Hilbert modules and modules over finite von Neumann algebras and applications to L^{2}-invariants

Wolfgang Lück

October 21, 1996

Abstract

Let A be a finite von Neumann algebra. We show that the category of finitely generated Hilbert A-modules is equivalent to the category of finitely generated projective A-modules and that the category of finitely presented A-modules is an abelian category. This enables us to translate functional analytic invariants such as L^{2}-Betti numbers and Novikov-Shubin invariants to an algebraic setting of finitely presented A-modules. This allows to apply standard homological algebra to questions about L^{2}- invariants. We will define refined L^{2}-invariants using the center-valued trace of A. We discuss applications. For instance we show for an extension 1 ⟶ Δ ⟶ Γ ⟶ π ⟶ 1 of groups such that Δ is finitely generated and infinite, Γ is finitely presented and π contains ℤ as subgroup that the deficiency of Γ is bounded by 1. We show that all L^{2}-Betti numbers of Thompson’s group F are trivial. This is a necessary condition for a positive answer to the still open question whether F is amenable. AMS-classification number: 57R99, 46L99 key words: Hilbert modules over finite von Neumann algebras, L^{2}-Betti numbers, Novikov-Shubin invariants

Introduction

Throughout this paper A is a finite von Neumann algebra and tr : A ⟶ ℂ is a finite normal faithful trace. Recall that a von Neumann algebra is finite if and only if it possesses such a trace. Let l^{2}(A) be the Hilbert space completion of A which is viewed as a pre-Hilbert space by the inner product ⟨a, b⟩ = tr(ab^{∗}). A finitely generated Hilbert A-module V is a Hilbert space V together with a left operation of A by ℂ-linear maps such that λ · 1_{A} acts by scalar multiplication with λ on V for λ ∈ ℂ and there exists a unitary A-embedding of V in l^{2}(A)^{n} = ⊕_{i=1}^{n}l^{2}(A). In the sequel A operates always from the left unless explicitly stated differently. A morphism of finitely generated Hilbert A-modules is a bounded A-operator. Let {fin. gen. Hilb. A-mod.} be the category of finitely generated Hilbert A-modules. This category plays an important role in the construction of L^{2}-invariants of finite connected CW -complexes such as L^{2}-Betti numbers and Novikov-Shubin invariants. For a survey on L^{2}-(co)homology we refer for instance to [18], [32], [44]. More information about L^{2}- invariants can be found for instance in [1], [5], [8], [9] [13], [14], [17], [19], [24], [25] [26], [29], [30], [31], [34], [40], [43]. These constructions of L^{2}-invariants use the rich functional analytic structure. However, it is a consequence of standard facts about von Neumann algebras that they can be interpreted purely algebraically. Namely, we will prove (see Theorem 2.2) if {fin. gen. proj. A-mod.} denotes the category of finitely generated projective A-modules

Theorem 0.1 There is an equivalence of categories compatible with the complex vector space structures on the set of morphisms and the direct sums

ν^{−1} : {fin. gen. Hilb. A-mod.} ⟶ {fin. gen. proj. A-mod.}.

This is quite convenient since A viewed as a ring has the following nice properties (see Corollary 2.4 and Theorem 1.2). For a A-module M define

TM := {x ∈ M | f(x) = 0 for all f ∈ hom_{R}(M, R)};
PM := M/TM.

Theorem 0.2 1. A finite von Neumann algebra A is semi-hereditary, i.e. any finitely generated submodule of a projective A-module is projective;

2. The category {fin. pres. A-mod.} of finitely presented A-modules is abelian, i.e. the kernel, the image and the cokernel of a A-linear map of finitely presented A-modules is again finitely presented;

3. A A-module is finitely presented if and only if it has a 1-dimensional finitely generated projective A-resolution;

4. Let M be a finitely presented A-module. Then PTM = TPM = 0 and M is finitely generated projective if and only if TM = 0. In particular PM is finitely generated projective and the exact sequence 0 ⟶ TM ⟶ M ⟶ PM ⟶ 0 splits.

Hence the general strategy is to read off L^{2}-invariants of a finitely generated Hilbert A-chain complex C from the homology of the finitely generated projective A-chain complex ν^{−1}(C) associated to C by Theorem 0.1. Namely, one can define for a finitely presented A-module M invariants

dim(M) ∈ ℝ^{≥0};
α(M) ∈ [0, ∞] ⨿ {∞^{+}};

and show (see Theorem 5.4)

Theorem 0.3 Let C be a finitely generated Hilbert A-chain complex. Then we get for the L^{2}-Betti numbers b_{p}^{(2)}(C) and the Novikov-Shubin invariants α_{p}(C):

b_{p}^{(2)}(C) = dim(PH_{p}(ν^{−1}(C)));
α_{p}(C) = α(TH_{p−1}(ν^{−1}(C))).

We will actually consider more refined invariants than the L^{2}-Betti numbers and Novikov-Shubin invariants by substituting the ℂ-valued trace tr by the center-valued trace tr^{u}. We obtain in section 3 for each finitely presented A-module M and finitely generated Hilbert A-chain complex C invariants

dim^{u}(M) ∈ Z(A);
b_{p}^{u}(C) ∈ Z(A);
ω^{u}(M) ∈ D(A);
ω^{u}(C) ∈ D(A).

where the definition of D(A) will be given in Definition 3.8. Theorem 0.3 extends to these refined invariants and the refined invariants determine the others. We use the refined invariants to show (see Corollary 3.2, Lemma 3.3 and Example 4.3)

Theorem 0.4 1. Two finitely generated projective A-modules P and Q are A-isomorphic if and only if dim^{u}(P) = dim^{u}(Q);

2. If A is of type II_{1}, for instance if A is the von Neumann algebra of a finitely generated group which does not contain ℤ^{n} as subgroup of finite index, then any element in Z(A)^{+} can be realized as dim^{u}(P) for a finitely generated projective A-module;

3. Let S ⊂ ℂ[ℤ] be the multiplicative subset of elements in ℂ[ℤ] which are invertible in L^{∞}(S^{1}). Let M and N be finitely generated S^{−1}ℂ[ℤ]-modules. Then M and N are isomorphic as S^{−1}ℂ[ℤ]-modules if and only if the finitely presented L^{∞}(S^{1})-modules M ⊗_{S^{−1}ℂ[ℤ]} L^{∞}(S^{1}) and N ⊗_{S^{−1}ℂ[ℤ]} L^{∞}(S^{1}) are isomorphic as L^{∞}(S^{1})-modules. The L^{∞}(S^{1})-isomorphism type of M ⊗_{S^{−1}ℂ[ℤ]} L^{∞}(S^{1}) resp. of TM ⊗_{S^{−1}ℂ[ℤ]} L^{∞}(S^{1}) is determined by its elementary ideals resp. by ω^{u}.

Let Γ be a discrete group and X be a finite Γ-CW -complex with finite isotropy groups. Examples for X are universal coverings of compact CW -complexes with Γ as fundamental group and smooth manifolds with smooth cocompact proper Γ-action. We will define L^{2}- Betti numbers b_{p}^{(2)}(X) = b_{p}^{(2)}(X; l^{2}(Γ)) and Novikov-Shubin invariants α_{p}(X) = α(X; l^{2}(Γ)) in section 6 using the cellular L^{2}-chain complex. In section 6 we will briefly discuss singular homology, universal coefficient spectral sequences, the Leray-Serre spectral sequence, Poincaré duality, Morse inequalities, L^{2}-torsion, the “Zero in the spectrum” conjecture and will show (see Theorem 6.3 and Example 4.3)

Theorem 0.5 1. Suppose that Γ is the free abelian group ℤ^{r} of rank r. Let ℂℤ_{(0)}^{r} be the quotient field of ℂℤ^{r}. Then

b_{p}^{(2)}(X; l^{2}(ℤ^{r})) = dim_{ℂℤ_{(0)}^{r}} (H_{p}(X; ℂ) ⊗_{ℂℤ^{r}} ℂℤ_{(0)}^{r});

2. Suppose that Γ is ℤ. Choose t ∈ ℂ[ℤ] such that the principal ideal generated by t is the annihilator of the torsion submodule of H_{p}(X; ℂ). Consider t as a polynomial in z and z^{−1}. If the torsion submodule is trivial or t has no roots on S^{1}, then α_{p}(X; l^{2}(ℤ)) = ∞^{+}. Otherwise we get for the the highest order a of all roots of t on S^{1}

α_{p}(X; l^{2}(ℤ)) = \frac{1}{a}.

John Lott has given in the case Γ = ℤ^{r} an expression of the Novikov-Shubin invariants in terms of Massey products [24, pages 495 - 496].

In section 7 we will generalize results of [30, Theorem 3.1]. We will prove in section 7

Theorem 0.6 Let Γ_{1} ⊂ Γ_{2} ⊂ … ⊂ Γ_{n+1} = Γ be a nested sequence of subgroups of Γ for an integer n ≥ 0. Suppose that Γ_{1} is infinite, Γ_{i} is normal in Γ_{i+1}, the quotient Γ_{i+1}/Γ_{i} contains ℤ as subgroup for 1 ≤ i ≤ n and BΓ_{i} has finite i-skeleton for 1 ≤ i ≤ n + 1. Then:

b_{p}^{(2)}(Γ) := b_{p}^{(2)}(EΓ; l^{2}(Γ)) = 0 for p ≤ n.

We mention the following consequence which follows from Theorem 0.6 using the same arguments as in the proofs of [30, Theorem 4.1, Theorem 5.1 and Corollary 6.2]. Recall that the deficiency def(Γ) of a finitely presented group Γ is the maximum over all differences g − r where g resp. r is the number of generators resp. relations in any presentation of Γ. It is known that this maximum always exists.

Theorem 0.7 Let 1 ⟶ Δ ⟶ Γ ⟶ π ⟶ 1 be an exact sequence of groups such that Δ is finitely generated and infinite, Γ is finitely presented and ℤ is a subgroup of π. Then:

1. b_{1}^{(2)}(Γ) = 0;

2. def(Γ) ≤ 1;

3. Let M be a connected closed orientable 4-manifold with Γ as fundamental group. Then we get for its signature sign(M) and Euler characteristic χ(M)

| sign(M)| ≤ χ(M).

We mention that in the version of this corollary in [30] the stronger assumption was needed that Δ is finitely presented. If M is a closed orientable 4-dimensional Einstein manifold, then the sharper inequality

\frac{3}{2} · | sign(M)| ≤ χ(M)

holds [20].

Finally we mention the following observation about Thompson’s group F. It is the group of orientation preserving dyadic PL-automorphisms of [0, 1] where dyadic means that all slopes are integral powers of 2 and the break points are contained in ℤ[1/2]. It has the presentation

F = ⟨x_{0}, x_{1}, x_{2}, … | x_{i}^{−1}x_{n}x_{i} = x_{n+1} for i < n⟩.

This group has some very interesting properties. It is not elementary amenable and does not contain a subgroup which is free on two generators [3], [7]. Hence it is a very interesting question whether F is amenable. Since BF is of finite type [4], the L^{2}-Betti numbers b_{p}^{(2)}(F) are defined for all p ≥ 0. We conclude from [9, Lemma 3.1 on page 203] and H^{i}(EF; ℂ) = 0 for i ≥ 1 that a necessary condition for F to be amenable is that b_{p}^{(2)}(F) vanishes for all p ≥ 0. This motivates the following result whose proof is given at the end of section 7:

Theorem 0.8 All L^{2}-Betti numbers b_{p}^{(2)}(F) of Thompson’s group F vanish.

The paper of Farber [15] is related to this paper as follows. Farber constructs a category E(A) which contains the category {fin. gen. Hilb. A-mod.} as subcategory. The main point is that it is an abelian category, it is a kind of abelian extension of {fin. gen. Hilb. A-mod.} in the sense of [16]. An object in E(A) is a map of finitely generated Hilbert A-modules (α : A′ ⟶ A). A morphism in E(A) from (α : A′ ⟶ A) to (β : B′ ⟶ B) is an equivalence class of maps f : A ⟶ B such that there exists a map g : A′ ⟶ B′ with f ∘ α = β ∘ g. Here f and f′ are called equivalent if and only if f − f′ = β ∘ h for some morphism h : A ⟶ B′. The embedding of {fin. gen. Hilb. A-mod.} into E(A) is used in [15] to prove an analogue of Theorem 0.3 and to give improved L^{2}-Morse inequalities. The two approaches are unified by the following result which will be proved at the end of section 2.

Theorem 0.9 There is an equivalence of abelian categories

\overline{ν^{−1}} : E(A) ⟶ \{fin. pres. A-mod.\}

which induces the equivalence appearing in Theorem 0.1

ν^{−1} : {fin. gen. Hilb. A-mod.} ⟶ {fin. gen. proj. A-mod.}.

The paper is organized as follows :

0. Introduction
1. Semi-hereditary rings
2. Finitely generated Hilbert A-modules and finitely generated projective A-modules
3. Isomorphism invariants of finitely presented A-modules
4. Abelian von Neumann algebras
5. L^{2}-Betti numbers and Novikov-Shubin invariants for chain complexes
6. L^{2}-Betti numbers and Novikov-Shubin invariants for spaces
7. L^{2}-Betti numbers, fibrations and deficiency of groups

References

1. Semi-hereditary rings

In this section we explain some elementary properties of semi-hereditary rings and of finitely presented modules over them.

Ring will always mean associative ring with unit and R-module will mean left R-module unless explicitly stated differently. An involution on R is a map ∗ : R ⟶ R r ↦ r^{∗} which satisfies (r + s)^{∗} = r^{∗} + s^{∗}, (rs)^{∗} = s^{∗}r^{∗} and (r^{∗})^{∗} = r and 1^{∗} = 1 for all r, s ∈ R. Everything in this section does also make sense without the involution if one is careful with left and right modules. We call R semi-hereditary if each finitely generated ideal is projective.

This property implies that each finitely generated submodule of a projective R-module is projective [6, Proposition I.6.2. on page 15]. Recall that the dual M^{∗} of a (left) R-module is the (left) R-module hom_{R}(M, R) where the R-multiplication is given by (rf)(x) = f(x)r^{∗} for f ∈ M^{∗}, x ∈ M and r ∈ R.

Definition 1.1 Let M be a R-submodule of N. Define the closure of M in N to be the R-submodule of N

\overline{M} = \{x ∈ N | f(x) = 0 for all f ∈ N^{∗} with M ⊂ ker(f)\}.

For a R-module M define the R-submodule TM and the R-quotient module PM by:

TM := {x ∈ M | f(x) = 0 for all f ∈ M^{∗}};
PM := M/TM.

Notice that TM is the closure of the trivial submodule in M. It can also be described as the kernel of the canonical map i(M) : M ⟶ (M^{∗})^{∗} which sends x ∈ M to the map M^{∗} ⟶ R f ↦ f(x)^{∗}. Notice that TPM = 0 and that PM = 0 is equivalent to M^{∗} = 0.

Let R be a commutative ring without zero-divisors and M be a finitely generated Rmodule. Then the torsion submodule of M coincides with TM since any torsion-free finitely generated R-module can be embedded into R^{n} for appropriate n [2, Prop. 3.3 in Chap. 9 on page 321]. If M is not finitely generated, this is not true in general as the example R = ℤ and M = ℚ shows because ℚ as abelian group is torsionfree and satisfies Tℚ = ℚ. If we suppose additionally that R is semi-hereditary then PM is projective, provided M is finitely generated. If R is commutative but has zero-divisors, then in general the torsion submodule of a finitely generated R-module M does not agree with TM. Our main example of a semihereditary ring will be any finite von Neumann algebra A (see Corollary 2.4). Notice that in general A has zero-divisors, is not noetherian and has finitely generated modules M which are not finitely-presented. An example is A = L^{∞}(S^{1}) and M = L^{∞}(S^{1})/ ⋃_{n≥1}(χ_{n}) where (χ_{n}) is the ideal generated by the characteristic function of the subset {exp(2πit) | t ∈ [0, 1 − 1/n]} of S^{1}. Namely, ⋃_{n≥1}(χ_{n}) cannot be finitely generated as (χ_{n}) ≠ (χ_{n+1}) holds for all n and hence M is not finitely presented because the kernel of an epimorphism of a finitely generated module onto a finitely presented module is always finitely generated.

Theorem 1.2 Let R be a semi-hereditary ring with involution. Then:

1. The following statements are equivalent for a R-module M:

(a) M is finitely presented;

(b) M has a 1-dimensional finitely generated projective R-resolution;

(c) M is of type FP, i.e. possesses a finite-dimensional finitely generated projective R-resolution;

(d) M is of type FP_{∞}, i.e. possesses a finitely generated projective R-resolution; 2. If f : M ⟶ N is a R-map of finitely presented R-modules, then its kernel, image and

cokernel are again finitely presented;

3. If M is a finitely presented R-module, then PM is a finitely generated projective Rmodule. If M is a finitely generated projective R-module, then PM = M.

\underline{\underline{Proof}} : 1.) is obvious.

2 .) If 0 ⟶ M_{1} ⟶ M_{2} ⟶ M_{3} ⟶ 0 is an exact sequence of R-modules and two of the three are of type FP, then all three are of type FP [28, Theorem 11.2.c on page 212]. Hence it suffices to show for a R-map f : M ⟶ N of finitely presented R-modules that its image is finitely presented. Let F_{1} ⟶^{g} F_{0} ⟶^{p} N ⟶ 0 be a finite presentation. Then F_{1} ⟶^{g} p^{−1}(im(f)) ⟶^{p} im(f) ⟶ 0 is exact. Since p^{−1}(im(f)) is a finitely generated submodule of a free R-module, it is projective. Let Q be a finitely generated projective R-module such that Q ⊕ p^{−1}(im(f)) = F′_{0} for a finitely generated free R-module F′_{0}. Then one easily constructs an exact sequence of R-modules F_{1} ⊕ F′_{0} ⟶ F′_{0} ⟶ im(f) ⟶ 0. Hence im(f) is finitely presented.

3.) Choose a finite presentation R^{m} ⟶^{f} R^{n} ⟶^{p} M ⟶ 0 of the R-module M. Then the sequence 0 ⟶ M^{∗} ⟶ (R^{n})^{∗} ⟶^{f^{∗}} (R^{m})^{∗} is exact. Since R is semi-hereditary the image of f^{∗} and hence M^{∗} are finitely generated projective. The projection pr : M ⟶ PM induces an isomorphism pr^{∗} : PM^{∗} ⟶ M^{∗}. Since TPM is trivial, the canonical map i(PM) : PM ⟶ (PM^{∗})^{∗} is an embedding of a finitely generated R-module into a finitely generated projective R-module. Hence PM is projective. This finishes the proof of Theorem 1.2.

Define for any chain complex C its dual R-cochain complex C^{∗} by the R-cochain complex whose n-th R-cochain module is C_{n}^{∗}. Given a R-module M, define M by Ext_{R}^{1}(M, R). If PM = 0 and M has a 1-dimensional projective resolution P then P^{∗} defines a 1-dimensional projective resolution for M and we get a canonical isomorphism M ⟶ (M ) .

Lemma 1.3 Let R be a semi-hereditary ring with involution. Let C be a finitely generated projective R-chain complex. Then there is a in C natural exact sequence

0 ⟶ (TH_{n−1}(C)) ⟶ H^{n}(C^{∗}) ⟶ (PH_{n}(C))^{∗} ⟶ 0.

The sequence splits (but not naturally). In particular we obtain in C natural R-isomorphisms (TH_{n−1}(C)) ⟶ TH^{n}(C^{∗}) and PH^{n}(C^{∗}) ⟶ (PH_{n}(C))^{∗}.

\underline{\underline{Proof}} : Notice that (TH_{n−1}(C)) is canonically isomorphic to Ext_{R}^{1}(H_{n−1}(C), R) and H_{n}(C)^{∗} is canonically isomorphic to (PH_{n}(C))^{∗}. Under these identifications the proof for instance in [45, Theorem 13.10 on page 240] for free abelian chain complexes goes through directly.

2. Finitely generated Hilbert A-modules and finitely generated projective A-modules

In this section we identify the categories of finitely generated Hilbert A-modules and of finitely generated projective A-modules and show that any finite von Neumann algebra is semi-hereditary.

For the rest of this paper let A be a finite von Neumann algebra and let tr : A ⟶ ℂ be some finite normal faithful trace. Recall that a von Neumann algebra is finite if and only if it possesses such a trace. Let l^{2}(A) be the Hilbert space completion of A which is viewed as a pre-Hilbert space by the inner product ⟨a, b⟩ = tr(ab^{∗}). A finitely generated Hilbert A-module V is a Hilbert space V together with a left operation of A by ℂ-linear maps such that λ · 1_{A} acts by scalar multiplication with λ on V for λ ∈ ℂ and there exists a unitary A-embedding of V in ⊕_{i=1}^{n}l^{2}(A) for some n. A morphism of finitely generated Hilbert A-modules is a bounded A-operator. Denote by {fin. gen. Hilb. A-mod.} the category of finitely generated Hilbert A-modules. For a survey on finite von Neumann algebras and Hilbert A-modules we refer for instance to [34, section 1].

A ℂ-category C is a category such that for each two objects the set of morphisms between them carries the structure of a complex vector space for which composition of morphisms is bilinear and that C has a (strict) sum which is compatible with the complex vector space structures above. A (strict) involution on a ℂ-category C is an assignment which associates to each morphism f : x ⟶ y a morphism f^{∗} : y ⟶ x and has the following properties

(f^{∗})^{∗} = f;
(λ · f + μ · g)^{∗} = \overline{λ} · f^{∗} + \overline{μ} · g^{∗};
(f ∘ g)^{∗} = g^{∗} ∘ f^{∗};
(f ⊕ g)^{∗} = f^{∗} ⊕ g^{∗};

where f,g are morphisms, λ and μ complex numbers. There is a canonical structure of a ℂ-category with involution on {fin. gen. Hilb. A-mod.} where the involution is given by taking adjoint operators. We call an endomorphism resp. isomorphism f in C selfadjoint resp. unitary if f = f^{∗} resp. f^{∗} = f^{−1} holds. A functor of ℂ-categories with involution is a functor compatible with the complex vector space structures on the morphisms, the sums and the involutions. A natural equivalence T of functors of ℂ-categories with involution is called unitary if the evaluation of T at each object is a unitary isomorphism. An equivalence of ℂ-categories with involution is a functor of such categories such that there is a functor of such categories in the other direction with the property that both compositions are unitarily naturally equivalent to the identity.

Given a finitely generated projective A-module P, an inner product on P is a map μ : P × P ⟶ A satisfying (cf [48, Definition 15.1.1 on page 232])

1. μ is A-linear in the first variable;

2. μ is symmetric in the sense μ(x, y) = μ(y, x)^{∗}; 3. μ is positive-definite in the sense that μ(p, p) is a positive element in A, i.e. of the

form a^{∗}a for some a ∈ A, and μ(p, p) = 0 ⟺ p = 0;

4. The induced map \overline{μ} : P ⟶ P^{∗} = hom_{A}(P, A) defined by \overline{μ}(y)(x) = μ(x, y) is bijective.

Notice that \overline{μ} is a A-isomorphism such that the composition P ⟶^{i(P)} (P^{∗})^{∗} ⟶^{\overline{μ}^{∗}} P^{∗} is \overline{μ}. Let {fin. gen. proj. A-mod. with ⟨ ⟩} be the ℂ-category whose objects are finitely generated projective A-modules with inner product (P, μ) and whose morphisms are A-linear maps. We get an involution on it if we specify f^{∗} : (P_{1}, μ_{1}) ⟶ (P_{0}, μ_{0}) for f : (P_{0}, μ_{0}) ⟶ (P_{1}, μ_{1}) by requiring μ_{1}(fx, y) = μ_{0}(x, f^{∗}(y)) for all x ∈ P_{0} and y ∈ P_{1}. In other words, we define f^{∗} := \overline{μ_{0}}^{−1} ∘ f^{∗} ∘ \overline{μ_{1}} where the second f^{∗} refers to the A-map f^{∗} = hom_{A}(f, id) : P_{1}^{∗} ⟶ P_{0}^{∗}. In the sequel we will use the symbol f^{∗} for both f^{∗} : P_{1} ⟶ P_{0} and f^{∗} : P_{1}^{∗} ⟶ P_{0}^{∗}.

Given a finitely generated projective A-module (P, μ) with inner product μ, we obtain a pre-Hilbert structure on P by tr ∘μ : P × P ⟶ ℂ. Let ν(P) be the associated Hilbert space. We will show later that this is a finitely generated Hilbert A-module and that any A-linear map P_{0} ⟶ P_{1} of finitely generated projective A-modules P_{i} with inner products μ_{i} extends to a morphism of Hilbert A-modules ν(f) : ν(P_{0}, μ_{0}) ⟶ ν(P_{1}, μ_{1}). Moreover, we will prove

Theorem 2.1 1. The functor

ν : {fin. gen. proj. A-mod. with ⟨ ⟩} ⟶ {fin. gen. Hilb. A-mod.}

is an equivalence of ℂ-categories with involutions;

2. Any finite generated projective A-module has an inner product. Two finitely generated projective A-modules with inner product are unitarily A-isomorphic if and only if the underlying A-modules are A-isomorphic.

\underline{\underline{Proof}} : Put A^{n} = ⊕_{i=1}^{n}A and l^{2}(A)^{n} = ⊕_{i=1}^{n}l^{2}(A). Let \{A^{n}\} resp. \{l^{2}(A)^{n}\} be the full subcategory of {fin. gen. proj. A-mod. with ⟨ ⟩} resp. {fin. gen. Hilb. A-mod.} whose objects are A^{n} resp. l^{2}(A)^{n} for n = 0, 1, 2 . . .. Here we equip A^{n} with the standard inner product

μ_{st}((a_{1}, a_{2}, …, a_{n}), (b_{1}, b_{2}, …, b_{n})) = ∑_{i=1}^{n} a_{i}b_{i}^{∗}.

Now ν as defined above yields a well-defined isomorphism of ℂ-categories with involutions

ν : {A^{n}} ⟶ {l^{2}(A)^{n}}

because the right regular representation A ⟶ B(l^{2}(A), l^{2}(A))^{A} from A into the space of bounded A-operators from l^{2}(A) to itself sending a ∈ A to the extension of the map A ⟶ A b ↦ ba^{∗} is known to be well-defined and bijective [12, Theorem 1 in I.5.2 on page 80, Theorem 2 in I.6.2 on page 99].

The idempotent completion Idem(C) of a ℂ-category C with involution has as objects (V, p) selfadjoint idempotents p : V ⟶ V. A morphism from (V_{0}, p_{0}) to (V_{1}, p_{1}) is a morphism f : V_{0} ⟶ V_{1} satisfying p_{1} ∘ f ∘ p_{0} = f. The identity on (V, p) is given by p : (V, p) ⟶ (V, p). The idempotent completion Idem(C) inherits from C the structure of a ℂ-category with involution in the obvious way. There are functors of ℂ-categories with involutions

IM : Idem({A^{n}}) ⟶ {fin. gen. proj. A-mod. with ⟨ ⟩};
IM : Idem({l^{2}(A)^{n}}) ⟶ {fin. gen. Hilb. A-mod.};

which send (A^{n}, p) resp. (l^{2}(A)^{n}, p) to the image of p where the inner product μ_{st} on im(p) is given by restricting the standard inner product μ_{st} on A^{n}.

Let P be a finitely generated projective A-module. It is isomorphic to im(q) for some idempotent q : A^{n} ⟶ A^{n}. Define p : A^{n} ⟶ A^{n} to be the idempotent for which ν(p) : l^{2}(A)^{n} ⟶ l^{2}(A)^{n} is the orthogonal projection onto the image of ν(q). Because of ν((id −q) ∘ p) = 0 we get (id −q) ∘ p = 0. Similiarly we get (id −p) ∘ q = 0. This implies that q and p have the same image. Hence P is isomorphic to im(p) for some selfadjoint idempotent p : A^{n} ⟶ A^{n}.

Next we prove the second assertion. Because of the argument above it suffices to check the claim for P = im(p) for a selfadjoint idempotent p : A^{n} ⟶ A^{n}. The standard inner product on A^{n} induces a standard inner product μ_{st} : P × P ⟶ A. It remains to show for a second inner product μ : P × P ⟶ A that there is a isometric A-isomorphism g : (P, μ) ⟶ (P, μ_{st}). Let f : P ⟶ P be the A-automorphism uniquely determined by the property that μ(x, y) = μ_{st}(f(x), y) holds for all x, y ∈ P. Let f^{∗} be the adjoint of f with respect to μ_{st}. Then the following calculation shows that f is positive with respect to μ_{st}:

μ_{st}(x, f(y)) = μ_{st}(f(y), x)^{∗} = μ(y, x)^{∗} = μ(x, y) = μ_{st}(f(x), y).
μ_{st}(f(x), x) = μ(x, x) ≥ 0.

Let g′ : A^{n} ⟶ A^{n} be defined by the property that ν(g′) : l^{2}(A)^{n} ⟶ l^{2}(A)^{n} is positive and ν(g′) ∘ ν(g′) = ν(i ∘ f ∘ p) where i : P ⟶ A^{n} is the inclusion which is the adjoint of p : A^{n} ⟶ P with respect to μ_{st}. Define g : P ⟶ P by p ∘ g′ ∘ i. Then g is selfadjoint with respect to μ_{st} and g^{2} = f. This implies

μ_{st}(g(x), g(y)) = μ_{st}(g^{2}(x), y) = μ_{st}(f(x), y) = μ(x, y).

This finishes the proof of the second assertion.

Next we can conclude for any A-linear map f : (P_{0}, μ_{0}) ⟶ (P_{1}, μ_{1}) of finitely generated projective A-modules that the Hilbert space completion ν(P_{i}, μ_{i}) is a finitely generated Hilbert A-module and f extends to a bounded A-operator ν(P_{0}, μ_{0}) ⟶ ν(P_{1}, μ_{1}). Because of assertion 2.) one can reduce the claim to the case (P_{i}, μ_{i}) = (A^{n_{i}}, μ_{st}) which we have already dealt with. Hence ν : {fin. gen. proj. A-mod. with ⟨ ⟩} ⟶ {fin. gen. Hilb. A-mod.} is well-defined. It is obviously a functor of ℂ-categories with involution. It remains to show that it is an equivalence of ℂ-categories with involutions.

The following diagram commutes

Idem({A^{n}})  ⟶^{Idem(ν)}  Idem({l^{2}(A)^{n}}
IM↓  ↓^{IM}
{fin. gen. proj. A-mod. with ⟨ ⟩} ⟶_{ν} {fin. gen. Hilb. A-mod.}

The upper horizontal arrow is induced by the isomorphism of ℂ-categories with involutions ν : {A^{n}} ⟶ {l^{2}(A)^{n}}. Hence it suffices to show that the vertical arrows are equivalences of ℂ-categories with involutions. This would follow if IM is full, i.e. induces an epimorphism on the set of unitary isomorphism classes of objects, and faithful, i.e. induces isomorphisms on the set of morphisms between to arbitrary objects onto the set of morphisms between the images of these objects (cf. [39, Theorem 1 on page 91]). Obviously IM is faithful in both cases. It is full in the second case of finitely generated Hilbert A-modules since by definition of a finitely generated A-module V there is a selfadjoint idempotent p : l^{2}(A)^{n} ⟶ l^{2}(A)^{n} whose image is unitarily A-isomorphic to V. It is full in the first case of finitely generated projective A-modules because of the second assertion. This finishes the proof of Theorem 2.1.

A sequence U ⟶^{f} V ⟶^{g} W of finitely generated Hilbert A-modules is weakly exact at V resp. exact at V if \overline{im(f)} = ker(g) resp. im(f) = ker(g) holds. The definition for finitely generated projective A-modules is analogous where the notion of closure of Definition 1.1 is used. In the sequel ν^{−1} is an inverse of ν which is well-defined up to unitary natural equivalence by Theorem 2.1. Theorem 2.1 implies

Theorem 2.2 The composition of ν^{−1} with the forgetful functor induces an equivalence of ℂ-categories

{fin. gen. Hilb. A-mod.} ⟶ {fin. gen. proj. A-mod.}.

Lemma 2.3 ν and ν^{−1} preserve weak exactness and exactness.

\underline{\underline{Proof}} : A sequence U ⟶^{f} V ⟶^{g} W of finitely generated Hilbert A-modules is weakly exact at V if and only if the following holds: g ∘ f = 0 and for any finitely generated Hilbert Amodules P and Q and morphisms u : V ⟶ P and v : Q ⟶ V with u ∘ f = 0 and g ∘ v = 0 we get u ∘ v = 0. It is exact at V if and only if the following holds: g ∘ f = 0 and for any finitely generated Hilbert A-module P and morphism v : P ⟶ V with g ∘ v = 0 there is a morphism u : P ⟶ U satisfying f ∘ u = v. The same is true if one considers finitely generated projective A-modules instead of finitely generated Hilbert A-modules. Now ν and ν^{−1} obviously preserve these criterions for weak exactness and exactness and the claim follows.

Corollary 2.4 A finite von Neumann algebra A is semi-hereditary.

\underline{\underline{Proof}} : Let M ⊂ A be a finitely generated ideal in A. Choose a A-map f : A^{n} ⟶ A whose image is M. It suffices to show that ker(f) is a direct summand. Let p : A^{n} ⟶ A^{n} be the A-map for which ν(p) is an idempotent with ker(ν(f)) as image. Because of Lemma 2.3 p is an idempotent with ker(f) as image.

Finally we give the promised proof of Theorem 0.9. We define

\overline{ν^{−1}} : E(A) ⟶ \{fin. pres. A-mod.\}

on objects by sending (α : A′ ⟶ A) to the cokernel of ν^{−1}(α). A morphism f in E(A) induces a A-map in the obvious way. Clearly an object of the shape 0 ⟶ A is sent to ν^{−1}(A). One easily checks using standard homological algebra that \overline{ν^{−1}} is full and faithful.

3. Isomorphism invariants of finitely presented A-modules

In this section we introduce some isomorphism invariants for finitely presented Amodules. We will completely classify finitely generated projective A-modules. This is a direct consequence of the following result which is taken from [21, Theorem 8.2.8 on page 517, Proposition 8.3.10 on page 525 and Theorem 8.4.3. on page 532].

Theorem 3.1 Let A be a finite von Neumann algebra on H. There is a map

tr^{u} = tr_{A}^{u} : A ⟶ Z(A)

into the center Z(A) of A called the center-valued trace or universal trace of A uniquely determined by the following properties:

1. tr^{u} is ℂ-linear;

2. If a ∈ A is positive, tr^{u}(a) is positive;

3. tr^{u}(ab) = tr^{u}(ba) for all a, b ∈ A;

4. tr^{u}(a) = a for all a ∈ Z(A).

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

5. If a ∈ A is positive and tr^{u}(a) = 0, then a = 0;

6. tr^{u} is continuous in the ultraweak topology;

7. || tr^{u}(a)|| ≤ ||a|| for a ∈ A;

8. tr^{u}(ab) = a tr^{u}(b) for all a ∈ Z(A) and b ∈ A; 9. Let p and q be projections in A. Then p ∼ q, i.e there is a partial isometry u ∈ A

satisfying p = uu^{∗} and q = u^{∗}u, if and only if tr^{u}(p) = tr^{u}(q);

10. Any bounded linear functional f : A ⟶ ℂ which is central, i.e. f(ab) = f(ba) for all a, b ∈ A, factories as

A ⟶^{tr^{u}} Z(A) ⟶^{f|_{Z(A)}} ℂ.

Define the center-valued von Neumann dimension of a finitely generated Hilbert Amodule V by

dim^{u}(V) = dim_{A}^{u}(V) := tr^{u}(p) ∈ Z(A)

where p : l^{2}(A)^{n} ⟶ l^{2}(A)^{n} is any A-projection whose image is isomorphic as finitely generated Hilbert A-module to V and tr^{u}(p) is the sum of the traces tr^{u}(A_{i,i}) of the diagonal entries of the (n, n)-matrix A with entries in A given by p. We define for a finitely presented A-module M

dim^{u}(M) := dim^{u}(ν(PM, μ)) ∈ Z(A)

for any inner product μ on PM. These two definitions are independent of the choices of p and μ by Theorem 2.1 and Theorem 3.1. We have dim^{u}(M) = dim^{u}(PM) by definition. Notice that the matrix algebra M_{k}(A) is again a finite von Neumann algebra. We will extend this notion to arbitrary modules over finite von Neumann algebras in [33]. We get from Theorem 3.1

Corollary 3.2 1. The following statements are equivalent for two finitely generated projective A-modules P and Q:

(a) P and Q are A-isomorphic;

(b) P and Q are stably A-isomorphic, i.e. P ⊕ V and Q ⊕ V are A-isomorphic for some finitely generated projective A-module V;

(c) dim^{u}(P) = dim^{u}(Q);

(d) [P] = [Q] in K_{0}(A);

2. The center-valued dimension induces an injection

dim^{u} : K_{0}(A) ⟶ Z(A)^{+} = {a ∈ Z(A) | a = bb^{∗} for some b ∈ A}.

If A is of type II, this map is an isomorphism (The image of dim^{u} is described in [21, Theorem 8.4.4 on page 533] in general).

Notice that for a finite group π and A the associated von Neumann algebra which is ℂπ in this case Corollary 3.2 reduces to the well-known fact that two finite-dimensional unitary π-representations are unitarily π-isomorphic if and only if they have the same character.

In this context we mention the computation of K_{1}(A) and K_{1}^{w}(A) (for any von Neumann algebra A) in [35] and the following lemma. Recall that a finitely generated group Γ is virtually abelian if and only if it contains ℤ^{r} as normal subgroup of finite index for some r ≥ 0. Let Γ_{f} be the normal subgroup of elements γ ∈ Γ for which the set (γ) of elements conjugated to γ is finite. The definition of type I and type II_{1} of a von Neumann algebra can be found in [21, Definition 6.5.1] and of type I_{f} means that the von Neumann algebra is finite and of type I.

Lemma 3.3 Let Γ be a finitely generated (discrete) group. Then:

1. The von Neumann algebra N (Γ) of Γ is a factor, i.e. its center is ℂ, if and only if Γ_{f} is trivial;

2. The von Neumann algebra N (Γ) is of type I_{f} if Γ is virtually abelian and of type II_{1} otherwise.

\underline{\underline{Proof}} : 1.) follows from [12, Proposition 4 in III.7.6 on page 319].

2.) For a subgroup H of Γ define its centralizer by

C_{H} = {γ ∈ Γ | γh = hγ for all h ∈ H} .

Given elements γ_{1},γ_{2}, … γ_{r} in Γ, we write ⟨γ_{1}, γ_{2}, … γ_{r}⟩ for the subgroup of Γ generated by these elements. We abbreviate C_{γ} = C_{⟨γ⟩} for γ ∈ Γ. Notice that the set (γ) of elements in Γ which are conjugated to γ is finite if and only if C_{γ} has finite index in Γ. Obviously C_{δ^{−1}γδ} = δ^{−1}C_{γ}δ. If H_{1} ⊂ H_{2} ⊂ Γ, then C_{H_{2}} ⊂ C_{H_{1}}. We have C_{⟨γ_{1},γ_{2},...γ_{r}⟩} = ∩_{i=1}^{r}C_{γ_{i}}.

Then Γ_{f} is the set of elements γ ∈ Γ for which C_{γ} has finite index. This is a normal subgroup of Γ. We get from [47] that N (Γ) is type I_{f} if and only if Γ has a (normal) abelian subgroup of finite index and from [41] that N (Γ) is type II_{1} if the index of Γ_{f} in Γ is infinite. See also [22] or [23]. Hence it remains to show for the finitely generated group Γ that the normal subgroup Γ_{f} has finite index if and only if Γ is virtually abelian.

Assume that Γ_{f} has finite index. A subgroup of finite index in a finitely generated group is again finitely generated. It suffices to prove this for a free group ∗_{r}ℤ of rank r. In this case, the claim follows from Schreier’s Theorem [37, Proposition 3.8 and 3.9 on page 16]. Choose a set of generators γ_{1}, γ_{2}, … γ_{r} for Γ_{f}. We obtain a normal subgroup of finite index Γ_{f} ∩ C_{Γ_{f}} = Γ_{f} ∩ (∩_{i=1}^{r}C_{γ_{i}}). By definition of the centralizer, this group is abelian. Hence Γ is virtually abelian.

Assume that Γ contains ℤ^{r} as a normal subgroup of finite index. Then for all γ ∈ ℤ^{r} the centralizer C_{γ} contains ℤ^{r} and hence has finite index in Γ. This shows ℤ^{r} ⊂ Γ_{f}. Hence Γ_{f} has finite index in Γ.

Recall from Theorem 1.2 and Corollary 2.4 that any finitely presented A-module M is isomorphic to TM ⊕ PM and that PM is finitely generated projective. In view of Corollary 3.2 we have a complete classification of finitely generated projective A-modules. It remains to investigate finitely presented A-modules M with PM = 0.

A map of finitely generated Hilbert A-modules f : V ⟶ W is called a weak isomorphism if and only if it is injective and has dense image. By dimension theory this is true if and only if f is injective and V and W are isomorphic (see [9, section 1]). Recall that a A-module M of type FP, i.e. M possesses a finite-dimensional finitely generated projective A-resolution, defines an element [M] ∈ K_{0}(A) by [M] = ∑_{n≥0}(−1)^{n}[P_{n}] for any choice of finite-dimensional finitely generated projective A-resolution P of M. The next lemma is a direct consequence of the results above, Lemma 2.3 and Corollary 3.2.

Lemma 3.4 The following assertions are equivalent for a finitely presented A-module M:

1. PM = 0;

2. dim^{u}(M) = 0;

3. [M] = 0 in K_{0}(A);

4. If P is a 1-dimensional finitely generated projective A-resolution, then P_{0} and P_{1} are A-isomorphic;

5. There is an exact sequence 0 ⟶ A^{n} ⟶^{c_{1}} A^{n} ⟶ M ⟶ 0 with c_{1}^{∗} = c_{1};

6. If 0 ⟶ P_{1} ⟶^{c_{1}} P_{0} ⟶ M ⟶ 0 is a 1-dimensional finitely generated projective Aresolution of M, then ν(c_{1}) is a weak isomorphism for each choice of inner products on P_{0} and P_{1}.

Now we can define an invariant for finitely presented A-modules M with PM = 0 under the assumption that A is of type I_{f}. Let

det_{norm} : M(k, k, A) ⟶ Z(A)

be the normalized determinant defined in [35, page 521]. If A is abelian, this is the ordinary determinant for commutative rings. Denote by Z(A)^{inv} the multiplicative group of units in the center of A. Denote by Z(A)^{w} the Grothendieck group of the multiplicative abelian semigroup of elements a ∈ Z(A) for which multiplication with a induces an injection Z(A) ⟶ Z(A). If we identify Z(A) with L^{∞}(X, μ) for some measure space (X, μ), we can identify Z(A)^{w} with the multiplicative group Inv(X, μ), whose elements are 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.

Definition 3.5 Suppose that A is of type I_{f}. Let M be a finitely presented A-module. For any exact sequence of A-modules 0 ⟶ A^{n} ⟶^{f} A^{n} ⟶ TM ⟶ 0 define

ρ^{u}(M) := det_{norm}(f) ∈ Z(A)^{w}/Z(A)^{inv}.

The existence of the exact sequence 0 ⟶ A^{n} ⟶^{f} A^{n} ⟶ TM ⟶ 0 follows from Lemma 3.4. Using [35, section 2] one can show that det_{norm}(f) takes value in Z(A)^{w}. By definition ρ^{u}(M) = ρ^{u}(TM) and ρ^{u}(M) is trivial if M is finitely generated projective. The independence of ρ^{u}(M) from the choice of f follows from the following lemma whose elementary proof is left to the reader.

Lemma 3.6 Let S be a ring and P_{1} ⟶^{f} P_{0} and and Q_{1} ⟶^{g} Q_{0} be 1-dimensional projective S-resolutions of the same S-module. Then there is a commutative square

P_{1} ⊕ Q_{1} ⟶^{f⊕id_{Q_{1}}} P_{0} ⊕ Q_{1}
u↓  ↓^{v}
Q_{1} ⊕ P_{1} ⟶_{g⊕id_{P_{1}}} Q_{0} ⊕ P_{1}

whose vertical arrows are isomorphisms.

Notice that the definition of ρ^{u}(M) was based on the existence of the normalized determinant and its basic properties. One could try to get another invariant in the same way using any other notion of determinant which has the same properties. However, this cannot give a finer invariant in the type I_{f}-case and leads always to a trivial invariant in the type II_{1}-case because of the results in [35, Theorem 2.1 on page 521 and Theorem 3.3 on page 525].

The next result follows from Theorem 1.2 and the standard properties of the normalized determinant and center-valued trace.

Lemma 3.7 Let 0 ⟶ M_{0} ⟶ M_{1} ⟶ M_{2} ⟶ 0 be an exact sequence of A-modules such that two of them are finitely presented. Then:

1. All three are finitely presented;

2. dim^{u}(M_{1}) = dim^{u}(M_{0}) + dim^{u}(M_{2}) ∈ Z(A);

3. PM_{1} = 0 ⟺ PM_{0} = PM_{2} = 0;

4. If A is of type I_{f} and PM_{1} = 0, then:

ρ^{u}(M_{1}) = ρ^{u}(M_{0}) · ρ^{u}(M_{2}) ∈ Z(A)^{w}/Z(A)^{inv}.

Next we construct invariants which are defined for all finite von Neumann algebras. Let f : U ⟶ V be a morphism of finitely generated Hilbert A-modules. Let {E_{λ}^{f^{∗}f} | λ ∈ ℝ} be the (right-continuous) family of spectral projections of the positive operator f^{∗}f. Define the center-valued spectral density function of f by

F_{f}^{u} : [0, ∞) ⟶ Z(A)^{+} λ ↦ dim^{u} (im(E_{λ^{2}}^{f^{∗}f})) = tr^{u}(E_{λ^{2}}^{f^{∗}f}).

Notice that F_{f}^{u} is a monotone increasing (right-continous) function. We call two monotone increasing functions G_{0}, G_{1} : [0, ∞) ⟶ Z(A)^{+} dilatationally equivalent if there are constants ϵ, C > 0 satisfying

G_{0}(C^{−1} · λ) ≤ G_{1}(λ) ≤ G_{0}(C · λ) for all λ ∈ [0, ϵ].

Definition 3.8 Denote by D(A) the abelian semi-group of dilatational equivalence classes [d] of monotone increasing functions d : [0, ∞) ⟶ Z(A)^{+} where the addition is given by [d] + [e] = [d + e]. For a morphism of finitely generated Hilbert A-modules f : U ⟶ V define

ω^{u}(f) := [F_{f}^{u}] ∈ D(A).

Lemma 3.9 1. If f : U ⟶ V is an isomorphism of finitely generated Hilbert A-modules,

then

ω^{u}(f) = 0;

2. ω^{u}(f ⊕ g) = ω^{u}(f) + ω^{u}(g);

3. If f,g and h are composable morphisms of finitely generated Hilbert A-modules and f and h are isomorphism, then:

ω^{u}(f ∘ g ∘ h) = ω^{u}(g).

\underline{\underline{Proof}} : 1.) and 2.) are obvious.

3.) The elementary proof in [26, Lemma 1.6 on page 21] for the complex-valued trace goes through word by word for the center-valued trace and shows

F_{g}^{u}(λ) ≤ F_{gh}^{u}(||h|| · λ);
F_{gh}^{u}(λ) ≤ F_{g}^{u}(||h^{−1}|| · λ);
F_{g}^{u}(λ) ≤ F_{fg}^{u}(||f|| · λ);
F_{fg}^{u}(λ) ≤ F_{g}^{u}(||f^{−1}|| · λ);

and the assertion follows.

Definition 3.10 Let M be a finitely presented A-module. For any choice of an exact sequence 0 ⟶ P_{1} ⟶^{f} P_{0} ⟶ M ⟶ 0 define

ω^{u}(M) := ω^{u}(ν(f)) ∈ D(A)

where ν(f) : ν(P, μ_{1}) ⟶ ν(P_{0}, μ_{0}) is the morphism of finitely generated Hilbert A-modules defined in section 2 after a choice of inner products μ_{i}.

We derive from Theorem 2.1, Lemma 3.6 and Lemma 3.9 that the definition above makes sense. We get for a finitely presented A-module M that i.) ω^{u}(M) = ω^{u}(TM), ii.) ω^{u}(M) is trivial if and only if M is finitely generated projective and iii.) M is trivial if and only if dim^{u}(M) and ω^{u}(M) are trivial.

Recall that we have specified a trace tr : A ⟶ ℂ. The definitions of von Neumann dimension and spectral density function of section 3 for the universal trace make also sense for the complex valued trace tr. This yields for a finitely generated Hilbert A-module V, a finitely presented A-module M and a morphism f : V ⟶ W of finitely generated Hilbert A-modules the von Neumann dimension (with respect to tr)

dim(V), dim(M) ∈ ℝ^{≥0};

the spectral density function (with respect to tr)

F_{f} : [0, ∞) ⟶ [0, ∞] λ ↦ dim (im(E_{λ^{2}}^{f^{∗}f})) = tr(E_{λ^{2}}^{f^{∗}f});

and the ω-invariants with respect to tr

ω(f), ω(M) ∈ D(ℂ).

We get from Theorem 3.1 that tr induces a map

tr : D(A) ⟶ D(ℂ) [d] ↦ [tr ∘d]

and dim(V), dim(M), ω(f) and ω(M) can be read off from dim^{u}(V), dim^{u}(M), ω^{u}(f) and ω^{u}(M) defined in section 3 by

dim(V) = tr(dim^{u}(V));
dim(M) = tr(dim^{u}(M));
ω(f) = tr(ω^{u}(f));
ω(M) = tr(ω^{u}(M)).

Of course one looses information by passing from the universal trace to the complex valued trace, but for most of the applications the invariants based on the standard complex valued trace for a von Neumann algebra of a group are sufficient mainly, because the vanishing of b_{p}^{(2)}(C) is equivalent to the vanishing of H_{p}^{(2)}(C). Moreover, one can assign in the complexvalued case an interesting real number to elements in D(ℂ). The Novikov-Shubin invariant of [d] ∈ D(ℂ) is

α([d]) = lim inf_{λ→0^{+}} \frac{ln(d(λ) − d(0))}{ln(λ)} ∈ [0, ∞],

provided that d(λ) > d(0) holds for all λ > 0. Otherwise, we put α([d]) = ∞^{+}. Here ∞^{+} is a new formal symbol which should not be confused with ∞.

Definition 3.11 Define the Novikov-Shubin invariant of a morphism of finitely generated Hilbert A-modules f : U ⟶ V by

α(f) := α(ω(f)) ∈ [0, ∞] ⨿ {∞^{+}}.

The Novikov-Shubin invariant α(M) of a finitely presented A-module M is defined by

α(M) := α(ω(M)) ∈ [0, ∞] ⨿ {∞^{+}}.

Let f : U ⟶ V be a morphism of finitely generated Hilbert A-modules. Since f and f^{∗}f have the same kernel we have dim(ker(f)) = F_{f}(0). We have α(f) = ∞^{+} if and only if f^{∗}f has a gap in the spectrum at zero, i.e. there exists ϵ > 0 such that F_{f}(λ) = F_{f}(0) for 0 < λ < ϵ. Moreover, f is an isomorphism if and only if f has trivial kernel, α(f) = ∞^{+} and dim^{u}(U) = dim^{u}(V) ∈ Z(A). A finitely presented A-module M is finitely generated projective if and only if α(M) = ∞^{+}. It is trivial if and only if dim(M) = 0 and α(M) = ∞^{+}.

Finally we prove the following two lemmas we will need later.

Lemma 3.12 Let C a finitely generated free A-chain complex. For a A-module M let μ(M) be the minimal numbers of generators. Then

dim(C_{p}) ≥ μ (H_{p}(C) ⊕ TH_{p−1}(C)) .

\underline{\underline{Proof}} : Choose a direct sum decomposition ker(c_{p}) ⊕ P_{1} = C_{p}. Let P_{0} ⊂ ker(c_{p−1}) be the preimage of TH_{p−1} under the canonical projection ker(c_{p−1}) ⟶ H_{n−1}(C). We get a 1-

dimensional finitely generated projective resolution 0 ⟶ P_{1} ⟶^{c_{p}|_{P_{1}}} P_{0} ⟶ TH_{p−1}(C) ⟶ 0. From Lemma 3.4 we conclude that P_{0} and P_{1} are A-isomorphic. Hence we can construct an epimorphism C_{p} ⟶ H_{p}(C) ⊕ TH_{p−1}(C).

Lemma 3.13 Let C be a A-chain complex and d be an integer such that C_{n} is finitely presented with PC_{n} = 0 for n ≤ d and C_{d+1} is finitely generated. Then H_{n}(C) is finitely presented with PH_{n}(C) = 0 for n ≤ d and H_{d+1}(C) is finitely generated.

\underline{\underline{Proof}} : From Lemma 3.7 we conclude that ker(c_{n}), im(c_{n}) and H_{n−1}(C) are finitely presented and P ker(c_{n}), P im(c_{n}) and PH_{n−1}(C) are trivial for n ≤ d. The obvious sequence 0 ⟶ im(c_{d+1}) ⟶ ker(c_{d}) ⟶ H_{d}(C) ⟶ 0 is exact, im(c_{d+1}) is finitely generated and ker(c_{d}) is finitely presented with P ker(c_{d}) = 0. Hence H_{d}(C) and im(c_{d+1}) are finitely presented and PH_{d}(C) = 0 by Lemma 3.7. As 0 ⟶ ker(c_{d+1}) ⟶ C_{d+1} ⟶^{c_{d+1}} im(c_{d+1}) ⟶ 0 is exact, C_{d+1} is finitely generated and im(c_{d+1}) is finitely presented, ker(c_{d+1}) is finitely generated. This finishes the proof of Lemma 3.13 .

4. Abelian von Neumann algebras

Next we consider the special case where A is abelian. Recall that any abelian von Neumann algebra A can be identified with L^{∞}(X, μ) for an appropriate measure space (X, μ) [12, Theorem 1 in I.7.3 on page 132]. We recall the classical notion of the k-th elementary ideal I_{k}(M) of a finitely presented S-module for any commutative ring S [11, Chapter VII]. Let S^{m} ⟶^{f} S^{n} ⟶ M be a finite presentation and A the (m, n)-matrix describing f by f(x) = xA for x ∈ S^{m}. A (l, l)-minor of A is the (ordinary) determinant (of commutative rings) of a (l, l)-submatrix of A. Define

I_{k}(M) := \{ \begin{matrix}0 & , if n − k > m or k < 0 \\ S & , if n − k ≤ 0 \\ ideal generated by all (n − k, n − k)-minors & , if 0 < n − k ≤ m. \end{matrix}

The following result was stated without proof in [35, Lemma 2.2. on page 522]. For the readers convenience we include a proof here.

Lemma 4.1 Let A = L^{∞}(X, μ) be an abelian von Neumann algebra and t : A^{n} ⟶ A^{n} be a normal morphism, i.e. t and t^{∗} commute. Then there exists a unitary isomorphism u : A^{n} ⟶ A^{n} such that u^{∗} ∘ t ∘ u is diagonal.

\underline{\underline{Proof}} : We first construct a measurable map

λ : M_{n}(ℂ) ⟶ ℂ

with the property that λ(A) is an eigenvalue of A for all A ∈ M_{n}(ℂ). Set-theoretically, the map is defined as follows. Let λ(A) be the eigenvalue of A with the following property. Let λ(A) = r · e^{2πit} for r ≥ 0 and t ∈ [0, 1[ be the polar decomposition of λ(A). If λ′ is any other eigenvalue of A with polar decomposition λ′ = r′ · e^{2πt′}, then either r′ < r, or r′ = r and t ≤ t′ holds. In other words, the norm of λ(A) is maximal and among those eigenvalues with the same norm as λ(A), the angle between the real axis and λ(A) with respect to the origin in the anticlockwise direction is minimal. The map λ is of course not continuous but, as we will see, measurable. Define the following sets for 1 ≤ k ≤ n:

S := {A ∈ M_{n}(ℂ) | λ(A) ∈ ℝ};
S_{k}^{c} := {A ∈ S^{c} | there are k eigenvalues of A satisfying |λ′| = |λ(A)|};

where we count the eigenvalues with multiplicity and S^{c} is the complement of S in M_{n}(ℂ). The eigenvalues of A are the roots of its characteristic polynomial and the map sending A to its characteristic polynomial is continuous. The roots of a polynomial depend continuously on the coefficients in the following sense. Given a polynomial p of degree n and ϵ > 0, there is δ > 0 such that for any polynomial q of degree n with the property that the difference of the i-th coefficients of p and q have norm less than or equal to δ for all i, there are numerations of the roots λ_{i}(p) of p and of the roots λ_{i}(q) of q, satisfying |λ_{i}(p) − λ_{i}(q)| ≤ ϵ for all i. This implies that λ is continuous on each set S_{k}^{c} and that the disjoint union ∐_{k=1}^{l} S_{k}^{c} is an open set for all 1 ≤ l ≤ n. The set S is closed. Hence each of the sets S_{k}^{c} and the set S are measurable and the restrictions of λ to these sets is measurable. Hence the map λ is measurable.

We leave it to the reader to verify that the function M_{n}(ℂ) ⟶ M_{n}(ℂ), sending A to the orthogonal projection pr_{im(A)} onto the image of A, is measurable.

Next, we construct a measurable map v_{1} : M_{n}(ℂ) ⟶ ℂ^{n} such that ∥v_{1}(A)∥ = 1 and Av_{1}(A) = λ(A)v_{1}(A) holds for all A in M_{n}(ℂ). Namely, define

v_{1}(A) = \frac{pr_{ker(A−λ(A)I)}(e_{i})}{∥ pr_{ker(A−λ(A)I)}(e_{i})∥}

if pr_{ker(A−λ(A)I)}(e_{i−1}) = 0 and pr_{ker(A−λ(A)I)}(e_{i}) ≠ 0, where e_{1}, e_{2} , … e_{n} are the standard basis vectors. Given v_{1}, one easily constructs measurable maps v_{2}, …, v_{n} from M_{n}(ℂ) to ℂ^{n} such that {v_{1}(A), v_{2}(A), …, v_{n}(A)} is an orthonormal basis for all A. Let U(A) be the unitary map having this orthonormal basis as columns. Provided that A is normal, the matrix U(A)^{−1}AU(A) is a diagonal block matrix.

U(A)^{−1}AU(A) = ( \begin{matrix}λ(A) & 0 \\ 0 & B(A) \end{matrix} )

and the (n − 1, n − 1)-matrix B(A) is normal again. Iterating this process, we can even assume that U(A)^{−1}AU(A) is a diagonal matrix. Since the composition of measurable functions is again measurable, we get for normal T ∈ M_{n}(L^{∞}(X, μ)) the desired unitary element U ∈ M_{n}(L^{∞}(X, μ)) by putting U(x) = U(T (x)). This finishes the proof of Lemma 4.1.

Example 4.2 Let A = L^{∞}(X, μ) be an abelian von Neumann algebra and let M be any finitely presented A-module M. Because of the polar decomposition and Lemma 4.1 there are elements t_{1}, t_{2}, …, t_{l} in A such that

M ≅ ⊕_{i=1}^{l}A/(t_{i}).

Let Zero(t_{i}) be {x ∈ X | t_{i}(x) = 0} and let χ_{i} be the characteristic function of Zero(t_{i}). Then:

PM = ⊕_{i=1}^{l}(χ_{i});
TM = ⊕_{i=1}^{l}A/(χ_{i} + t_{i});
dim^{u}(M) = ∑_{i=1}^{l} χ_{i};

ρ^{u}(M) = ∏_{i=1}^{l}(χ_{i} + t_{i});

I_{k}(M) = (∏_{i∈I} t_{i} | I ⊂ {1, 2, …, n}, |I| = l − k) for 0 ≤ k ≤ l.

In particular A/(t) and A/(s) are A-isomorphic if and only if Zero(t) = Zero(s) (up to sets of measure zero) and t/s and s/t are essentially bounded outside Zero(t) = Zero(s).

Example 4.3 Next we consider the von Neumann algebra N (ℤ) = L^{∞}(S^{1}) with its standard trace tr : L^{∞}(S^{1}) ⟶ ℂ given by integration and want to classify L^{∞}(S^{1})-modules of the shape

⊕_{i=1}^{n}L^{∞}(S^{1})/(p_{i})

where p_{i}(z) is a polynomial in z. Since ℂ[ℤ] is a principal ideal domain [2, Proposition V.5.8 on page 151 and Corollary V.8.7 on page 162] any such L^{∞}(S^{1})-module is of the shape

N ⊗_{ℂ[ℤ]} L^{∞}(S^{1})

for some finitely generated ℂ[ℤ]-module N. Let S ⊂ ℂ[ℤ] be the multiplicative subset of elements in ℂ[ℤ] which become invertible in L^{∞}(S^{1}). This is the set of elements which can be written as finite products of non-zero complex numbers and elements of the form z − a with |a| ≠ 1. The localization S^{−1}ℂ[ℤ] is again a principal ideal domain [2, Corollary V.8.7 on page 162] and we have

N ⊗_{ℂ[ℤ]} L^{∞}(S^{1}) ≅ S^{−1}N ⊗_{S^{−1}ℂ[ℤ]} L^{∞}(S^{1}).

Hence it suffices to investigate for a finitely generated S^{−1}ℂ[ℤ]-module M the L^{∞}(S^{1})-module

M^{(2)} := M ⊗_{S^{−1}ℂ[ℤ]} L^{∞}(S^{1}).

From [2, Theorem X.5.7 on page 370] we get non-negative integers n, l and elements t_{1}, t_{2}, …, t_{l} in S^{−1}ℂ[ℤ] such that no t_{i} is zero or a unit, t_{i} divides t_{i+1} and

M = S^{−1}ℂ[ℤ]^{n} ⊕ ⊕_{i=1}^{l}S^{−1}ℂ[ℤ]/(t_{i})

and n, l and the ideals (t_{i}) are uniquely determined by this property. We can also write M in the form

M = S^{−1}ℂ[ℤ]^{n} ⊕ ⊕_{i=1}^{k}S^{−1}ℂ[ℤ]/((z − a_{i})^{r_{i}})

where n, k, r_{i} are integers with n, k ≥ 0 and r_{i} ≥ 1 and a_{i} ∈ S^{1}. For a = exp(iψ) ∈ S^{1} and r ∈ ℤ, r ≥ 1 define

f_{a,r} : [0, ∞) ⟶ L^{∞}(S^{1})^{+} λ ↦ χ_{{cos(ϕ)+i sin(ϕ) | λ≥|ϕ−ψ|^{r}}}.

We claim

dim^{u}(M) = n · 1_{L^{∞}(S^{1})};
ρ^{u}(M) = ∏_{i=1}^{l} t_{i};
I_{k}(M) = 0 for k < n
I_{k}(M) = ( ∏_{i=1}^{n+l−k} t_{i}) for n ≤ k;

ω^{u}(M) = ∑_{i=1}^{k}[f_{a_{i},r_{i}}];

α(M) = \{ \begin{matrix}min\{\frac{1}{r_{i}} | i = 1, … k\} & , if k > 0 \\ ∞^{+} & , if k = 0. \end{matrix}

The first four equations follow directly from Example 4.2. We get from Lemma 3.9

ω^{u}(M) = ∑_{i=1}^{k} ω^{u}(L^{∞}(S^{1})/((z − a_{i})^{r_{i}})).

Hence it suffices to check for the fourth equation the special case for b ∈ S^{1} and r ∈ ℤ, r ≥ 1

ω^{u}(L^{∞}(S^{1})/((z − b)^{r})) = [f_{b,r}].

Since the group of isometries on S^{1} acts transitively on S^{1}, it suffices to treat the case b = 1. The spectral density function of the morphism L^{∞}(S^{1}) ⟶ L^{∞}(S^{1}) given by multiplication with (z − 1)^{r} assigns to λ ∈ [0, ∞) the characteristic function of the set

{z ∈ S^{1} | λ ≥ |z − 1|^{r}} = {cos(ϕ) + i sin(ϕ) | λ ≥ |2 − 2 cos(ϕ)|^{r/2}}

This proves the fifth equation. Since

lim_{ϕ→0} \frac{2 − 2 cos(ϕ)}{ϕ^{2}} = 1

the sixth equation follows.

The last equation follows from the fourth using [26, Lemma 1.10 on page 23]

α([d] + [e]) = min{α([d]), α([e])}  for [d], [e] ∈ D(ℂ);
tr([f_{a,r}]) = [λ^{1/r}]  ∈ D(ℂ);
α([λ^{1/r}]) = \frac{1}{r}.

We mention the consequence of these equations that the elementary ideals I_{k}(M^{(2)}) for k ≥ 0 determine the isomorphism type of both M and M^{(2)} and that ω^{u}(M^{(2)}) determines the isomorphism type of both Tors(M) and TM^{(2)} .

Namely, the computations above show that the elementary ideals I_{k}(M^{(2)}) determine both n and ∏_{i=1}^{k} t_{i} up to multiplication with units in L^{∞}(S^{1}). Hence n and ∏_{i=1}^{k} t_{i} for all k are up to multiplication with units in S^{−1}ℂ[ℤ] uniquely determined by the isomorphism type of M^{(2)}. This implies that the isomorphism type of M^{(2)} determines n and the S^{−1}ℂ[ℤ]-ideals (t_{i}) and hence the S^{−1}ℂ[ℤ]-isomorphism type of M itself.

In order to show that ω^{u}(M^{(2)}) determines the isomorphism type of Tors(M), we must show the following. Given non-negative integers n and q, pairwise disjoint elements a_{i} ∈ S^{1} for i = 1, 2, …, q and non-negative integers m_{i,j} for i = 1, 2, …, q and j ∈ ℤ, j ≥ 1 for which only finitely many are different from zero, the class

∑_{i=1}^{q} ∑_{j≥1} m_{i,j} · [f_{a_{i},j}] ∈ D(A)

determines q, a_{i} and m_{i,j} up to permutation of the indices i. We have already introduced an (additive) semi-abelian group structure on D(A) by [d + e] = [d] + [e]. Analogously multiplication defines a (multiplicative) structure of an abelian semi-group by

[d] · [e] := [d · e].

Let χ_{i_{0},ϵ} be the characteristic function for the set {cos(ϕ) + i sin(ϕ) | ϵ ≥ |ϕ − ψ|} if we write a_{i_{0}} = cos(ψ) + i sin(ψ). Denote the corresponding constant function [0, ∞) ⟶ L^{∞}(S^{1}) in the same way. Then one easily checks for small enough ϵ:

(∑_{i=1}^{q} ∑_{j≥1} m_{i,j} · [f_{a_{i},j}]) · [χ_{i_{0},ϵ}] = ∑_{j≥1} m_{i_{0},j} · [f_{a_{i_{0}},j}].

Since the group of isometries on S^{1} acts transitively on S^{1}, it suffices to show for a sequence of non-negative integers m_{j} for j = 1, 2, … for which only finite many are different from zero that

∑_{j≥1} m_{j} · [f_{1,j}] ∈ D(A)

determines the integers m_{j}. It suffices to show for all positive integers j_{0} that the sums ∑_{j≥j_{0}} m_{j} are determined. For [d], [e] ∈ D(A) we write

[d] ≤ [e]

if and only if there are representatives d and e and constants C, ϵ > 0 with the property that d(λ) ≤ e(C · λ) holds for all λ ∈ [0, ϵ). It suffices to prove

∑_{j≥j_{0}} m_{j} = max { k ∈ ℤ | k ≥ 0 and k · [f_{1,j_{0}}] ≤ ∑_{j≥1} m_{j} · [f_{1,j}]} .

Consider k ∈ ℤ with k ≥ 0, 0 < ϵ < 1 and C > 1 such that for λ ∈ [0, ϵ]

k · f_{1,j_{0}}(λ) ≤ ∑_{j≥1} m_{j} · f_{1,j}(C · λ)

holds. Fix λ ∈ [0, ϵ] satisfying λ^{(j_{0})^{−1}}/2 ≥ (C · λ)^{j^{−1}} for all j < j_{0}. Let χ be the characteristic function of the set {cos(ϕ) + i sin(ϕ) | ϕ ∈ [λ^{j_{0}}/2, λ^{j_{0}}]}. Multiplying the inequality of elements in L^{∞}(S^{1})^{+} above with χ yields

k · χ ≤ ∑_{j≥j_{0}} m_{j} · χ.

This implies k ≤ ∑_{j≥j_{0}} m_{j}. Since [f_{1,j_{0}}] ≤ [f_{1,j}] for j ≥ j_{0}, the claim follows.

5. L^{2}-Betti numbers and Novikov-Shubin invariants for chain complexes

In this section we introduce L^{2}-invariants such as L^{2}-homology, L^{2}-Betti numbers and Novikov-Shubin invariants for finitely generated Hilbert A-chain complexes and express them in terms of the homology of finitely generated projective A-chain complexes which is associated to it by Theorem 2.1. In the sequel ν^{−1} is an inverse of ν which is well-defined up to unitary equivalence as described in Theorem 2.1.

Definition 5.1 If C is a chain complex of finitely generated Hilbert A-chain modules, define its p-th L^{(2)}-homology to be the finitely generated Hilbert A-module

H_{p}^{(2)}(C) = ker(c_{p})/\overline{im(c_{p+1})},

its center-valued p-th L^{2}-Betti number by

b_{p}^{u}(C) := dim^{u}(H_{p}^{(2)}(C)) ∈ Z(A),

its p-th L^{2}-Betti number by

b_{p}^{(2)}(C) := dim(H_{p}^{(2)}(C)) ∈ ℝ^{≥0},

its p-th ω-invariant by

ω_{p}^{u}(C) := ω(c_{p} : C_{p} ⟶ C_{p−1}) ∈ D(A),

and its p-th Novikov-Shubin invariant by

α_{p}(C) := α(c_{p} : C_{p} ⟶ C_{p−1}) ∈ [0, ∞] ⨿ {∞^{+}}.

Lemma 5.2 Let C be a finitely generated Hilbert A-chain complex. Then there is a in C natural isomorphism

h(C) : ν^{−1}(H_{p}^{(2)}(C)) ⟶ PH_{p}(ν^{−1}(C)).

\underline{\underline{Proof}} : We define h(C) by the following commutative diagram whose columns are exact and whose middle and lower vertical arrows are isomorphisms by Lemma 2.3

0  0
↑  ↑
ν^{−1}(H_{p}^{(2)}(C)) ⟶^{h(C)} PH_{p}(ν^{−1}(C))
ν^{−1}(q)↑  ^{r}↑
ν^{−1}(ker(c_{p})) ⟶^{ν^{−1}(i)} ker(ν^{−1}(c_{p}))
ν^{−1}(j)↑  ^{l}↑
ν^{−1}(\overline{im(c_{p+1})}) ⟶^{ν^{−1}(k)} \overline{im(ν^{−1}(c_{p+1}))}
↑  ↑
0  0

where i, j ,k and l are the obvious inclusions and q and r the obvious projections. Then h(C) is an isomorphism by the five-lemma.

Lemma 5.3 If f : C ⟶ D is a chain homotopy equivalence of finitely generated Hilbert A-chain complexes, then we get for all p

ω_{p}^{u}(C) = ω_{p}^{u}(D).

\underline{\underline{Proof}} : We will need the following fact for an exact sequence of finitely generated Hilbert A-

chain complexes 0 ⟶ C ⟶ D ⟶ E^{p}⟶ 0: If E is contractible then there is a chain map s : E ⟶ D with p ∘ s = id_{E}. Namely, choose for any n ≥ 0 a morphism σ_{n} : E_{n} ⟶ D_{n} with p_{n} ∘ σ_{n} = id_{E_{n}}. If γ is a chain contraction for E, define

s_{n} = d_{n+1} ∘ σ_{n+1} ∘ γ_{n} + σ_{n} ∘ γ_{n−1} ∘ e_{n}.

If cone(f) resp. cyl(f) is the mapping cone resp. mapping cylinder of f : C ⟶ D (see [28, page 213]), there are canonical exact sequences

0 ⟶ C ⟶ cyl(f) ⟶ cone(f) ⟶ 0

and

0 ⟶ D ⟶ cyl(f) ⟶ cone(C) ⟶ 0.

In both sequence the third chain complex is contractible. Hence we obtain from the fact above an isomorphism of A-chain complexes

C ⊕ cone(f) ≅ D ⊕ cone(C).

The ω-invariants are invariant under isomorphisms of A-chain complexes by Lemma 3.9. Hence it suffices to show for finitely generated projective A-chain complexes C and D that C and C ⊕ D have in all dimensions the same ω-invariants, provided that D is contractible. A A-chain complex is elementary if for some n it is concentrated in two consecutive dimensions n and n−1 and the n-th differential is an isomorphism. Since D is contractible, D is a sum of elementary contractible finitely generated Hilbert A-chain complexes. Hence we can assume without loss of generality that D is elementary. But then the claim follows Lemma 3.9.

The next theorem enables us to read of the center-valued L^{2}-Betti numbers and ωinvariants of a finitely generated Hilbert A-chain complex C from the homology of the A-chain complex H_{p}(ν^{−1}(C)).

Theorem 5.4 Let C be a finitely generated Hilbert A-chain complex. Then:

b_{p}^{u}(C) = dim^{u}(PH_{p}(ν^{−1}(C)));
b_{p}^{(2)}(C) = dim(PH_{p}(ν^{−1}(C)));
ω_{p}^{u}(C) = ω^{u}(TH_{p−1}(ν^{−1}(C)));
α_{p}(C) = α(TH_{p−1}(ν^{−1}(C))).

\underline{\underline{Proof}} : It suffices to prove the claim for the invariants based on the center-valued trace since they determine the others. The assertion about the center-valued L^{2}-Betti numbers follows from Lemma 5.2. We will now and later need the following general observation which follows from the fact that a chain map of projective chain complexes is a homotopy equivalence if and only if it induces an isomorphism on homology. Let Σ^{p} denote the p-fold suspension ΣΣ … Σ.

Lemma 5.5 Let S be a ring and C be a projective S-chain complex. Suppose that for each p there is a 1-dimensional projective S-resolution P [p] of H_{p}(C). Then there is a S-chain map j[p] : Σ^{p}P [p] ⟶ C which induces the identity on the p-th homology. The S-chain map

⊕_{p≥0}j[p] : ⊕_{p≥0}Σ^{p}P [p] ⟶ C

is a S-chain homotopy equivalence.

Because of Theorem 1.2 and Lemma 3.4 we can find a 1-dimensional finitely generated projective A-resolution P [p] of TH_{p}(ν^{−1}(C)) and PH_{p}(ν^{−1}(C)) is finitely generated projective. Let p[PH_{p}(ν^{−1})] be the obvious A-chain complex concentrated in dimension p. From Lemma 5.5 we get a chain homotopy equivalence

⊕_{p≥0}Σ^{p}P [p] ⊕ p[PH_{p}(ν^{−1}(C))] ⟶ ν^{−1}(C).

From Lemma 5.3 we conclude

ω^{u}(TH_{p−1}(ν^{−1}(C))) = ω_{p}^{u}(Σ^{p−1}ν(P [p − 1]))
= ω_{p}^{u}(Σ^{p−1}ν(P [p − 1]) ⊕ (p − 1)[ν(PH_{p−1}(ν^{−1}(C)))])
= ω^{u}(⊕_{p≥0}Σ^{p}ν(P [p]) ⊕ p[ν(PH_{p}(ν^{−1}(C)))])
= ω_{p}^{u}(ν(ν^{−1}(C)))
= ω_{p}^{u}(C)

This finishes the proof of Theorem 5.4.

6. L^{2}-Betti numbers and Novikov-Shubin invariants for spaces

In this section we will extend the notions and results of section 4 for A-Hilbert chain complexes to proper Γ-CW -complexes of finite type and discuss applications.

Let Γ be a discrete group and X be a proper Γ-CW -complex of finite type. Finite type means that all its skeleta are finite. Recall that a Γ-CW -complex is finite resp. of finite type if and only if Γ\X is finite resp. of finite type, and is proper if and only if all isotropy groups of the Γ-action are finite [28, Theorem 1.23 on page 18]. An example of a proper Γ-CW -complex which is finite resp. of finite type is the universal covering of a CW -complex which is finite resp. of finite type with fundamental group Γ. Let A be a finite von Neumann algebra and V be a finitely generated Hilbert A-module together with a unitary representation μ : Γ ⟶ U_{A}(V)^{op} into the (opposite of the) group of unitary A-automorphism of V. In particular V is a A-ℤΓ-bimodule. Then the cellular L^{2}-chain complex of X with coefficients in V is the finitely generated Hilbert A-chain complex

C^{(2)}(X; V) = V ⊗_{ℤΓ} C^{c}(X)

where C^{c}(X) is the cellular ℤΓ-chain complex. Its L^{2}-homology is the L^{2}-homology of X with coefficients in V and denoted by H_{p}^{(2)}(X; V). For details about Γ-CW -complexes and the Hilbert A-module structure on C^{(2)}(X; V) we refer to [34, section 3]. If we specify an inverse ν^{−1} as described in Theorem 2.1, then V and the unitary representation μ determine a A-ℤΓ-bimodule Q which is finitely generated projective over A. The cellular A-chain complex of X with coefficients in Q is the finitely generated A-chain complex

C^{Γ}(X; Q) = Q ⊗_{ℤΓ} C^{c}(X).

Its homology is the homology of X with coefficients in Q and denoted by H_{p}^{Γ}(X; Q). The center-valued L^{2}-Betti number b_{p}^{u}(X; V), the L^{2}-Betti number b_{p}^{(2)}(X; V), the ω-invariant ω_{p}^{u}(C) and the Novikov-Shubin invariant α_{p}(X; V) are defined as the corresponding invariants for C^{(2)}(X; V) (see Definition 5.1). Everything above has also a cohomological analogue which gives the same information [34, Lemma 3.10 on page 231]. Now Theorem 5.4 applied to this situation gives:

Theorem 6.1 Under the conditions above we get

b_{p}^{u}(X; V) = dim^{u}(PH_{p}^{Γ}(X, Q));
b_{p}^{(2)}(X; V) = dim(PH_{p}^{Γ}(X, Q));
ω_{p}^{u}(C) = ω^{u}(TH_{p−1}^{Γ}(X, Q));
α_{p}(C) = α(TH_{p−1}^{Γ}(X, Q)).

The advantage of this theorem is that it reduces the computations of L^{2}-invariants of the L^{2}-chain complex of X to the study of the homology of X with coefficents in a A-ℤΓ-bimodule Q and that A is semi-hereditary. In particular all standard results for homology with coefficients over a semi-hereditary ring apply to H^{Γ}(X; Q). Next we give a list of these tools and their applications.

1. Singular homology

If we want to compute H^{Γ}(X, Q) we can also use the singular chain complex C^{s}(X) of X which is (up to ℤΓ-chain homotopy naturally) ℤΓ-homotopy equivalent to the cellular one [28, Proposition 13.10 on page 264]. There are problems in the L^{2}-setting. For instance, for an arbitrary free Γ-CW -complex the differentials in the L^{2}-chain complex need not to be bounded operators. Notice that using the singular chain complex one can define H^{Γ}(X; Q) for all Γ-spaces X. Sometimes it can be useful for the computation of the homology of a proper Γ-CW -complex of finite type to consider also the homology of more general spaces. We will make use of this for instance in section 7. We will explain in a different paper how to use this approach to give a convenient reformulation of the notion of singular L^{2}-cohomology in [9, section 2].

2. Universal coefficient spectral sequences If one knows the homology with complex coefficients H_{n}(X; ℂ) of X, then there are spectral sequences computing the homology resp. cohomology of X with coefficients in Q (see [38, Theorem 12.1 on page 400], [42, Theorem 3.14 on page 73]). Namely, there is a spectral sequence converging to H_{p+q}^{Γ}(X; Q) whose E^{2}-term is

E_{p,q}^{2} = Tor_{p}^{ℂΓ}(Q, H_{q}(X; ℂ))

and a spectral sequence converging to H_{Γ}^{p+q}(X; Q) whose E_{2}-term is

E_{2}^{p,q} = Ext_{ℂΓ}^{p}(H_{q}(X; ℂ); Q).

3. Leray-Serre spectral sequence

Let F ⟶ E ⟶ B be a fibration such that B is a CW -complex. Assume for simplicity that F, E and B are connected and possess universal coverings. We abbreviate Γ = π_{1}(E) and π = π_{1}(B). Let S be a ring and Q be a S-ℤΓ-bimodule. Let Ẽ and B̃ be the universal coverings of E and B and let \overline{F} be the covering of F associated to the map π_{1}(F) ⟶ Γ induced by the inclusion. The composition q : Ẽ ⟶ B of p with the universal covering of E is a Γ-equivariant fibration with fiber \overline{F} if we equip B with the trivial Γ-action. The equivariant fiber transport yields a homomorphism π ⟶ ([\overline{F} , \overline{F}]^{Γ})^{op} into the opposite of the monoid of Γ-homotopy classes of Γ-maps from \overline{F} to itself. Thus we get the structure of a S-ℤπ-bimodule on H_{q}^{Γ}(\overline{F}; Q). The Leray-Serre spectral sequence converges to H_{p+q}^{Γ}(Ẽ; Q) and has as E^{2}-term

E_{p,q}^{2} = H_{p}^{π}(B̃; H_{q}^{Γ}(\overline{F}; Q)).

Of course there is also a cohomological version. The fiber transport of a Γ-fibration is explained in [27, section 1 and Theorem 6.1]. The construction of the Leray-Serre spectral sequence and the identification of its E^{2}-term is for instance given in [42, section 5.1] in the language of local coefficient systems, provided that Γ acts trivially on Q. We will expain in a different paper the rather straightforward extension of the proof to the general case if one uses [27, section 7]. The L^{2}-Leray-Serre spectral sequence for a fibration of CW -complexes of finite type was constructed in [46].

4. Poincaré duality

Suppose that X is a smooth orientable manifold of dimension m with a smooth proper Γaction such that Γ\M is compact. Then there are smooth Γ-equivariant triangulations so that we get a finite proper Γ-CW -complex structure and we can talk about the cellular ℤΓchain complex C^{c}(X). There is a fundamental class [X] in H_{m}(X, ∂X; ℚ^{w}) where w is the homomorphism w : Γ ⟶ {±1} sending γ ∈ Γ to 1 resp. −1 if γ acts orientation preserving resp. reversing and ℚ^{w} is the rational numbers with the right Γ-action given by rγ = w(γ)r for r ∈ ℚ and γ ∈ Γ. The A-chain complex C^{m−p}(X, ∂X;^{w}Q) has as p-th-chain module hom_{ℤπ}(C_{m−p}^{c}(X),^{w}Q) where ^{w}Q is the left ℤΓ-module obtained from Q by γq = w(γ)qγ^{−1} for γ ∈ Γ and q ∈ Q and carries the same left A-operation as Q. Then we obtain a A-chain homotopy equivalence unique up to A-chain homotopy

∩[X] : C^{m−p}(X, ∂X;^{w}Q) ⟶ C_{p}(X; Q).

Details of these facts above can be found in [34, section 5]. Let Q^{w} be the A-ℤΓ-bimodule obtained from the A-ℤΓ-bimodule Q by changing the Γ-action to qγ = w(γ)qγ for γ ∈ Γ and q ∈ Q. We conclude from Lemma 1.3 that for all p there are natural identifications

H^{m−p}(X, ∂X;^{w}Q) = H_{p}(X; Q);
(PH_{m−p}(X, ∂X; Q^{w}))^{∗} = PH_{p}(X; Q);
(TH_{m−1−p}(X, ∂X; Q^{w})) = TH_{p}(X; Q).

Notice for any finitely presented A-module M that (PM)^{∗} resp. (TM) and PM resp. TM are (not canonically) isomorphic. In particular we get back the Poincaré duality assertions for the L^{2}-Betti numbers and Novikov-Shubin invariants [26, Proposition 3.2 on page 33].

5. Morse inequalities

Assume that Γ acts freely on X so that Γ\X is a CW -complex of finite type. Let i_{X}(p) be the number of p-cells in Γ\X. Then we get from Lemma 3.12 (see also [15]):

i_{X}(p) ≥ μ(H_{p}^{Γ}(X; Q) ⊕ TH_{p−1}^{Γ}(X; Q)).

Since the left side of the equation is equal or greater than the p-th L^{2}-Betti number b_{p}^{(2)}(X; V), this improves the Morse inequalities of [43]. In particular one gets if α_{p}(X; V) ≠ ∞^{+} the inequality

i_{X}(p) > b_{p}^{(2)}(X).

6. Deficiency of groups

It is shown in [30, Theorem 6.1 on page 212] for a group group Γ whose classifying space BΓ = Γ\EΓ has finite 3-skeleton that the deficiency of Γ satisfies

def(Γ) ≤ 1 − b_{0}^{(2)}(EΓ; V) + b_{1}^{(2)}(EΓ; V) − b_{2}^{(2)}(EΓ; V).

Suppose that α_{3}(EΓ; V) ≠ ∞^{+} what is equivalent to TH_{2}^{Γ}(EΓ; Q) ≠ 0. Then this inequality must be a strict inequality. This follows from an elementary modification of the proof in [30, Theorem 6.1 on page 212] which is based on the following observation. If X is the universal covering of a finite 2-dimensional CW -complex with fundamental group Γ, then the classifying map is 2-connected and induces an epimorphism H_{2}^{Γ}(X; Q) ⟶ H_{2}^{Γ}(EΓ; Q) and TH_{2}^{Γ}(X; Q) = 0. If TH_{2}^{Γ}(EΓ; Q) is not trivial, we conclude:

b_{2}^{(2)}(X; V) > b_{2}^{(2)}(EΓ; V).

7. L^{2}-torsion

Using for instance [5] or [34] one can also translate the definition of L^{2}-torsion for finitedimensional finitely generated Hilbert A-chain complexes to the setting of finite-dimensional finitely generated projective A-chain complexes using the same pattern as above for the L^{2}-Betti numbers and Novikov-Shubin invariants. This allows for instance to carry over the results in [36] about torsion invariants and fibrations from ordinary torsion to L^{2}-torsion. Analogously to the extension of L^{2}-Betti numbers and Novikov-Shubin invariants one can try to use the center-valued trace to define a refined Fuglede-Kadison determinant and L^{2}-torsion taking values in K_{1}^{w}(A) or Z(A)^{w}. This has been done in [34]. However, in the II_{1}-case this requires a suitable center-valued version of the condition necessary for the definition of the real-valued L^{2}-torsion that the Novikov-Shubin invariants are positive. The details have so far not yet been carried out. Without such additional conditions one would always get a trivial invariant in the type II_{1}-case because of the computations of K_{1}^{w}(A) in [35, Theorem 2.1 on page 521 and Theorem 3.3 on page 525].

8. The “Zero in the spectrum” conjecture

The “Zero in the spectrum” conjecture says that there is no contractible closed Riemannian manifold M with an isometric proper cocompact action of a discrete group Γ such that for all p zero is not in the spectrum of the Laplace operator in dimension p [18, page 238], [25]. The author does not even know an example of a finite proper Γ-CW -complex X such that for all p zero is not in the spectrum of the combinatorial Laplace operator in dimension p. Notice that the last condition on X is equivalent to the purely algebraic statement that H_{p}^{Γ}(X; N (Γ)) is trivial for all p, or equivalently C^{Γ}(X; N (Γ)) is contractible.

9. Homological computations

In some special situations H^{(2)}(X; V) depends only on the homology of X with complex coefficients H(X; ℂ). Namely, suppose that H_{p}(X; ℂ) has a 1-dimensional finitely generated projective ℂΓ-resolution P [p] for all p ≥ 0. Let Q be the A-ℤΓ-bimodule associated to V as explained in section 4. Then we get from Lemma 5.5 a ℂΓ-chain homotopy equivalence

⊕_{p≥0}j[p] : ⊕_{p≥0}Σ^{p}P [p] ⟶ C^{c}(X; ℂ).

It induces a chain homotopy equivalence of A-chain complexes

⊕_{p≥0}Q ⊗_{ℂΓ} Σ^{p}P [p] ⟶ C^{Γ}(X; Q).

We get isomorphisms of A-modules

Q ⊗_{ℂΓ} H_{p}(X; ℂ) ⊕ Tor_{1}^{ℂΓ}(Q, H_{p−1}(X; ℂ)) ⟶ H_{p}^{Γ}(X; V).

Notice that Tor_{1}^{ℂΓ}(Q, H_{p−1}(X; ℂ)) is finitely generated projective as A is semi-hereditary. Hence we get

Lemma 6.2 Suppose that H_{p}(X; ℂ) has a 1-dimensional finitely generated projective ℂΓresolution for all p ≥ 0. Then we obtain isomorphisms of A-modules

P (Q ⊗ H_{p}(X; ℂ)) ⊕ Tor_{1}^{ℂΓ}(Q, H_{p−1}(X; ℂ)) ⟶ PH_{p}^{Γ}(X; V);
T (Q ⊗ H_{p}(X; ℂ)) ⟶ TH_{p}^{Γ}(X; V).

In particular H_{p}(X; ℂ) determines the L^{2}-Betti numbers and Novikov-Shubin invariants.

Notice that the assumption on H_{p}(X) is always satisfied if Γ is a finitely generated free group. Namely, the complex group ring of a finitely generated free group is a fir, i.e free ideal ring [10, Corollary 3 on page 68]. This phenomenon was already observed in [15].

One can make computations more explicite in the following case.

Theorem 6.3 Suppose that Γ is the free abelian group ℤ^{r} of rank r. Let ℂℤ_{(0)}^{r} be the quotient field of ℂℤ^{r}. Then

b_{p}^{(2)}(X; l^{2}(ℤ^{r})) = dim_{ℂℤ_{(0)}^{r}} (H_{p}(X; ℂ) ⊗_{ℂℤ^{r}} ℂℤ_{(0)}^{r}) .

\underline{\underline{Proof}} : We abbreviate C = ℂ ⊗_{ℤ} C^{c}(X), C^{(2)} = l^{2}(ℤ^{r}) ⊗_{ℂℤ^{r}} C and C_{(0)} = ℂℤ_{(0)}^{r} ⊗_{ℂℤ^{r}} C. We first treat the case where C_{(0)} has trivial homology. Then we can find a ℂ[ℤ^{r}]_{(0)}- chain contraction γ′. Choose u ∈ ℂℤ^{r} with u ≠ 0 and maps γ_{n} : C_{n} ⟶ C_{n+1} such that l_{u} ∘ γ′_{n} = (γ_{n})_{(0)} holds for all n where l_{u} is multiplication with u. Then γ is a chain homotopy of ℂ[ℤ^{r}]-chain maps l_{u} ≃ 0 : C ⟶ C. This induces a chain homotopy of chain maps of finite Hilbert N (ℤ^{r})-chain complexes l_{u} ≃ 0 : C^{(2)} ⟶ C^{(2)}. Hence multiplication with u induces the zero map on the L^{2}-homology of C^{(2)}. This is only possible if the L^{2}-homology is trivial and hence all L^{2}-Betti numbers of C^{(2)} vanish.

Next we treat the general case. Put b_{n} = dim_{ℂ[ℤ^{r}]_{(0)}}(H_{n}(C_{(0)})). Then there is a ℂ[ℤ^{r}]_{(0)}- isomorphism

⊕_{i=1}^{b_{n}}ℂ[ℤ^{r}]_{(0)} ⟶ H_{n}(C_{(0)}) = H_{n}(C) ⊗_{ℂ[ℤ^{r}]} ℂ[ℤ^{r}]_{(0)}.

By composing it with a map given by multiplication with a suitable element in ℂ[ℤ^{r}] one can construct a ℂ[ℤ^{r}]-map

i_{n} : ⊕_{i=1}^{b_{n}}ℂ[ℤ^{r}] ⟶ H_{n}(C)

such that (i_{n})_{(0)} is a ℂ[ℤ^{r}]_{(0)}-isomorphism. Let D be the finite free ℂ[ℤ^{r}]-chain complex whose n-th chain module is D_{n} = ⊕_{i=1}^{b_{n}}ℂ[ℤ^{r}] and whose differentials are all trivial. Choose a ℂ[ℤ^{r}]-chain map j : D ⟶ C which induces on the n-th homology the map i_{n}. Let cone(j) be its mapping cone. There is a canonical exact sequence

0 ⟶ C ⟶ cone(j) ⟶ ΣD ⟶ 0.

It remains exact under the passage from C to C_{(0)} or C^{(2)}. We conclude from the long exact homology sequence that cone(j)_{(0)} is acyclic. Hence the L^{2}-homology of cone(j)^{(2)} is trivial by the first step. We conclude from the long weakly exact L^{2}-homology sequence [8, Theorem 2.1 on page 10]

b_{n}^{(2)}(C^{(2)}) = b_{n}^{(2)}(D^{(2)}) = dim_{N (ℤ^{r})}(D_{n}^{(2)}) = b_{n} = dim_{ℂℤ_{(0)}^{r}}(H_{n}(C_{(0)}))

and the claim follows.

7. L^{2}-Betti numbers, fibrations and deficiency of groups

In this section we generalize [30, Theorem 3.1] using the algebraic description of L^{2}- homology as described in this paper. We mention that the proof presented here seems to be more conceptual than the one in [30]. We will show

Theorem 7.1 Let d ≥ 0 be an integer and let F ⟶ E ⟶ B be a fibration of spaces such that F resp. E has the homotopy type of a connected CW -complex with finite d-skeleton resp. d + 1-skeleton. Let \overline{F} be the covering of F associated to the map π_{1}(F) ⟶ π_{1}(E) induced by the inclusion. Suppose that b_{p}^{(2)}(\overline{F}; l^{2}(π_{1}(E))) = 0 for p ≤ d − 1 and π_{1}(B) contains an element of infinite order. Then we get

b_{p}^{(2)}(Ẽ) := b_{p}^{(2)}(Ẽ; l^{2}(π_{1}(E)) = 0 for p ≤ d.

Before we give the proof of Theorem 7.1 we prove the next lemma which is probably well-known but for whose second assertion we could not find a reference.

Lemma 7.2 Let F ⟶ E ⟶^{p} B be a fibration of path-connected spaces and let d ≥ 0 be an integer. Then:

1. If both F and B are homotopy equivalent to finite CW -complexes resp. finite-dimen-sional CW -complexes resp. CW -complexes of finite type resp. CW -complexes, then the same is true for E.

2. Suppose that F is homotopy equivalent to a CW -complex with finite d-skeleton resp. CW -complex of finite type, E is homotopy equivalent to a CW -complex with finite d+1-skeleton resp. CW -complex of finite type and B has the homotopy type of a CW - complex. Then B is homotopy equivalent to a CW -complex with finite d + 1-skeleton resp. CW -complex of finite type.

\underline{\underline{Proof}} : 1.) This is done by induction over the skeletons of B and the fact that any fibration over D^{n} is fiber homotopy equivalent to the trivial fibration. More details can be found for instance in [36, section 1].

2.) We only treat the case where F has finite d-skeleton and E has finite d + 1-skeleton. If d = 0, the claim follows from the fact that a connected space X of the homotopy type of a CW -complex is homotopy equivalent to a CW -complex with finite 1-skeleton if and only if π_{1}(X) is finitely generated. Hence it remains to treat the case d ≥ 1.

We construct inductively for n ≥ 1 a max{2, n}-dimensional CW -complex X_{n} together with a n-connected map f_{n} : X_{n} ⟶ B such that π_{1}(f_{n}) is an isomorphisms, X_{1} is finite, f_{n+1} extends f_{n} and, provided that n ≤ d, the space X_{n+1} is obtained from X_{n} by attaching finitely many cells of dimension n + 1. Then the direct limit yields a CW -complex X with finite d + 1-skeleton and a homotopy equivalence f : X ⟶ B.

The induction begin n = 1 is done as follows. Since π_{0}(F) is trivial, π_{1}(F) is finitely generated and π_{1}(E) is finitely presented, we conclude from the long homotopy sequence of the fibration that π_{1}(B) is finitely presented. Let X_{1} be the finite 2-dimensional CW -complex associated to some finite presentation of π_{1}(B). Obviously there is a map f_{1} : X_{1} ⟶ B which induces an isomorphism on the fundamental groups.

The induction step from n ≥ 1 to n + 1 is done as follows. Let p_{n} : Y_{n} ⟶ X_{n} be the pull back fibration of p with respect to f_{n}:

Y_{n} ⟶^{g_{n}} E
p_{n}↓  ↓^{p}
X_{n} ⟶_{f_{n}} B

Let B̃ ⟶ B be the universal covering of B with π = π_{1}(B) as group of deck transformations. We obtain the following π-equivariant pull back by pulling back the universal covering of B using the square above:

\overline{Y_{n}} ⟶^{\overline{g_{n}}} \overline{E}
\overline{p_{n}}↓  \sqrt{p}
X̃_{n} \overrightarrow{f̃_{n}} B̃

Notice that X̃_{n} is the universal covering of X_{n}, as π_{1}(f_{n}) is bijective. The coverings \overline{Y_{n}} and \overline{E} are not-necessarily the universal coverings but they have connected total spaces as F is path-connected. The following diagram commutes for k ≥ 1 where the horizontal maps are Hurewicz homomorphisms and the vertical maps are induced by the square above:

π_{k}(\overline{g_{n}}) ⟶ H_{k}(\overline{g_{n}})
↓  ↓
π_{k}(f̃_{n}) ⟶ H_{k}(f̃_{n})

The left vertical arrow is an isomorphism for k ≥ 1 by [49, Corollary 8.8 on page 187]. In particular \overline{g_{n}} is n-connected because f̃_{n} is n-connected. Since \overline{Y_{n}} and \overline{E} are path-connected, we conclude from [49, Corollary 7.9 on page 180] that H_{k}(\overline{g_{n}}) is zero for k ≤ n. The lower horizontal arrow is an isomorphism for k ≤ n + 1 by the Hurewicz isomorphism since X̃_{n} and B̃ are simply-connected and f̃_{n} is n-connected [49, Corollary 7.10 on page 181]). Hence the right vertical arrow is surjective for k = n + 1.

Next we show that H_{n+1}(\overline{g_{n}}) is a finitely generated ℤπ-module provided n ≤ d. As F and X_{n} have finite n-skeletons up to homotopy, the same is true for Y_{n} by the first assertion. By assumption E has finite n + 1-skeleton up to homotopy. Hence we can assume without loss of generality, that the cellular ℤπ-chain complexes of \overline{Y_{n}} resp. \overline{E} have finitely generated free ℤπ-modules in dimensions ≤ n resp. ≤ n + 1. Hence the mapping cone D of the ℤπ-chain map C(\overline{g_{n}}) has finitely generated free ℤπ-modules as chain modules in dimensions ≤ n + 1. By definition H_{k}(D) and H_{k}(\overline{g_{n}}) agree so that H_{k}(D) is trivial for k ≤ n as shown above. Hence there is a long exact sequence of ℤπ-modules:

{0} ⟶ ker(d_{n+1}) ⟶ D_{n+1} ⟶^{d_{n+1}} D_{n} ⟶^{d_{n}} … ⟶^{d_{1}} D_{0} ⟶ {0}

Since D_{k} is finitely generated free for all k ≤ n + 1, we conclude that ker d_{n+1} is a finitely generated projective ℤπ-module. This implies that H_{n+1}(D) = H_{n+1}(\overline{g_{n}}) is finitely generated over ℤπ. Since H_{n+1}(f̃_{n}) is a quotient of H_{n+1}(\overline{g_{n}}), it is finitely generated over ℤπ. This shows that π_{n+1}(f̃_{n}) is finitely generated over ℤπ provided n ≤ d.

Given a set of generators of the ℤπ-module π_{n}(f_{n}) ≅ π_{n}(f̃_{n}), there is the standard procedure of attaching cells to X_{n}, one for each generator, to obtain X_{n+1} such that f_{n} extends to a (n + 1)-connected map f_{n+1} : X_{n+1} ⟶ B. This finishes the proof of Lemma 7.2.

The second assertion 2.) in Lemma 7.2 becomes false if one substitutes of finite type by homotopy equivalent to a finite CW -complex or by homotopy equivalent to a finitedimensional CW -complex. Here is a counterexample. Realize the exact sequence of abelian groups 0 ⟶ ℤ ⟶^{·2} ℤ ⟶ ℤ/2 ⟶ 0 by a fibration of Eilenberg-MacLane spaces of type 1. Notice that K(ℤ, 1) is homotopy equivalent to a circle and K(ℤ/2, 1) is not-homotopy equivalent to a finite-dimensional CW -complex because its cohomology ring with ℤ/2-coefficients is a free polynomial algebra in one generator. One can say nothing about the fiber if one has only information about the total and the base space. For instance the homotopy fiber of the map S^{1} ∨ S^{1} to S^{1} which is the identity on each component is an infinite wedge of S^{1}-s and hence not homotopy equivalent to a CW -complex of finite type. Other examples come from path fibrations ΩX ⟶ P X ⟶ X over finite CW -complexes since P X is contractible and ΩX has often homology in all dimensions.

Now we are ready to give the proof of Theorem 7.1. Because of Lemma 7.2 we can assume that both E and B are connected CW -complexes with finite d+1-skeletons and F is a connected CW -complex with finite d-skeleton. We abbreviate Γ = π_{1}(E) and π = π_{1}(B). Recall from section 4 that the E^{2}-term of the Leray-Serre spectral sequence which converges to H_{p+q}^{Γ}(Ẽ; N (Γ)) is

E_{p.q}^{2} = H_{p}^{π}(B̃; H_{q}^{Γ}(\overline{F}; N (Γ))).

Next we show for all r ≥ 2 that E_{p,q}^{r} is finitely presented with PE_{p,q}^{r} = 0 if p + q ≤ d and q ≠ d and and is finitely generated if p + q = d + 1 and q ≤ d − r. Notice that then the same is true for r = ∞ since the Leray-Serre spectral sequence is a first quadrant spectral sequence.

Since F has finite d-skeleton and b_{q}^{(2)}(\overline{F}; N (Γ)) = 0 for q ≤ d − 1, the N (Γ)-module H_{q}(\overline{F}; N (Γ)) is finitely presented with PH_{q}(\overline{F}; N (Γ)) = 0 for q ≤ d − 1 because of Lemma 3.7 and Theorem 6.1. Hence E_{p,q}^{2} is finitely presented with PE_{p,q}^{2} = 0 for p ≤ d and q ≤ d − 1 and is finitely generated for p = d + 1 and q ≤ d − 1 since B has finite d + 1-skeleton. This finishes the induction begin r = 2. The induction step follows from an iterated application of Lemma 3.13 using the fact that the E^{r+1}-term is the homology of the E^{r}-term and the r-th differentials are d_{p,q}^{r} : E_{p.,q}^{r} ⟶ E_{p−r,q+r−1}^{r}.

We get from Lemma 3.7 and the Leray-Serre spectral sequence PH_{n}^{Γ}(Ẽ; N (Γ)) = 0 for n ≤ d − 1 and that there is an exact sequence of N (Γ)-modules

H_{0}^{π}(B̃; H_{d}^{Γ}(\overline{F}; N (Γ))) ⟶ H_{d}^{Γ}(Ẽ; N (Γ)) ⟶ M ⟶ 0

where H_{d}^{Γ}(Ẽ; N (Γ)) and M are finitely presented N (Γ)-modules and PM = 0. Recall from Theorem 6.1 that b_{p}^{(2)}(Ẽ) := b_{p}^{(2)}(Ẽ; l^{2}(Γ)) vanishes for all p ≤ d if and only if PH_{p}^{Γ}(Ẽ; N (Γ)) vanishes for all p ≤ d. Hence it suffices because of Lemma 3.7 to construct a finitely presented N (Γ)-module N with PN = 0 such that there is an epimorphism from N onto H_{0}^{π}(B̃; H_{d}^{Γ}(\overline{F}; N (Γ))).

Let f : S^{1} ⟶ B be a map which induces an injection on the fundamental groups. Let F ⟶ E_{0} ⟶^{p_{0}} S^{1} and f_{0} : E_{0} ⟶ E be given by the pull back construction. The Leray-Serre spectral sequence for p_{0} yields an exact sequence

0 ⟶ H_{0}^{ℤ}(S^{1}; H_{d}^{Γ}(\overline{F}; N (Γ))) ⟶ H_{d}^{Γ}(\overline{E_{0}}; N (Γ)) ⟶ H_{1}^{ℤ}(S^{1}; H_{d−1}^{Γ}(\overline{F}; N (Γ))) ⟶ 0

where we identify π_{1}(S^{1}) = ℤ and \overline{E_{0}} ⟶ E_{0} is the covering given by π_{1}(f_{0}) : π_{1}(E_{0}) ⟶ Γ. Let g : F ⟶ F be a cellular map whose homotopy class is given by the fiber transport for p_{0} with a generator in ℤ. Let T_{g} be the mapping torus of g which is obtained from the cylinder over F by identifying the bottom and the top by g. There is a homotopy equivalence h : T_{g} ⟶ E_{0} such that its composition with p_{0} is homotopic to the canonical projection T_{g} ⟶ S^{1}. Let g_{d} : F_{d} ⟶ F_{d} be the restriction of g to the d-skeleton of F. Denote by j : T_{g_{d}} ⟶ T_{g} the inclusion. Let \overline{T_{g_{d}}} and \overline{T_{g}} be the coverings associated to the homomorphisms from π_{1}(T_{g_{d}}) and π_{1}(T_{g})) to Γ which are induced by j, h and f_{0}. Since T_{g} is obtained from T_{g_{d}} by attaching cells of dimensions ≥ d + 1, the inclusion j induces an epimorphism

j_{∗} : H_{d}^{Γ}(\overline{T_{g_{d}}}; N (Γ)) ⟶ H_{d}^{Γ}(\overline{T_{g}}; N (Γ)).

Let Δ be the image of π_{1}(f_{0}) : π_{1}(E_{0}) ⟶ Γ. Then the map π_{1}(T_{g_{d}}) ⟶ ℤ induced by the canonical projection T_{g_{d}} ⟶ S^{1} factorizes over the epimorphism

π_{1}(T_{g_{d}}) ⟶^{π_{1}(j)} π_{1}(T_{g}) ⟶^{π_{1}(h)} π_{1}(E_{0}) ⟶^{π_{1}(f_{0})} Δ.

Let \overline{\overline{T_{g_{d}}}} be the covering of T_{g_{d}} associated to the map π_{1}(T_{g_{d}}) ⟶ Δ above. Since T_{g_{d}} is finite, we conclude from [30, Lemma 1.2.3 on page 205 and Theorem 2.1. on page 207]

b_{d}^{(2)}(\overline{T_{g_{d}}}; N (Γ)) = b_{d}^{(2)}(\overline{\overline{T_{g_{d}}}}; N (Δ)) = 0.

Hence H_{d}^{Γ}(\overline{T_{g_{d}}}; N (Γ)) is finitely presented and PH_{d}^{Γ}(\overline{T_{g_{d}}}; N (Γ)) is trivial. Since the N (Γ)- module H_{1}^{ℤ}(S^{1}; H_{d−1}^{Γ}(\overline{F}; N (Γ))) is finitely presented and PH_{1}^{ℤ}(S^{1}; H_{d−1}^{Γ}(\overline{F}; N (Γ))) = 0, the kernel N of the composition

H_{d}^{Γ}(\overline{T_{g_{d}}}; N (Γ)) ⟶^{j_{∗}} H_{d}^{Γ}(\overline{T_{g}}; N (Γ)) ⟶^{h_{∗}} H_{d}^{Γ}(\overline{E_{0}}; N (Γ)) ⟶ H_{1}^{ℤ}(S^{1}, H_{d−1}^{Γ}(\overline{F}; N (Γ)))

is finitely generated and satisfies PN = 0 because of Lemma 3.7. The epimorphism h_{∗} ∘ j_{∗} induces an epimorphism

N ⟶ H_{0}^{ℤ}(S^{1}; H_{d}^{Γ}(\overline{F}; N (Γ))).

The map f induces a surjective homomorphism

H_{0}^{ℤ}(S^{1}; H_{d}^{Γ}(\overline{F}; N (Γ))) ⟶ H_{0}^{π}(B; H_{d}^{Γ}(\overline{F}; N (Γ)))

as it can be identified with the projection ℂ ⊗_{ℂℤ} H_{d}^{Γ}(\overline{F}; N (Γ)) ⟶ ℂ ⊗_{ℂπ} H_{d}^{Γ}(\overline{F}; N (Γ)) where ℤ and π act trivially on ℂ. This finishes the proof Theorem 7.1.

Next we give the promised proof of Theorem 0.6 and Theorem 0.8. Theorem 0.6 follows by induction over n from Theorem 7.1 applied to the fibrations BΓ_{i} ⟶ BΓ_{i+1} ⟶ BΓ_{i+1}/Γ_{i} since b_{p}^{(2)}(Γ_{i}) = b_{p}^{(2)}(\overline{BΓ_{i}}; l^{2}(Γ_{i+1})) holds for the covering \overline{BΓ_{i}} associated to the inclusion Γ_{i} ⟶ Γ_{i+1} [30, Lemma 1.2.3 on page 205]. The induction begin follows from b_{0}^{(2)}(Γ) = |Γ|^{−1} [30, Lemma 1.2.5 on page 205].

Next we prove Theorem 0.8. There is a subgroup F_{1} ⊂ F together with a monomorphism Φ : F_{1} ⟶ F_{1} such that F_{1} is isomorphic to F and F is the HNN-extension of F_{1} with respect to Φ with one stable letter [4, Proposition 1.7 on page 370]. From the topological description of HNN-extensions [37, page 180] we conclude that F is the fundamental group of the mapping torus T_{BΦ} of the map BΦ : BF_{1} ⟶ BF_{1} induced by Φ. The inclusion BF_{1} ⟶ BF induces on the fundamental groups the inclusion of F_{1} in F. The argument in [30, page 207] shows that the cellular ℤF -chain complex of the universal covering T̃_{BΦ} of T_{BΦ} is the mapping cone of a certain ℤF -chain map from ℤF ⊗_{ℤF_{1}} C(EF_{1}) to itself. Since ℤF is free over ℤF_{1}, we conclude for p ≥ 1

H_{p}(ℤF ⊗_{ℤF_{1}} C(EF_{1})) = ℤF ⊗_{ℤF_{1}} H_{p}(C(EF_{1})) = 0.

This implies H_{p}(T̃_{BΦ}; ℤ) = 0 for p ≥ 2. Hence T_{BΦ} is a model for BF. Now the claim follows from [30, Theorem 2.1. on page 207].

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[24] Lott, J.: “Heat kernels on covering spaces and topological invariants”, J. of Diff. Geom. 35, 471 - 510 (1992)

[25] Lott, J.: “The Zero-in-the-Spectrum Question”, to appear in L’Enseignement Mathématique preprint (1996)

[26] Lott, J. and Lück, W.: “L^{2}-topological invariants of 3-manifolds”, Inventiones Math. 120, 15 -60 (1995)

[27] Lück, W.: “The transfer maps induced in the algebraic K_{0}- and K_{1}-groups by a fibration I”, Math. Scand. 59, 93 - 121 (1986)

[28] Lück, W.: “Transformation groups and algebraic K-theory”, Lecture Notes in Mathematics 1408 (1989)

[29] Lück, W.: “L^{2}-torsion and 3-manifolds”, Conference Proceedings and Lecture Notes in Geometry and Topology Volume III “Low-dimensional topology”, Knoxville 1992, editor: Klaus Johannson, International Press, 75- 107 (1994)

[30] Lück, W.: “L^{2}-Betti numbers of mapping tori and groups”, Topology 33, 203 - 214 (1994)

[31] Lück, W.: “Approximating L^{2}-invariants by their finite-dimensional analogues”, Geometric and Functional Analysis 4, 455 - 481 (1994)

[32] Lück, W.: “L^{2}-invariants of regular coverings of compact manifolds and CW - complexes”, to appear in “handbook of geometry”, Elsevier (1997)

[33] Lück, W.: “Dimension theory of arbitrary modules over finite Neumann algebras and applications to L^{2}-invariants”, preprint, in preparation

[34] Lück, W. and Rothenberg, M.: “Reidemeister torsion and the K-theory of von Neumann algebras”, K-theory 5, 213 - 264 (1991)

[35] Lück, W. and Rørdam, M.: “Algebraic K-theory of von Neumann algebras”, Ktheory 7, 517 - 536 (1993)

[36] Lück, W., Schick, T. and Thielmann, T.: “Torsion and fibrations”, preprint, Mainz (1995)

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[44] Pansu, P.: “Introduction to L^{2} Betti numbers”,in “Riemannian Geometry”, editors: Lovric, M., Min Oo, M. and Wang, McK. ed., Fields Institute Monographs 4, Amer. Math. Soc., 53 - 86 (1996).

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Current address Wolfgang Lück Fachbereich Mathematik und Informatik Westfälische Wilhelms-Universität Münster Einsteinstr. 62 48149 Münster Bundesrepublik Deutschland email: lueck@math.uni-muenster.de