Algebraic K-theory of von Neumann algebras

Wolfgang Lück^{∗}  Mikael Rørdam

November 8, 1996

Abstract

To every von Neumann algebra one can associate a (multiplicative) determinant defined on the invertible elements of the algebra with range a subgroup of the abelian group of the invertible elements of the center of the von Neumann algebra. This determinant is a normalization of the usual determinant for finite von Neumann algebras of type I, for the type II_{1}-case it is the Fuglede-Kadison determinant, and for properly infinite von Neumann algebras the determinant is constant equal to 1. It is proved that every invertible element of determinant 1 is a product of a finite number of commutators. This extends a result of T. Fack and P. de la Harpe for II_{1}-factors. As a corollary it follows that the determinant induces an injection from the algebraic K_{1}-group of the von Neumann algebra into the abelian group of the invertible elements of the center. Its image is described. Another group, K_{1}^{w}(A), which is generated by elements in matrix algebras over A that induce injective right multiplication maps is also computed. We use the Fuglede-Kadison determinant to detect elements in the Whitehead group W h(G).

∗Partially supported by NSF Grant DMS - 9103327

Introduction

The purpose of this paper is to compute the algebraic K_{1}-groups K_{1}(A) and K_{1}^{w}(A) for a von Neumann algebra A. We give their definitions in section 1. One motivation for their study comes from the construction of Reidemeister von Neumann torsion for a compact Riemannian manifold in Lück-Rothenberg [12] which takes values in these K_{1}-groups for the von Neumann algebra of the fundamental group. Recall that the topological K_{1}-group of a von Neumann algebra is trivial.

In section 2 we treat von Neumann algebras of type I_{f}. Since they can be written as product of matrix algebras over abelian von Neumann algebras, the ordinary determinant for commutative rings extends to a normalized determinant det_{norm} : M_{k}(A) ⟶ Z(A) into the center Z(A). We can write Z(A) as the algebra L^{∞}(X; ν) of essentially bounded measurable functions from X to ℂ ∪ {∞} for some compact space X with positive finite measure ν. Let L^{∞}(X; ν)_{inv} be the abelian group of invertible elements and Inv(X; ν) be the (multiplicative) abelian group of measurable functions from X to ℂ∪{∞} whose preimage of both 0 and ∞ is a zero-set. We prove in theorem 2.1 that the normalized determinant induces isomorphisms:

det_{norm} : K_{1}(A) ⟶ L^{∞}(X; ν)_{inv}

det_{norm} : K_{1}^{w}(A) ⟶ Inv(X; ν)

Section 3 is devoted to the type II_{1}-case. The Fuglede-Kadison determinant of an invertible element A ∈ M_{n}(A) is defined by

det_{FK}(A) = exp (\frac{1}{2} · tr(log(A^{∗}A))) ∈ Z(A)_{inv}^{+}

where Z(A)_{inv}^{+} denotes the group of positive invertible elements in Z(A). We show in theorem 3.3 that the Fuglede-Kadison determinant induces an isomorphism:

det_{FK} : K_{1}(A) ⟶ Z(A)_{inv}^{+}

and that K_{1}^{w}(A) is trivial. The main technical ingredients are proposition 3.1, which is a variation of a technique of Broise [2] and is a kind of “Eilenberg swindle”, and proposition 3.9. This result about K_{1}(A) was proved by Fack and de la Harpe [4] for a II_{1}-factor. We avoid disintegration theory in our extension of the proof to general von Neumann algebras of type II_{1}.

We prove in theorem 4.2 in section 4 that K_{1}(A) and K_{1}^{w}(A) are trivial if A is properly infinite.

If an invertible matrix represents zero in K_{1}(A), we obtain a bound which depends only on the type of the von Neumann algebra on the number of commutators needed to write the matrix as a product of commutators.

In section 5 we detect elements in the Whitehead group W h(G) by the Fuglede-Kadison determinant. Namely, we show in theorem 5.1 for a normal finite subgroup H of a countable discrete group G that the map W h(H)^{G} ⟶ W h(G) induced by induction is rationally injective where the G-action on W h(H) comes from the conjugation action of G on H. This computation is compatible with the much more general isomorphism conjecture on algebraic K-groups by Farrell and Jones [5].

Our interest in K_{1}(A) and K_{1}^{w}(A) arises from the construction of Reidemeister von Neumann torsion in Lück-Rothenberg [12]. It is a generalization of classical Reidemeister torsion from finite to infinite groups. From the nature of its definition Reidemeister von Neumann torsion does not take values in K_{1}(A) but in K_{1}^{w}(A). Namely, the combinatorial Laplace operator acting on the complement of its kernel is a weak isomorphism, but not necessarily an isomorphism. In particular the relevant K-theory cannot be described by invertible matrices and some of the known techniques involving commutators have to be modified in order to apply them to other K-groups like K_{1}^{w}(A).

The analytic counterpart of Reidemeister von Neumann torsion is the analytic L^{2}- torsion defined by Lott [9] for a closed manifold M. Its definition requires the assumption that the Novikov-Shubin invariants of M are positive. Conjecture 9.1 in Lott-Lück [10] says that this assumption always holds. At the first glance the combinatorial definition does not need the assumption. However, if the von Neumann algebra is of type II_{1}, then we show that K_{1}^{w}(A) is trivial. This indicates that one has to make also in the combinatorial case an assumption on the Novikov-Shubin invariants of the operators coming from the cellular chain complex in order to guarantuee that the K_{1}-group they take values in are nontrivial. Namely, for weak isomorphisms with positive Novikov-Shubin invariants one can define a generalized Fuglede-Kadison determinant and obtains a non-trivial Reidemeister von Neumann torsion with values in the real numbers, called in this context combinatorial L^{2}-torsion. This is carried out in Lück [11] and it is conjectured that combinatorial and analytic L^{2}-torsion agree (see [11, conjecture 3.1] [10, conjecture 9.7]). We mention that the von Neumann algebra of a finitely generated group π is of type I_{f} if π is virtually a finitely generated free abelian group, (i.e. if π contains a normal finitely generated free abelian group of finite index) and is of type II_{f} otherwise and that the conjectures mentioned above are known to be true for virtually finitely generated free abelian groups.

We refer to the survey article of Rosenberg [15] for information about connections between topology and algebraic K-theory of operator algebras.

1. Definition of K-groups of a von Neumann algebra

In this section we define the various algebraic K-groups we want to study. Throughout this section let R be an associative ring with unit.

Definition 1.1 Let K_{1}(R) and K_{1}^{w}(R) be the abelian groups generated by conjugation classes of bijective, respectively, injective R-endomorphisms of finitely generated free R-modules satisfying the following relations • [f] + [h] = [g] , if there is an exact sequence of bijective, respectively, injective

R-endomorphisms 0 ⟶ (U, f) ⟶^{i} (V, g) ⟶^{p} (W, h) ⟶ 0;

• [gf] = [f] + [g] , if f and g are bijective resp. injective R-endomorphisms of the same finitely generated free R-module;

• [id : V ⟶ V] = 0 , if V is a finitely generated free R-module.

The group K_{1}(R) can be identified with the abelianization GL(R)_{ab} of the general linear group GL(R) = lim_{n→∞} GL(n, R). The identification is given by interpreting an invertible (n, n)-matrix as an automorphism of R^{n} and vice versa. The generators of K_{1}^{w}(R) can be identified with elements of M(n, R) for which the corresponding endomorphism of R^{n} given by right multiplication is injective. The description in Definition 1.1 with generators and relations is more natural, but for the computations in this paper the second description will be used.

Remark 1.2 The group K_{1}(R) can be identified with the abelianization GL(R)_{ab} of the general linear group GL(R) = lim_{n→∞} GL(n, R). The identification is given by interpreting an invertible (n, n)-matrix as an automorphism of R^{n} and vice versa. A direct description of K_{1}^{w}(R) by groups of matrices is not available since injective R-endomorphisms are not necessarily bijective. If R is a finite von Neumann algebra A, generators of K_{1}^{w}(A) have the following description in terms of Hilbert A-modules.

Assume that A is a finite von Neumann algebra with a faithful normal normalized trace tr. Let L^{2}(A, tr) be the corresponding Hilbert space which is the completion of A with respect to the inner product < a, b >= tr(b^{∗}a). Then ⊕_{i=1}^{n}L^{2}(A, tr) is an A − M(n, A) bimodule, and every bounded A-endomorphism of ⊕_{i=1}^{n}L^{2}(A, tr) is given by right multiplication by an element of M(n, A). An A-endomorphism of ⊕_{i=1}^{n}L^{2}(A, tr) is said to be a weak isomorphism if its kernel is zero and its image is dense. Since A is finite it follows that an endomorphism of ⊕_{i=1}^{n}L^{2}(A, tr) is a weak isomorphism if and only if the corresponding element in M(n, A) induces an injective endomorphism on A^{n}. Hence the generators of K_{1}^{w}(A) are weak isomorphisms of ⊕_{i=1}^{n}L^{2}(A, tr) for n ∈ ℕ. It is important for the construction of Reidemeister von Neumann torsion to allow weak isomorphisms and not only isomorphisms (see Lück-Rothenberg [12]).

We have the following type decomposition theorem for von Neumann algebras. See [6, 6.5.2].

Theorem 1.3 Given a von Neumann algebra A, there is a natural unique decomposition:

A = A_{I_{f}} × A_{I_{∞}} × A_{II_{1}} × A_{II_{∞}} × A_{III}

into von Neumann algebras of type I_{f}, I_{∞}, II_{1}, II_{∞} and III. In particular there are natural isomorphisms induced by the projections:

K_{1}(A) = K_{1}(A_{I_{f}}) × K_{1}(A_{I_{∞}}) × K_{1}(A_{II_{1}}) × K_{1}(A_{II_{∞}}) × K_{1}(A_{III})

and similiarly for K_{1}^{w}(A).

This theorem reduces the computation of the various K_{1}-groups of a Neumann algebra to the computation in the case where A is of type I_{f}, I_{∞}, II_{1}, II_{∞} and III. Notice that a von Neumann algebra is properly infinite if and only if its natural decomposition does not contain pieces of type I_{f} and II_{1}.

2. The type I_{f}-case

This section contains the computations of the K_{1}-groups of a von Neumann algebra of type I_{f}. We begin by recalling the structure of these von Neumann algebras.

Let B_{n} be a von Neumann algebra on a Hilbert space H_{n} for n ∈ ℕ. Let H = ⊕_{i=1}^{∞}H_{n} be the Hilbert space direct sum. The product von Neumann algebra ∏_{n=1}^{∞} B_{n} is the von Neumann algebra on H, which elements are sequences (B_{n} ∈ B_{n} | n ∈ ℕ), such that there exists a number K (depending on the sequence, but not on n), with ∥B_{n} ∥ ≤ K for all n ∈ ℕ. The embedding of ∏_{n=1}^{∞} B_{n} in B(H) sends such a sequence to the sum of the operators B_{n} : H_{n} ⟶ H_{n}.

Every von Neumann algebra A of type I_{f} is of the form

∏_{n=1}^{∞} A_{n},

where A_{n} is a von Neumann algebra of type I_{n}. Furthermore, A_{n} is isomorphic to M_{n}(Z_{n}), where Z_{n} is the center of A_{n}. The center Z(A) of A is

∏_{n=1}^{∞} Z_{n}.

Let η_{n} : Z_{n} → Z_{n} be the map which sends ua into ua^{1/n} when u, a ∈ Z_{n}, u is unitary and a is positive. Note that η_{n} is multiplicative. Let

det : M_{k}(A_{n}) = M_{k}(M_{n}(Z_{n})) = M_{kn}(Z_{n}) → Z_{n}

be the usual determinant, and set

det_{norm} = η_{n} ∘ det : M_{kn}(A_{n}) → Z_{n}

Then det_{norm} is multiplicative and

det_{norm}(U) = U^{n}, U ∈ (Z_{n})_{inv}, U^{∗}U = I
det_{norm}(A) = A, A ∈ (Z_{n})_{inv}, A^{∗} = A
∥ det_{norm}(A) ∥ ≤ ∥ A ∥^{k}, A ∈ M_{k}(A_{n}).

Define

det_{norm} : M_{k}(A) → Z(A), k ∈ ℕ,

by the product of the determinants det_{norm} for the A_{n}-s.

Let Z(A)_{inv} be the multiplicative group of invertible elements in Z(A). Denote by Z(A)^{w} the Grothendieck group of the abelian semigroup of elements a ∈ Z(A), for which multiplication with a induces an injection m_{a} : Z(A) ⟶ Z(A ). If we identify Z(A) with L^{∞}(X; ν), for some measure space (X; ν), we can identify Z(A)^{w} with Inv(X; ν), i.e. the space of measurable functions from X to ℂ ∪ {∞}, for which the preimages of 0 and ∞ are zero sets. In particular, the canonical map

Z(A)_{inv} ⟶ Z(A)^{w}

is injective. The next theorem was proved for A abelian in Lück-Rothenberg [12, section 2].

Theorem 2.1 The normalized determinant induces isomorphisms for a von Neumann algebra A of type I_{f}.

det_{norm} : K_{1}(A) ⟶ Z(A)_{inv};

det_{norm} : K_{1}^{w}(A) ⟶ Z(A)^{w}.

Notice that the proof of theorem 2.1 is straightforward for K_{1} if A is a finite product ∏_{i=1}^{N} A_{n}. In this case the product of the von Neumann algebras is an ordinary product of rings and we get from Morita equivalence and Milnor [13, section 7]

K_{1}(A) = K_{1}(∏_{n=1}^{N} A_{n}) = ∏_{i=1}^{N} K_{1}(A_{n}) = ∏_{i=1}^{N} K_{1}(M_{n}(Z_{n})) = ∏_{i=1}^{N} K_{1}(Z_{n}) = ∏_{i=1}^{N}(Z_{n})_{inv} = Z(A)_{inv}.

In order to handle the general case and K_{1}^{w}(A), we outline a different proof. Namely, theorem 2.1 follows directly from proposition 2.4. The proof of proposition 2.4 uses the following two lemmas whose fairly straightforward proofs are omitted.

Lemma 2.2 Let (X; ν) be a measure space. Let T ∈ M_{n}(L^{∞}(X; ν)) be normal, i.e., T and T^{∗} commute. Then there is a unitary U ∈ M_{n}(L^{∞}(X; ν)) such that U^{∗}T U is diagonal.

Lemma 2.3 Let f_{i} ∈ L^{∞}(X; ν), 1 ≤ i ≤ n, be positive functions. Suppose that their product equals 1. Then there is a measurable map σ : X ⟶ Σ_{n} into the discrete group of permutations of {1, 2, …, n}, such that for all x ∈ X and 1 ≤ k ≤ n,

min{f_{i}(x) | 1 ≤ i ≤ n} ≤ ∏_{i=1}^{k} f_{σ(i)(x)} ≤ max{f_{i}(x) | 1 ≤ i ≤ n}.

Proposition 2.4 Let T ∈ M_{n}(L^{∞}(X; ν)) be unitary or positive. Assume that the normalized determinant of T equals 1. Then there exist a unitary element U ∈ M_{n}(L^{∞}(X; ν)) and a unitary, respectively, an invertible, positive element A ∈ M_{n}(L^{∞}(X; ν)), satisfying:

T = AUA^{−1}U^{∗};
∥A∥ ≤ ∥T ∥;
∥A^{−1} ∥ ≤ ∥T^{−1} ∥ .

\underline{\underline{Proof}} : By the polar decomposition theorem and lemma 2.2, we may assume that T is either positive or unitary and that T is diagonal. Let t_{1}, t_{2}, …, t_{n} be the diagonal entries. Denote by A the diagonal matrix having as (i, i)-th entry the product ∏_{j=1}^{i} t_{j}.

We first treat the case when T is unitary. Let U be the matrix representing the cyclic permutation in Σ_{n}, sending n to 1 and i to i + 1 for 1 ≤ i < n. Then A and U are unitaries satisfying T = AUA^{−1}U^{−1}.

Suppose that T is positive. By lemma 2.3 there is a measurable function σ : X ⟶ Σ_{n} such that for all x ∈ X and 1 ≤ k ≤ n, the following inequality

min{t_{i} | 1 ≤ i ≤ n} ≤ ∏_{i=1}^{k} t_{σ(i)} ≤ max{t_{i} | 1 ≤ i ≤ n}

holds. Let P be the permutation matrix associated to σ. Then P^{−1}T P is again a diagonal matrix which is obtained from the diagonal matrix T by permuting the diagonal entries according to σ. Hence we can assume without loss of generality that

min{t_{i} | 1 ≤ i ≤ n} ≤ ∏_{i=1}^{k} t_{i} ≤ max{t_{i} | 1 ≤ i ≤ n}.

Since t_{i} is essentially bounded from above for each 1 ≤ i ≤ n and ∏_{i=1}^{n} t_{i} = 1 holds by assumption, 1/t_{i} is essentially bounded from above. This shows that T is invertible. Now we can proceed as in the unitary case because the inequality above gives the desired bounds on ∥ A ∥ and ∥ A^{−1} ∥.

Proposition 2.4 shows for T ∈ M_{k}(A), satisfying det_{norm}(T) = 1, that T can be written as a product of two commutators of invertible elements in M_{k}(A).

3. The type II_{1}-case

In this section we compute K_{1}(A) for a von Neumann algebra of type II_{1} using the Fuglede-Kadison determinant. This has essentially been proved by Fack and de la Harpe [4] in the case when A is a II_{1}-factor. The present proof generalizes Fack and de la Harpe’s proof to the non-factor case. For the convenience of the reader, we recall the technique introduced by Broise [2] to express elements as products of commutators.

Proposition 3.1 Let A be a von Neumann algebra on H. Let {P_{n} | n ∈ ℕ} be a sequence of mutually orthogonal projections P_{n} in A such that ∑_{n=1}^{∞} P_{n} = 1. Let {T_{n} | n ∈ ℕ}, {A_{n} | n ∈ ℕ} and {B_{n} | n ∈ ℕ} be sequences of elements in A ⊂ B(H), satisfying for all n ∈ ℕ:

T_{n} = P_{n}T_{n}P_{n} + (1 − P_{n});
A_{n} = (P_{n} + P_{n+1})A_{n}(P_{n} + P_{n+1}) + (1 − P_{n} − P_{n+1});
B_{n} = (P_{n} + P_{n+1})B_{n}(P_{n} + P_{n+1}) + (1 − P_{n} − P_{n+1});
B_{n}A_{n}T_{n} = A_{n}B_{n}T_{n+1}.

Suppose there is a number K such that

∥A_{n} ∥, ∥B_{n} ∥, ∥T_{n} ∥ ≤ K

for all n ∈ ℕ. Then:

1. The following products are strongly convergent:

A_{odd} = A_{1}A_{3}A_{5} …;

A_{ev} = A_{2}A_{4}A_{6} …;

B_{odd} = B_{1}B_{3}B_{5} …;

B_{ev} = B_{2}B_{4}B_{6} …;

T_{ev} = T_{2}T_{4}T_{6} …;

T_{ext} = T_{3}T_{5}T_{7} ….

2. If T_{n}, A_{n} and B_{n} are invertible and

∥A_{n}^{−1} ∥, ∥B_{n}^{−1} ∥, ∥T_{n}^{−1} ∥ ≤ K

holds for n ∈ ℕ, then the operators A_{odd}, A_{ev}, B_{odd}, B_{ev}, T_{ev} and T_{ext} are invertible.

3. If T_{n}, A_{n} and B_{n} induce injections A ⟶ A for n ∈ ℕ, then the same is true for the operators A_{odd}, A_{ev}, B_{odd}, B_{ev}, T_{ev} and T_{ext}.

4. B_{odd}A_{odd}T_{1}T_{ext} = A_{odd}B_{odd}T_{ev};

B_{ev}A_{ev}T_{ev} = A_{ev}B_{ev}T_{ext}.

\underline{\underline{Proof}} : 1.) and 2.). Each of the sequences clearly converges on the dense subspace of H spanned by P_{n}(H), n ∈ ℕ. The boundedness of the sequences ∥A_{n}∥, ∥B_{n}∥ and ∥T_{n}∥ now implies that the partial products are also bounded. Hence the six products converge on every vector in H. The inverses of these products are the products of the inverses in reversed order which are strongly convergent if ∥A_{n}^{−1}∥, ∥B_{n}^{−1}∥ and ∥T_{n}^{−1}∥ are bounded.

3.) We give the proof for A_{ev} and the other cases follow in a similar way. Let x ∈ ker A_{ev}. Note that A_{2k} and A_{ev} commute with P_{2n} + P_{2n+1} for all n, k ∈ N. It follows that

0 = (P_{2n} + P_{2n+1})A_{ev}x = A_{ev}(P_{2n} + P_{2n+1})x
= A_{2n}(P_{2n} + P_{2n+1})x.

Because A_{2n} is injective, we conclude that (P_{2n} +P_{2n+1})x = 0, and since n is arbitrary, x = 0 follows.

4.) is easily verified on elements in P_{n}(H) for all n ∈ ℕ. As these elements span a dense subspace of H, the claim follows.

We will use this proposition to show that the class [T_{1}] of T_{1} in some K_{1}-group vanishes. This would follow if each of the operators A_{odd}, A_{ev}, B_{odd}, B_{ev}, T_{ev}, T_{ext} and T_{1} define a class in the relevant K_{1}-group since [AB] = [A] + [B] holds. Notice that we avoid to speak of commutators in order to make sure that this lemma also applies in the case where we are dealing with weak isomorphisms. If all maps involved are isomorphisms, the conclusion of the lemma above would be that T_{1} is a product of two commutators.

Definition 3.2 Let A be a von Neumann algebra of type II_{1} and let tr denote the center valued trace on A. Extend tr the standard way to M_{n}(A) so that tr takes values in Z(A) for all n ∈ ℕ. The Fuglede-Kadison determinant is in [6] defined to be

det_{FK}(A) = exp(\frac{1}{2}tr(log(A^{∗}A))) ∈ Z(A)_{inv}^{+},

for A ∈ GL(n, A).

It is proved in [6] (see also [8, I.6.10]) that

det_{FK}(AB) = det_{FK}(A) det_{FK}(B), A, B ∈ GL(n, A)

det_{FK} ( \begin{matrix}A & 0 \\ 0 & B \end{matrix} ) = det_{FK}(A) det_{FK}(B), A ∈ GL(n, A), B ∈ GL(m, A)

det_{FK}(A) = A, A ∈ Z(A)_{inv}^{+}.

In particular, the determinant of every element in the derived group [GL(A), GL(A)] is 1, and so we get a homomorphism

det_{FK} : K_{1}(A) → Z(A)_{inv}^{+},

which is surjective by the last identity above.The following theorem is the main result of this section.

Theorem 3.3 Let A be a von Neumann algebra of type II_{1}.

1. The Fuglede-Kadison determinant defines an isomorphism:

det_{FK} : K_{1}(A) ⟶ Z(A)_{inv}^{+};

2. The weak K-group K_{1}^{w}(A) vanishes.

The rest of the section is devoted to proving that det_{FK} is injective and that K_{1}^{w}(A) is trivial. We prove the former by showing that if det_{FK}(A) = 1, then A is a product of at most nine commutators. Notice that we cannot define the homomorphism above for K_{1}^{w}(A), since for a weak isomorphism A the logarithm log(A^{∗}A) is not necessarily a bounded operator.

Recall that our Hilbert spaces are assumed to be separable. In particular all von Neumann algebras we consider are countably decomposable. Hence we get from Kadison-Ringrose [8], exercise 6.9.20 on page 447, and 6.9.27 on page 448:

Lemma 3.4 Let B be a von Neumann algebra with no central portion of type I_{n} with n odd. Let A ⊂ B be a maximal abelian subalgebra. Then A contains a projection P satisfying P ∼ 1 − P.

Lemma 3.5 Let A be a von Neumann algebra either of type II_{1} or properly infinite.

1. If T ∈ A is normal, then there is a projection P ∈ A commuting with T and elements T_{1} and T_{2} in A satisfying:

P ∼ 1 − P;

T = T_{1}T_{2};

T_{1} = P T_{1}P + (1 − P);

T_{2} = (1 − P)T_{2}(1 − P) + P.

If T is positive, unitary, or induces an injection r_{T} : A ⟶ A, or invertible, then the same is true for T_{1} and T_{2}.

2. Let T in A be positive and invertible such that det_{FK}(T) = 1. Then there are positive invertible operators T_{1} and T_{2}, invertible operators A and B and a projection P ∈ A commuting with T satisfying:

P ∼ 1 − P;
T = T_{1}T_{2}ABA^{−1}B^{−1};
T_{1} = P T_{1}P + (1 − P);
T_{2} = (1 − P)T_{2}(1 − P) + P;
det_{FK}(T_{1}) = det_{FK}(T_{2}) = 1.

\underline{\underline{Proof}} : By lemma 3.4 there is a projection P commuting with T satisfying P ∼ 1 − P. Set

T̃_{1} = P T P + (1 − P);
T̃_{2} = (1 − P)T (1 − P) + P.

Then T_{1} = T̃_{1} and T_{2} = T̃_{2} will satisfy 1.). In 2.) let C ∈ Z(A) be det_{FK}(T̃_{1}) and choose a partial isometry V satisfying P = V^{∗}V and 1 − P = V V^{∗}. Now define:

T_{1} = P C^{−2}T P + (1 − P);
T_{2} = (1 − P)C^{2}T (1 − P) + P;
A = P C + (1 − P)C^{−1};
B = V + V^{∗}.

Lemma 3.6 Let A be of type II_{1} and P be a projection satisfying P ∼ 1 − P. Then there is a sequence of projections {P_{n} | n ∈ ℕ} satisfying

1. The projections P_{n} are mutually orthogonal;

2. P_{1} = P;

3. tr(P_{n}) = 2^{−n};

4. ∑_{n=1}^{∞} P_{n} = 1.

\underline{\underline{Proof}} : Use [8], Theorems 8.4.3 and 8.4.4 to find projections P_{n}, n ≥ 2, in A such that 1.) and 3.) are satisfied. Since tr is normal,

tr(∑_{n=1}^{∞} P_{n}) = ∑_{n=1}^{∞} tr(P_{n}) = 1,

and so 4.) holds.

The following lemma is a consequence of theorem 8.4.4 in [8] on page 533.

Lemma 3.7 Let A be a von Neumann algebra of type II_{1}. Let P_{0} and P_{1} be projections in A and C ∈ Z(A) be such that P_{0} ≤ P_{1} and tr(P_{0}) ≤ C ≤ tr(P_{1}). Then there is a projection P ∈ A such that P_{0} ≤ P ≤ P_{1} and tr(P) = C.

Lemma 3.8 Let A be of type II_{1} and T ∈ A be selfadjoint. Then there is a selfadjoint element C ∈ Z(A) and a projection F ∈ A satisfying:

1. F T ≤ F C;

2. (1 − F)T ≥ (1 − F)C;

3. F ∼ (1 − F);

4. F commutes with T.

\underline{\underline{Proof}} : Let E_{λ} for λ ∈ ℝ be the spectral projection of T in A for the interval ] − ∞, λ]. Denote by P_{λ} the spectral projection of tr(E_{λ}) in Z(A) corresponding to the interval [1/2, ∞[. Then {P_{λ} | λ ∈ ℝ} is a spectral family in Z(A) and we can define a selfadjoint element C in Z(A) by

C = ∫_{ℝ} λdP_{λ}.

Since we have 1/2 · P_{μ} ≤ tr(E_{μ})P_{μ} and 1/2 · (1 − P_{λ}) ≥ tr(E_{λ})(1 − P_{λ}), we get for λ ≤ μ:

tr(E_{λ})(P_{μ} − P_{λ}) ≤ \frac{1}{2} · (P_{μ} − P_{λ}) ≤ tr(E_{μ})(P_{μ} − P_{λ}).

Let G_{λ} be the spectral projection for T − C in A corresponding to the interval ] − ∞, λ]. Denote by G_{<0} the spectral decomposition for T − C in A corresponding to the interval ] − ∞, 0[. Notice for the sequel that P_{λ} and C are central and E_{λ} commutes with G_{μ} for all λ, μ ∈ ℝ. Since we have γG_{γ} ≥ G_{γ}(T − C) and γ(1 − G_{γ}) ≤ (1 − G_{γ})(T − C), we obtain for λ, μ, γ ∈ ℝ with λ < μ:

γ(1 − E_{λ})(P_{μ} − P_{λ})G_{γ} ≥ (1 − E_{λ})(P_{μ} − P_{λ})G_{γ}(T − C)
γE_{μ}(P_{μ} − P_{λ})(1 − G_{γ}) ≤ E_{μ}(P_{μ} − P_{λ})(1 − G_{γ})(T − C)

As (1 − E_{λ})T ≥ λ(1 − E_{λ}), E_{μ}T ≤ μE_{μ}, P_{μ}C ≤ μP_{μ} and (1 − P_{λ})C ≥ λ(1 − P_{λ}) hold, we get for λ, μ, γ ∈ ℝ with λ < μ:

(1 − E_{λ})(P_{μ} − P_{λ})G_{γ}(T − C) ≥ (λ − μ)(1 − E_{λ})(P_{μ} − P_{λ})G_{γ};
E_{μ}(P_{μ} − P_{λ})(1 − G_{γ})(T − C) ≤ (μ − λ)E_{μ}(P_{μ} − P_{λ})(1 − G_{γ}).

This implies for λ, μ, γ ∈ ℝ with λ < μ:

γ(1 − E_{λ})(P_{μ} − P_{λ})G_{γ} ≥ (λ − μ)(1 − E_{λ})(P_{μ} − P_{λ})G_{γ};
γE_{μ}(P_{μ} − P_{λ})(1 − G_{γ}) ≤ (μ − λ)E_{μ}(P_{μ} − P_{λ})(1 − G_{γ}).

We conclude:

(P_{μ} − P_{λ})G_{γ} ≤ E_{λ}(P_{μ} − P_{λ})G_{γ} , if γ < λ − μ ≤ 0;

(P_{μ} − P_{λ})(1 − G_{γ}) ≤ (1 − E_{μ})(P_{μ} − P_{λ})(1 − G_{γ}) , if γ > μ − λ ≥ 0.

Hence we obtain:

(P_{μ} − P_{λ})G_{γ} ≤ E_{λ}(P_{μ} − P_{λ}) , if γ < λ − μ ≤ 0;

(P_{μ} − P_{λ})(1 − G_{γ}) ≤ (1 − E_{μ})(P_{μ} − P_{λ}) , if γ > μ − λ ≥ 0.

Combining these inequalities with the first inequality appearing in this proof yields:

tr(G_{γ})(P_{μ} − P_{λ}) ≤ \frac{1}{2} · (P_{μ} − P_{λ}) , if γ < λ − μ ≤ 0;

tr(1 − G_{γ})(P_{μ} − P_{λ}) ≤ \frac{1}{2} · (P_{μ} − P_{λ}) , if γ > μ − λ ≥ 0.

This implies:

tr(G_{γ}) ≤ \frac{1}{2} · 1 for γ < 0;
tr(G_{γ}) ≥ \frac{1}{2} · 1 for γ > 0.

Hence we get:

tr(G_{<0}) ≤ \frac{1}{2} · 1 ≤ tr(G_{0}).

By lemma 3.7 there is a projection F ∈ A satisfying:

tr(F) = \frac{1}{2} · 1 and G_{<0} ≤ F ≤ G_{0}.

By construction we have

F ∼ 1 − F;
F (T − C) ≤ 0;
(1 − F)(T − C) ≥ 0.

As F = G_{<0} + (F − G_{<0}), (T − C)(F − G_{<0}) = (F − G_{<0})(T − C) = 0 and G_{<0} commutes with T − C, F commutes with T − C and hence with T.

Proposition 3.9 Let A be a von Neumann algebra of type II_{1} and let T ∈ A. Suppose P is a projection such that P ∼ 1 − P and T = P T P + (1 − P).

1. Assume that T is positive, invertible and det_{F K}(T) = 1. Then there are sequences of invertible operators T_{n}, A_{n} and B_{n} for n ∈ ℕ such that the assumptions of proposition 3.1 are satisfied. In particular, T is the product of two commutators of invertible elements in A and its class [T] in K_{1}(A) is trivial.

2. Assume that T is unitary. Then there are sequences of unitary operators T_{n}, A_{n} and B_{n} for n ∈ ℕ such that the assumptions of proposition 3.1 are satisfied. In particular, T is the product of two commutators in unitary elements in A and its class [T] in K_{1}(A) is trivial.

3. Assume that T is positive, 0 ≤ T ≤ 1 and T induces an injection A ⟶ A. Then there are sequences of operators T_{n}, A_{n} and B_{n} for n ∈ ℕ, inducing injections A ⟶ A such that the assumptions of proposition 3.1 are satisfied. In particular, the class [T] in K_{1}^{w}(A) is trivial.

\underline{\underline{Proof}} : We first prove assertion 1.) Choose a sequence of projections P_{n} as in lemma 3.6. Choose α ≥ 1 such that α^{−1} ≤∥T^{−1} ∥^{−1}≤∥T ∥≤ α holds. Now we will define sequences of invertible operators

T_{n}, A_{n}, B_{n} ∈ A;

K_{n}, L_{n} ∈ Z(P_{n}AP_{n})

with the following properties:

(a) T_{n} = P_{n}T_{n}P_{n} + (1 − P_{n});

(b) A_{n} = (P_{n+1} + P_{n})A_{n}(P_{n+1} + P_{n}) + (1 − P_{n} − P_{n+1}); B_{n} = (P_{n+1} + P_{n})B_{n}(P_{n+1} + P_{n}) + (1 − P_{n} − P_{n+1});

(c) B_{n}A_{n}T_{n} = A_{n}B_{n}T_{n+1};

(d) ∥T_{n} ∥, ∥T_{n}^{−1} ∥≤ α^{2}; ∥A_{n} ∥, ∥A_{n}^{−1} ∥≤ α^{2}; ∥B_{n} ∥, ∥B_{n}^{−1} ∥≤ α

(e) L_{n} = α^{2} · K_{n};

(f) K_{n} ≤ P_{n}T_{n}P_{n} ≤ L_{n};

(g) det_{F K}(T_{n}) = 1.

Notice that if we have completed this construction, then assertion 1.) will follow from proposition 3.1.

We construct the operators T_{n}, A_{n−1}, B_{n−1}, K_{n} and L_{n} inductively. Put:

T_{1} = T;

K_{1} = α^{−1} · P_{1};

L_{1} = α · P_{1}.

Next we prove the induction step from n to n + 1. Apply lemma 3.8 to P_{n}T_{n}P_{n} ∈ P_{n}AP_{n} to obtain a projection F_{n} ∈ P_{n}AP_{n} and an invertible positive element C_{n} ∈ Z(P_{n}AP_{n}) satisfying:

F_{n}T_{n} ≤ F_{n}C_{n};

(P_{n} − F_{n})T_{n} ≥ (P_{n} − F_{n})C_{n}.

F_{n} ∼ P_{n} − F_{n} ∼ P_{n+1};

F_{n} commutes with T_{n};

Find partial isometries V_{n}, W_{n} in P_{n}AP_{n} satisfying:

F_{n} = V_{n}^{∗}V_{n};

P_{n} − F_{n} = W_{n}^{∗}W_{n};

P_{n+1} = V_{n}V_{n}^{∗} = W_{n}W_{n}^{∗}.

We define:

R_{n} = V_{n}T_{n}V_{n}^{∗};

S_{n} = W_{n}T_{n}W_{n}^{∗}.

Then we get:

V_{n}K_{n}V_{n}^{∗} ≤ R_{n} ≤ V_{n}C_{n}V_{n}^{∗};

W_{n}C_{n}W_{n}^{∗} ≤ S_{n} ≤ W_{n}L_{n}W_{n}^{∗}.

Since V_{n}^{∗}W_{n} ∈ P_{n}AP_{n}, C_{n}, K_{n}, and L_{n} commute with V_{n}^{∗}W_{n} and also with W_{n}^{∗}V_{n}. This implies:

V_{n}C_{n}V_{n}^{∗} = W_{n}C_{n}W_{n}^{∗} ∈ Z(P_{n+1}AP_{n+1});

V_{n}K_{n}V_{n}^{∗} = W_{n}K_{n}W_{n}^{∗} ∈ Z(P_{n+1}AP_{n+1});

V_{n}L_{n}V_{n}^{∗} = W_{n}L_{n}W_{n}^{∗} ∈ Z(P_{n+1}AP_{n+1}).

We conclude:

V_{n}K_{n}V_{n}^{∗} ≤ R_{n} ≤ V_{n}C_{n}V_{n}^{∗} = W_{n}C_{n}W_{n}^{∗} ≤ S_{n} ≤ W_{n}L_{n}W_{n}^{∗}.

Now we define

T_{n+1} = R_{n}^{1/2}S_{n}R_{n}^{1/2} + (1 − P_{n+1});

K_{n+1} = V_{n}K_{n}C_{n}V_{n}^{∗};

L_{n+1} = V_{n}L_{n}C_{n}V_{n}^{∗}.

Then we get:

K_{n+1} ≤ Rn^{\frac{1}{2}}S_{n}Rn^{\frac{1}{2}} = P_{n+1}T_{n+1}P_{n+1} ≤ L_{n+1};

L_{n+1} = α^{2} · K_{n+1}.

The following 3 × 3 matrix calculation from [4]

( \begin{matrix}0 & 0 & 1 \\ s^{\frac{1}{2}} r^{\frac{1}{2}} & 0 & 0 \\ 0 & s^{\frac{1}{2}} r^{\frac{1}{2}} & 0 \end{matrix} ) ( \begin{matrix}0 & s^{−1} & 0 \\ 0 & 0 & 1 \\ s^{− \frac{1}{2}} r^{− \frac{1}{2}} & 0 & 0 \end{matrix} ) ( \begin{matrix}r & 0 & 0 \\ 0 & s & 0 \\ 0 & 0 & 1 \end{matrix} )

= ( \begin{matrix}0 & s^{−1} & 0 \\ 0 & 0 & 1 \\ s^{− \frac{1}{2}} r^{− \frac{1}{2}} & 0 & 0 \end{matrix} ) ( \begin{matrix}0 & 0 & 1 \\ s^{\frac{1}{2}} r^{\frac{1}{2}} & 0 & 0 \\ 0 & s^{\frac{1}{2}} r^{\frac{1}{2}} & 0 \end{matrix} ) ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & r^{\frac{1}{2}} sr^{\frac{1}{2}} \end{matrix} )

shows that B_{n}A_{n}T_{n} = A_{n}B_{n}T_{n+1} for some A_{n}, B_{n} ∈ A satisfying (b), and

∥A_{n}∥ ≤ max\{∥T_{n}^{−1}∥, ∥T_{n+1}^{−1}∥^{\frac{1}{2}} \}, ∥B_{n}∥ ≤ ∥T_{n+1}∥^{\frac{1}{2}} ,
∥A_{n}^{−1}∥ ≤ max\{∥T_{n}∥, ∥T_{n+1}∥^{\frac{1}{2}} \}, ∥B_{n}^{−1}∥ ≤ ∥T_{n+1}^{−1}∥^{\frac{1}{2}} .

In particular we get:

det_{FK}(T_{n}) = det_{FK}(T_{n+1}) = 1,

which, together with K_{n+1} + (1 − P_{n+1}) ≤ T_{n+1} ≤ L_{n+1} + (1 − P_{n+1}), implies

K_{n+1} ≤ P_{n+1} ≤ L_{n+1}.

Since L_{n+1} = α^{2} · K_{n+1}, we get:

α^{−2}P_{n+1} ≤ K_{n+1} ≤ P_{n+1}T_{n+1}P_{n+1} ≤ L_{n+1} ≤ α^{2}P_{n+1}.

As T_{n+1} = P_{n+1}T_{n+1}P_{n+1} + (1 − P_{n+1}) holds by definition, we conclude:

∥T_{n+1} ∥, ∥T_{n+1}^{−1} ∥ ≤ α^{2}.

This implies:

∥A_{n} ∥, ∥A_{n}^{−1} ∥ ≤ α^{2}, and ∥B_{n} ∥, ∥B_{n}^{−1} ∥≤ α.

This finishes the proof of assertion 1.) of proposition 3.9.

To prove 2.) and 3.) we construct sequences of operators P_{n}, T_{n}, A_{n} and B_{n} satisfying conditions (a), (b) and (c). The projections P_{n} are chosen as in the proof of 1.). Assume F_{n} ≤ P_{n} is a projection which commutes with T_{n} and satisfies F_{n} ∼ P_{n} − F_{n}. Consider the following 3 × 3 matrix identity:

( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ) ( \begin{matrix}0 & r & 0 \\ 0 & 0 & rs \\ 1 & 0 & 0 \end{matrix} ) ( \begin{matrix}r & 0 & 0 \\ 0 & s & 0 \\ 0 & 0 & 1 \end{matrix} )

= ( \begin{matrix}0 & r & 0 \\ 0 & 0 & rs \\ 1 & 0 & 0 \end{matrix} ) ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ) ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & rs \end{matrix} ) .

It follows that operators A_{n}, B_{n} and T_{n+1} can be found so that B_{n}A_{n}T_{n} = A_{n}B_{n}T_{n+1}, where

T_{n+1} = V_{n}T_{n}V_{n}^{∗}W_{n}T_{n}W_{n}^{∗} + (1 − P_{n+1}),

V_{n}^{∗}V_{n} = F_{n}, W_{n}^{∗}W_{n} = P_{n} − F_{n} and V_{n}V_{n}^{∗} = W_{n}W_{n}^{∗} = P_{n+1}. Note that A_{n}, B_{n} and T_{n+1} are unitaries if T_{n} is unitary. If T_{n} induces an injection and ∥T_{n}∥ ≤ 1, then A_{n}, B_{n} and T_{n+1} induce injections and ∥A_{n}∥, ∥B_{n}∥, ∥T_{n}∥ ≤ 1.

The projection F_{n} can be found using lemma 4.7 if T_{n} is unitary, and this completes the proof of 2.). In the case 3.) use 4.7 repeatedly to find projections F (λ) ≤ P = P_{1} for each dyadic rational λ ∈ [0, 1], so F (λ) commutes with T, F (λ′) ≥ F (λ) if λ′ ≥ λ, and trF (λ) = λtr P. Set F_{1} = F (1/2) and choose V_{1} and W_{1} so that

V_{1}F (λ)V_{1}^{∗} = W_{1}(F (λ + 1/2) − F (1/2))W_{1}^{∗}, 0 ≤ λ ≤ 1/2.

Then F_{2} = V_{1}F (1/4)V_{1}^{∗} has the desired properties. This algorithm can be continued to obtain F_{n} for all n. .

Now we can finish the proof of theorem 3.3. It remains to show that the map

det_{FK} : K_{1}(A) ⟶ Z(A)_{inv}^{+}

is injective and K_{1}^{w}(A) = {0}.

Consider η in K_{1}(A) satisfying det_{F K}(η) = 1. Choose an invertible S ∈ M_{k}(A) satisfying η = [S]. Then S has polar decomposition S = UT, with U unitary and T invertible and positive. By Lemma 3.7 there are projections E and F, positive invertible elements T_{1} and T_{2} with Fuglede-Kadison determinant 1, invertibles A and B, and unitaries U_{1} and U_{2} satisfying:

T = T_{1}T_{2}ABA^{−1}B^{−1}, U = U_{1}U_{2}, E ∼ 1 − E, F ∼ 1 − F,
T_{1} = ET_{1}E + (1 − E), T_{2} = (1 − E)T_{2}(1 − E) + E
U_{1} = F U_{2}(1 − F), U_{2} = (1 − F)U_{2}(1 − F) + F.

Proposition 3.11 now implies that S is a product of nine commutators. In particular η = 0.

Assume S ∈ A is injective. Then S = UT where U is unitary and T is positive and injective. From above we have that U is a product of four commutators in A, so [U] = 0 in K_{1}^{w}(A). As in Lemma 3.5, T = T_{1}T_{2} where T_{1} and T_{2} are positive and injective and α_{j}T_{j} satisfy the conditions of proposition 3.9 3.) if α_{j} ∈ R is chosen so that 0 < α_{j} ≤ 1 and 0 ≤ α_{j}T_{j} ≤ 1. It follows that [α_{j}T_{j}] and [α_{j}^{−1}] vanish in K_{1}^{w}(A). Hence [S] = 0 in K_{1}^{w}(A) which completes the proof.

4. The properly infinite case

In this section we show that K_{1}(A) and K_{1}^{w}(A) are trivial for a properly infinite von Neumann algebra A. This follows for K_{1}(A) already from de la Harpe-Skandalis [7, Theorem 7.5].

Lemma 4.1 Let A be a von Neumann algebra and T ∈ A such that T is invertible, respectively right multiplication with T, induces an injection r_{T} : A ⟶ A. Suppose that there is a countable sequence of mutually orthogonal projections P_{1}, P_{2}, … satisfying:

P_{n} ∼ P_{n+1};
∑_{n=1}^{∞} P_{n} = 1;
T = P_{1}T P_{1} + (1 − P_{1}).

Then the class [T] of T in K_{1}(A), respectively, K_{1}^{w}(A) vanishes.

\underline{\underline{Proof}} : Choose partial isometries V_{n} ∈ A such that P_{n} = V_{n}^{∗}V_{n} and P_{n+1} = V_{n}V_{n}^{∗}. Put:

T_{1} = T;
T_{n+1} = V_{n}T_{n}V_{n}^{∗} + (1 − P_{n+1});
A_{n} = T_{n+1};
B_{n} = V_{n} + V_{n}^{∗} + (1 − P_{n} − P_{n+1}).

Then B_{n} is invertible and:

B_{n}A_{n}T_{n} = A_{n}B_{n}T_{n+1}
∥A_{n} ∥∥T_{n} ∥ = ∥T ∥
∥B_{n} ∥ = ∥B_{n}^{−1} ∥ = 1

If T is invertible, then A_{n} is invertible and

∥A_{n}^{−1} ∥ = ∥T^{−1} ∥

If r_{T} : A ⟶ A is injective, then r_{T_{n}} and r_{A_{n}} are injective. Now we derive from proposition 3.1 that the class [T] in K_{1}(A), respectively K_{1}^{w}(A), vanishes. This finishes the proof of lemma 4.1.

Theorem 4.2 Let A be a properly infinite von Neumann algebra. Then K_{1}(A) and K_{1}^{w}(A) are trivial.

\underline{\underline{Proof}} : Consider η in K_{1}(A), respectively K_{1}^{w}(A). In view of lemma 3.5 we can assume that η is represented by T ∈ A, such that T is invertible or right multiplication with T induces an injection r_{T} : A ⟶ A, and that there is a projection P satisfying:

T = P T P + (1 − P);
P ∼ 1 − P.

Put P_{1} = P. Since 1 − P is properly infinite there is a sequence of projections P_{2}, P_{3}, ⋯ in A satisfying

P_{2}, P_{3}, … satisfying:
T = P_{1}T P_{1} + (1 − P_{1});
P_{n} ∼ P_{n+1} for n ≥ 1;
∑_{n=1}^{∞} P_{n} = 1.

Now the claim follows from lemma 4.1.

Notice that we have shown that any invertible T ∈ M_{k}(A) can be written as the product of four commutators of invertible elements in M_{k}(A), if A is properly infinite.

5. Detecting elements in W h(G)

Let G be a countable (discrete) group. The Whitehead group W h(G) is the quotient of K_{1}(ℤG) by the subgroup of trivial units {±g | g ∈ G}. Denote by W h′(G) the quotient of W h(G) by its torsion subgroup. We want to detect elements in W h′(G) using the Fuglede-Kadison determinant.

Recall that Z(R) denotes the center and R_{inv} denotes multiplicative group of invertible elements if R is a ring. We equip ℂG and ℤG with the involution of rings sending ∑_{g∈G} λ_{g}g to ∑_{g∈G} \overline{λ_{g}}g^{−1}. It induces involutions on Z(ℤG), Z(ℂG), K_{1}(ℂG), W h(G) and W h′(G). This involution corresponds to taking adjoints on operator level. Let Z(ℤG)^{ℤ/2} be the fixed point set under this involution and Z(ℤG)_{+} ⊂ Z(ℤG)^{ℤ/2} be the positive elements, i.e, elements of the shape aa^{∗} for a ∈ Z(ℤG). Consider a normal subgroup H of G. Then G acts on H by conjugation and this action induces G-actions on W h′(H). The fixed point set is denoted by W h′(H)^{G}. The main result of this section is

Theorem 5.1 For a finite normal subgroup H ⊂ G the map

i_{3} : W h′(H)^{G} ⟶ W h′(G)

induced by induction is injective.

A homomorphism f : A ⟶ B of abelian groups is rationally injective, respectively, bijective, i.e., f ⊗_{ℤ} id_{ℚ} : A ⊗_{ℤ} ℚ ⟶ B ⊗_{ℤ} ℚ is injective, respectively, bijective if and only if the kernel, respectively, both the kernel and the cokernel are torsion. For the proof of theorem 5.1 we need the following lemma:

Lemma 5.2 Let A and B be ℤG-modules. Let f : A ⟶ B be a ℤG-homomorphism. If f is rationally injective, respectively, bijective, the same holds for the induced map f^{G} : A^{G} ⟶ B^{G}

\underline{\underline{Proof}} : For any ℤG -module M there is a natural map

T (M) : hom_{ℤG}(M, A) ⊗_{ℤ} ℚ ⟶ hom_{ℚG}(M ⊗_{ℤ} ℚ, A ⊗_{ℤ} ℚ)

assigning to f ⊗_{ℤ} r the ℚG-map M ⊗_{ℤ} ℚ ⟶ A ⊗_{ℤ} ℚ sending m ⊗_{ℤ} s to f(m) ⊗_{ℤ} rs. Obviously T (⊕_{i∈I}M_{i}) = ⊕_{i∈I}T (M_{i})) holds and T (ℤG) is bijective. Hence T (M) is bijective for any projective ℤG-module M. Let P_{∗} be a projective ℤG-resolution for the trivial ℤGmodule ℤ. Since ℚ is flat as an ℤ-module, P_{∗} ⊗_{ℤ} ℚ is a projective ℚG-resolution for the trivial ℚG-module ℚ = ℤ ⊗_{ℤ} ℚ. We obtain a chain isomorphism

T (P_{∗}) : hom_{ℤG}(P_{∗}, A) ⊗_{ℤ} ℚ ⟶ hom_{ℚG}(P_{∗} ⊗_{ℤ} ℚ, A ⊗_{ℤ} ℚ)

Since ℚ is flat as ℤ-module, the natural map

H^{p}(hom_{ℤG}(P_{∗}, A)) ⊗_{ℤ} ℚ ⟶ H^{p}(hom_{ℤG}(P_{∗}, A) ⊗_{ℤ} ℚ)

is an isomorphism. We obtain a natural isomorphism

H^{p}(hom_{ℤG}(P_{∗}, A)) ⊗_{ℤ} ℚ ⟶ H^{p}(hom_{ℚG}(P_{∗} ⊗_{ℤ} ℚ, A ⊗_{ℤ} ℚ))

There are natural identifications

H^{0}(hom_{ℤG}(P_{∗}, A) = hom_{ℤG}(ℤ, A) = A^{G}

and

H^{0}(hom_{ℚG}(P_{∗} ⊗_{ℤ} ℚ, A ⊗_{ℤ} ℚ)) = hom_{ℚG}(ℚ, A ⊗_{ℤ} ℚ) = (A ⊗_{ℤ} ℚ)^{G}

Hence the natural map

A^{G} ⊗_{ℤ} ℚ ⟶ (A ⊗_{ℤ} ℚ)^{G}

is an isomorphism. Therefore f^{G} ⊗_{ℤ} id_{ℚ} is injective, respectively, bijective, if and only if (f ⊗_{ℤ} id_{ℚ})^{G} is. This finishes the proof of lemma 5.2.

Now we give the proof of theorem 5.1. Consider the following commutative diagram:

Z(ℂH)_{inv}^{G×ℤ/2}  \underrightarrow{s} Z(N (G))_{inv}^{+}
[figure: k^{G×ℤ/2}; det_{F K}]

K_{1}(ℂH)^{G×ℤ/2}  \underrightarrow{i_{1}}  K_{1}(ℂG)
[figure: j_{1}^{G×ℤ/2}; j_{2}]

K_{1}(ℤH)^{G×ℤ/2}  \underrightarrow{i_{2}}  K_{1}(ℤG)
[figure: p_{1}^{G×ℤ/2}; p_{2}]

W h′(H)^{G}  \underrightarrow{i_{3}}  W h′(G)

The G × ℤ/2-action comes from the G-action and the involution described above which are compatible with one another. Notice that the Fuglede-Kadison determinant det_{F K} sends the class of an element in K_{1}(ℂG) represented by an element a ∈ Z(ℂG)_{inv} to | a |= (aa^{∗})^{1/2} ∈ Z(N (G))_{inv}^{+}. Hence the map s is the composition of the injection

Z(N (G))_{inv}^{+} ⟶ Z(N (G))_{inv}^{+} a ↦ \sqrt{a},

the inclusion

Z(ℂG)_{inv}^{+} ⟶ Z(N (G))_{inv}^{+},

the inclusion

(Z(ℂH)_{inv}^{G})^{+} ⟶ Z(ℂG)_{inv}^{+}

and the rational isomorphism

Z(ℂH)_{inv}^{G×ℤ/2} ⟶ (Z(ℂH)_{inv}^{G})^{+}

sending a to aa^{∗} = a^{2}. The map k : Z(ℂH)_{inv} ⟶ K_{1}(ℂH) is the canonical map. The maps j_{1} and j_{2} are change of rings homomorphisms and the maps p_{1} and p_{2} are the natural projections. Next we show:

Lemma 5.3 1. The map s is a rationally injective;

2. The map k is rationally bijective;

3. The map j_{1}^{G×ℤ/2} is rationally injective;

4. The map p_{1}^{G×ℤ/2} is rationally bijective;

5. det_{F K} ∘j_{2} maps the kernel of p_{2} to the torsion subgroup of Z(N (G))_{inv}^{+}.

\underline{\underline{Proof}} : 1.) follows from the description of s as a composition of maps which are rationally injective.

2.) Since H is finite ℂH is a von Neumann algebra of type I_{f}. Composing k with the isomorphism det_{norm} of theorem 2.1 yields an epimorphism with finite kernel. Now apply lemma 5.2.

3.) Wall [16] has shown for finite H that the kernel SK_{1}(ℤH) of the change of rings map

K_{1}(ℤH) ⟶ K_{1}(ℚH)

is finite and maps under the canonical projection K_{1}(ℤH) ⟶ W h(H) bijectively onto the torsion subgroup of W h(H) (see also Oliver [14] page 5 and page 180). The change of rings map

K_{1}(ℚH) ⟶ K_{1}(ℂH)

is injective (see Oliver [14] page 5 and page 43). Hence

j_{1} : K_{1}(ℤH) ⟶ K_{1}(ℂH)

is rationally injective. Now apply lemma 5.2

4.) The map p_{1} is surjective and its kernel is a torsion subgroup, since the subgroup of trivial units {±h | h ∈ H} in K_{1}(ℤH) is finite. We derive from lemma 5.2 that p_{1}^{G×ℤ/2} is rationally bijective. Since the involution on W h′(H) is trivial (see Wall [16], Oliver [14] page 182) we have W h′(H)^{G} = W h′(H)^{G×ℤ/2} and the claim follows.

5.) Obviously det_{F K} ∘j_{2} maps a trivial unit to 1 since a trivial unit represents a unitary operator. The kernel of the projection p′_{2} : K_{1}(ℤG) ⟶ W h(G) is the subgroup of trivial units and the kernel of the projection p′′_{2} : W h(G) ⟶ W h′(G) is torsion. Since p_{2} is the composition p′′_{2} ∘ p′_{2} the claim follows.

Now we can finish the proof of theorem 5.1. Since W h′(H) is torsion-free, it suffices to show that the map

i_{3} : W h′(H)^{G} ⟶ W h′(G)

is rationally injective. This follows from lemma 5.3 and the diagram above.

Corollary 5.4 Let G be a countable group. If W h(G) is trivial, then any finite subgroup of the center of G is isomorphic to a product of finitely many copies of ℤ/2 and ℤ/3 or of ℤ/2 and ℤ/4.

\underline{\underline{Proof}} : Suppose W h(G) is trivial. Consider a finite subgroup H of the center of G. We derive from theorem 5.1 that W h′(H)^{G} is trivial. Since the G-action on W h′(H) is trivial, W h′(H) must vanish. The rank of the free abelian group W h′(H) is the difference r − q, where r is the number of ℝ-conjugacy classes and q the number of ℚ-conjugacy classes in G (see Oliver [14] page 49). Two elements g and h in the abelian group H are ℝ-conjugated if and only if g = h or g = h^{−1} holds and they are ℚ-conjugated if they generate the same cyclic subgroup. Hence W h′(H) is trivial if and only if r = q holds. A finite abelian group H satisfies r = q if and only any cyclic subgroup satisfies this condition. A non-trivial cyclic group ℤ/n satisfies r = q if and only if n = 2, 3, 4, 6. Hence H is a finite product of copies of ℤ/2, ℤ/3 or of ℤ/2 and ℤ/4.

References

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Current addresses

Wolfgang Lück Mikael Rørdam Fachbereich Mathematik Matematisk Institut Johannes Gutenberg-Universität Odense Universitet Saarstraße 21 Campusvej 55 55099 Mainz 5230 Odense Bundesrepublik Deutschland Danmark lueck@topologie.mathematik.uni-mainz.de mikael@imada.ou.dk