C_{G}C} ⊗ id}

Λ ⊗_{ℤ} K_{0}(∗) ⊗_{ℤ} Rep_{ℂ}(C) id ⊗i∘asmb^{{1}} ⊗ id ⟶ Λ ⊗_{ℤ} K_{0}(N ({1}) ⊗_{ℤ} Rep_{ℂ}(C)

The composition

K_{0}(N ({1})⊗_{ℤ}Rep_{ℂ}(C) ⟶^{ind_{{1}}^{C_{G}C} ⊗ id} K_{0}(N (C_{G}C))⊗_{ℤ}Rep_{ℂ}(C) ⟶^{μ} K_{0}(N (C_{G}C×C)) ⟶^{ind_{m_{C}}} K_{0}(N (G))

agrees with the composition

K_{0}(N ({1})) ⊗ Rep_{ℂ}(C) ⟶^{α} Rep_{ℂ}(C) = K_{0}(N (C)) ⟶^{ind_{C}^{G}} K_{0}(N (G)).

We conclude that the following diagram commutes for any cyclic subgroup C ∈ F(X)

Λ ⊗_{ℤ} K_{0}^{C_{G}C}(EG × X^{C}) ⊗_{ℤ} K^{C}(∗) ⟶^{(id ⊗i∘asmb^{C_{G}C})∘ν∘μ} Λ ⊗_{ℤ} K_{0}(N (G))
id ⊗(i∘asmb^{{1}} ∘K_{0}(pr)∘ind_{C_{G}C→{1}})⊗j↓  ↑^{id ⊗ ind_{C}^{G}}
Λ ⊗_{ℤ} K_{0}(N ({1})) ⊗_{ℤ} Rep_{ℂ}(C)  ⟶^{id ⊗α}  Λ ⊗_{ℤ} Rep_{ℂ}(C)

Hence the following diagram commutes for any cyclic subgroup C ∈ F(X)

Λ ⊗_{ℤ} K_{p}(C_{G}C\X^{C}) ⊗_{ℤ} Rep_{ℂ}(C) id ⊗(α∘(i∘asmb^{{1}} ∘K_{0}(pr))⊗id) ⟶ Λ ⊗_{ℤ} Rep_{ℂ}(C)

id ⊗\underline{ch}_{0}^{G}(X)(C)↓  ^{id ⊗ ind_{C}^{G}}↓
Λ ⊗_{ℤ} K_{0}^{G}(X)  ⟶^{id ⊗(i∘asmb^{G})}  Λ ⊗_{ℤ} K_{0}(N (G))

Now Theorem 5.4 (and hence also Theorem 0.8) follow.

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