“Approximating L^{2}-invariants by their finite-dimensional analogues” by Wolfgang Lück

Abstract: Let X be a finite connected CW -complex. Suppose that its fundamental group π is residually finite, i.e., there is a nested sequence … ⊂ Γ_{m+1} ⊂ Γ_{m} ⊂ … ⊂ π of in π normal subgroups of finite index whose intersection is trivial. Then we show that the p-th L^{2}-Betti number of X is the limit of the sequence b_{p}(X_{m})/[π : Γ_{m}] where b_{p}(X_{m}) is the (ordinary) p-th Betti number of the finite covering of X associated with Γ_{m}.

Introduction and statement of results

Let X be a finite connected CW -complex with fundamental group π and A ⊂ X be a CW -subcomplex. Let b_{p}^{(2)}(X, A) be the p-th L^{2}-Betti number. Suppose π is residually finite, i.e., there is a nested sequence of in π normal subgroups … ⊂ Γ_{m+1} ⊂ Γ_{m} ⊂ … ⊂ π such that the index [π : Γ_{m}] is finite for all m ≥ 0 and the intersection ∩_{m≥0}Γ_{m} is the trivial group. Examples of residually finite groups are fundamental groups of compact Haken 3-manifolds and finitely generated groups possessing a faithful representation into GL(n, F) for some field F. Consider any such sequence (Γ_{m})_{m≥0}. Let p_{m} : X_{m} ⟶ X be the covering of X associated with Γ_{m} ⊂ π and put A_{m} = p_{m}^{−1}(A). Denote by b_{p}(X_{m}, A_{m}) the (ordinary) p-th Betti number of (X_{m}, A_{m}). The following theorem answers a query of Gromov [14, pages 13, 153].

Theorem 0.1 (Kazhdan’s equality) Under the conditions above

lim_{m→∞} \frac{b_{p}(X_{m}, A_{m})}{[π : Γ_{m}]} = b_{p}^{(2)}(X, A).

The inequality lim sup_{m→∞} \frac{b_{p}(X_{m},A_{m})}{[π:Γ_{m}]} ≤ b_{p}^{(2)}(X, A) for X a closed manifold and empty A is discussed by Gromov [14, pages 13, 153] and is essentially due to Kazhdan [20].

There are estimates on the sum of the ordinary Betti numbers by the volume and by the simplicial volume for a closed Riemannian manifold which satisfies certain pinching conditions on its curvature [12, page 12]. Provided that the fundamental group is residually finite, these hold also for the sum of the L^{2}-Betti numbers by Theorem 0.1.

Suppose that M is a closed Riemannian manifold. Let L^{2}Ω^{k}(M̃) be the Hilbert space of square integrable p-forms on the universal covering M̃. The Laplace operator Δ̃^{p} acting on L^{2}Ω^{k}(M̃) is essentially self-adjoint. Hence the spectral theorem applies and there is a spectral family {P_{λ}^{p}|λ ∈ [0, ∞)} associated to Δ̃^{p}. Let P_{λ}^{p}(x, y) be a Schwartz kernel of P_{λ}^{p}. Denote by tr_{ℝ} (P_{λ}^{p}(x, x)) the trace of the linear endomorphism of the finite-dimensional real vector space ∧^{p}T_{x}^{∗}M̃. Let F ⊂ M̃ be a fundamental domain for the π-action. Define the analytic p-th spectral density function of M by

F^{p} : [0, ∞) ⟶ [0, ∞) λ ↦ ∫_{F} tr_{ℝ} (P_{λ}^{p}(x, x)) · dvol.

Notice that F^{p}(0) = b_{p}^{(2)}(M) and that the spectrum of Δ̃^{p} has a gap at zero if and only if F^{p}(λ) − F^{p}(0) = 0 for some λ > 0.

Suppose that π = π_{1}(M) is residually finite with a nested sequence of normal subgroups of finite index (Γ_{m})_{m≥0} with trivial intersection as above. Fix a CW -structure on M. Equip all finite coverings M_{m} with the induced CW -structure. The cellular ℂ[Γ_{m}\π]- cochain complex C^{p}(M_{m}; ℂ) inherits a ℂ-basis and in particular a Hilbert space structure. Hence the adjoint (c^{p})^{∗} of its differential c^{p} is defined. The combinatorial Laplace operator Δ_{m}^{p} : C^{p}(M_{m}; ℂ) ⟶ C^{p}(M_{m}; ℂ) is defined by c^{p−1}(c^{p−1})^{∗} + (c^{p})^{∗}c^{p}. Denote by N_{m}^{p}(λ) the number of eigenvalues μ of Δ_{m}^{p} satisfying μ ≤ λ counted with multiplicity. Notice that the spectrum of Δ̃^{p} has a gap at zero if and only if F^{p}(λ) = F^{p}(0) for some λ > 0. We get under the conditions above

Theorem 0.2 (Kazhdan’s criterion) The spectrum of Δ̃^{p} has a gap at zero if and only if there is a λ > 0 such that

lim_{m→∞} \frac{N_{m}^{p}(λ) − N_{m}^{p}(0)}{[π : Γ_{m}]} = 0.

Theorem 0.3 (Logarithmic spectral density estimate) There are constants C > 0 and ϵ > 0 (depending on M but not on λ) such that for all λ ∈ (0, ϵ)

F^{p}(λ) − F^{p}(0) ≤ \frac{C}{− ln(λ)}.

Notice that zero is not in the spectrum of Δ̃^{p} if and only if F^{p}(0) = b_{p}^{(2)}(M) = 0 and the spectrum of Δ̃^{p} has a gap at zero, or equivalently, if and only if F^{p}(λ) = 0 for some λ > 0. Hence Theorem 0.1 and Theorem 0.2 together imply the necessary and the sufficient Kazhdan’ criterion in [14, page 14] for the question whether zero is not in the spectrum of Δ̃^{p}. Theorem 0.3 gives some evidence for [22, Conjecture 9.1] that the Novikov-Shubin invariants of M are always positive. This means that one can improve the inequality in Theorem 0.3 for some α > 0 to

F^{p}(λ) − F^{p}(0) ≤ Cλ^{α}.

All we have said about a closed Riemannian manifold M can be extended to compact Riemannian manifolds with boundary by imposing absolute boundary conditions.

We give two applications to “middle” algebraic K-groups of ℤπ, provided that π is countable and residually finite. Let N (π) = B(l^{2}(π), l^{2}(π)^{π} be the von Neumann algebra of π and denote by tr_{N (π)}^{c} its unique center valued trace.

Theorem 0.4 (Swan’ theorem for countable residually finite groups)

1. Let p ∈ M(n, n, ℤπ) be an idempotent. If I_{1} is the unit element in N (π) and r is the rank of the abelian group ℤ ⊗_{ℤπ} im(p), then

tr_{N (π)}^{c}(p) = r · I_{1}.

2. The change of rings homomorphism

K̃_{0}(ℤπ) ⟶ K̃_{0}(N (π))

is trivial.

Roughly speaking, Theorem 0.4 means for a finitely generated projective ℤπ-module P that l^{2}(π) ⊗_{ℤπ} P is π-isometrically isomorphic to the Hilbert N (π)-module mbox⊕_{i=1}^{r}l^{2}(π) for r the rank of ℤ ⊗_{ℤπ} P. Theorem 0.4 was already proven by Swan [31, Theorem 8.1] for finite π. Notice that for finite π the von Neumann algebra N (π) reduces to ℂπ and |π| · tr_{N (π)}^{c}(p) for an idempotent p ∈ M(n, n, ℤπ) is the character of the complex π-representation ℂ ⊗_{ℤ} im(p). Bass [2, section 9] shows for a torsionfree linear finitely generated group Γ the even stronger statement that the Hattori-Stallings rank of a finitely generated projective ℤΓ-module is the the Hattori-Stallings rank of the free ℤΓ-module of rank dim_{ℚ}(ℚ ⊗_{ℤΓ} P).

Secondly, we show Theorem 0.5 (Homotopy invariance of combinatorial L^{2}-torsion)

1. If π is a countable residually finite group, then

Φ : Wh(π) ⟶ ℝ^{>0}

given by the Fuglede Kadison determinant is trivial.

2. The combinatorial L^{2}-torsion of a finite CW -complex with residually finite fundamental group whose L^{2}-Betti numbers are all trivial and whose Novikov-Shubin invariants are all positive is a homotopy invariant.

Combinatorial L^{2}-torsion was introduced and studied in [4],[23] and [25]. The equivalence of the two assertions in Theorem 0.5 is proven in [23, Theorem 1.4]. In this context we mention the conjecture that the combinatorial L^{2}-torsion agrees with the analytic L^{2}-torsion defined by Lott [21] and Mathai [28]. The analytic L^{2}-torsion is known to be a diffeomorphism invariant of closed Riemannian manifolds whose L^{2}-Betti numbers all vanish and whose Novikov-Shubin invariants are all positive. The considerations above make it plausible that the analytic L^{2}-torsion is even a homotopy invariant. Notice that the isomorphism conjectures by Farrell and Jones [10] imply both Theorem 0.4 and Theorem 0.5.

Let M be a compact connected orientable irreducible 3-manifold with infinite fundamental group which is Haken, Seifert or hyperbolic. Suppose that its boundary is empty or a disjoint union of incompressible tori. Let M_{1}, … M_{r} be the hyperbolic pieces in the decomposition by incompressible tori into Seifert and hyperbolic pieces. There is the [23, Conjecture 2.3] for the combinatorial L^{2}-torsion ρ(M)

ln(ρ(M)) = \frac{−1}{3π} · ∑_{i=1}^{r} V ol(M_{i}).

This implies in particular that ln(ρ(M)) is always non-positive. We can prove this last statement in

Theorem 0.6 (Bound on combinatorial L^{2}-torsion for 3-manifolds) Let M be a 3manifold satisfying the conditions above or let M be a connected finite 2-dimensional CW - complex with residually finite fundamental group whose L^{2}-Betti numbers are all trivial and whose Novikov-Shubin invariants are all positive. Then we get for the combinatorial L^{2}- torsion

ρ(M) ≤ 1.

The paper is organized as follows :

1. Review of L^{2}-Betti numbers and residually finite groups
2. Proof of Kazhdan’s equality
3. Density functions and their invariants
4. Extending Swan’s theorem from finite to residually finite groups

References

1. Review of L^{2}-Betti numbers and residually finite groups

We give the basic definitions of L^{2}-Betti numbers and of residually finite groups and state some important facts. For the remainder of this section let X be a finite connected CW -complex with fundamental group π and A ⊂ X be a CW -subcomplex. Let p : X̃ ⟶ X be the universal covering and put Ã = p^{−1}(A). We let operate π from the left on the universal covering and on its cellular chain complex.

We recall the basic definitions about L^{2}-Betti numbers. Let l^{2}(π) be the Hilbert space of formal sums ∑_{g∈π} λ_{g} · g with complex coefficients λ_{g} satisfying ∑_{g∈π} |λ_{g}|^{2} < ∞. The von Neumann algebra of π

N (π) = B(l^{2}(π), l^{2}(π))^{π}

is the algebra of bounded π-equivariant operators from l^{2}(π) to l^{2}(π). The von Neumann trace of an element f ∈ N (π) is defined by

tr_{N (π)}(f) = ⟨f(e), e⟩

for e ∈ π the unit element. For a bounded π-equivariant operator f : ⊕_{i=1}^{n}l^{2}(π) ⟶ ⊕_{i=1}^{n}l^{2}(π) define

tr_{N (π)}(f) = ∑_{i=1}^{n} tr_{N (π)} (f_{i,i}) .

A finitely generated Hilbert N (π)-module P is a Hilbert space with isometric π-action such there exists an isometric π-equivariant embedding into ⊕_{i=1}^{r}l^{2}(π) (which is not part of the structure) for some r ∈ ℕ. Let pr : ⊕_{i=1}^{r}l^{2}(π) ⟶ ⊕_{i=1}^{r}l^{2}(π) be a projection whose image is isometrically π-equivariantly isomorphic to P. The von Neumann dimension of P is defined by

dim_{N (π)}(P) = tr_{N (π)}(pr).

If we tensor the ℤπ-chain complex C(X̃, Ã) with l^{2}(π), we obtain a chain complex C^{(2)}(X, A) of finitely generated Hilbert N (π)-modules with bounded π-equivariant operators c_{p}^{(2)} as differentials. Define the p-th L^{2}-homology H_{p}^{(2)}(X, A) by the quotient ker(c_{p}^{(2)})/clos(im(c_{p+1}^{(2)})). Since we divide by the closure of the image, this is again a finitely generated Hilbert N (π)- module. Its von Neumann dimension is the p-th L^{2}-Betti number of (X, A). Notice that L^{2}-homology and L^{2}-cohomology are the same up to isometric π-isomorphism. We give some basic facts about L^{2}-Betti numbers. For more information we refer to [14, section 8].

• The L^{2}-Betti numbers are homotopy invariants.

• If χ(X, A) is the Euler characteristic, then [1]

χ(X, A) = ∑_{p≥0}(−1)^{p} · b_{p}^{(2)}(X, A).

• Let M be a compact Riemannian manifold. The space of harmonic L^{2}-integrable pforms on M̃ satisfying absolute boundary conditions is isometrically π-isomorphic to H_{p}^{(2)}(M) and its von Neumann dimension agrees with b_{p}^{(2)}(M) [8] for closed M, [22, section 6].

• The von Neumann dimension is additive under exact sequences of finitely generated Hilbert N (π)-modules. The von Neumann dimension of P is zero if and only if P is zero. The von Neumann dimension is continuous, i.e., if P_{1} ⊃ P_{2} ⊃ … is a nested sequence of finitely generated Hilbert A-modules then

dim_{A} (⋂_{n=1}^{∞} P_{n}) = lim_{n→∞} dim_{A}(P_{n}).

• If the compact manifold M fibers over S^{1}, then b_{p}(M) = 0 for all p ≥ 0 [24].

• If π_{1}(X) is an extension of a finitely presented group which contains ℤ as a subgroup by an infinite finitely presented group, then b_{1}^{(2)}(X) = 0 [24].

• If π contains a normal infinite amenable subgroup and Bπ is of finite type, then b_{p}^{(2)}(Bπ) = 0 for all p ≥ 0 [5].

• Let M be a closed Riemannian manifold of dimension n with sectional curvature K pinched by

−1 ≤ K ≤ −c^{2} < 1 − (\frac{n − 2}{n − 1})^{2} .

Then the L^{2}-Betti number b_{p}(M) vanishes if |p − n/2| ≥ 1 [7].

• If M is a symmetric space of non-compact type, then the L^{2}-cohomology vanishes possibly except for the middle dimension.

• If M is Kähler hyperbolic in the sense of [13], then the L^{2}-cohomology vanishes except for the middle dimension where it is non-trivial [13].

• Let M be a compact 3-manifold with infinite fundamental group whose prime decomposition M = M_{1}♯M_{2}♯ … ♯M_{r} consists of non-exceptional manifolds M_{i} (i.e., which are finitely covered by a manifold which is homotopy equivalent to a Haken, Seifert or hyperbolic manifold). Then [22]

b_{0}^{(2)}(M) = 0
b_{1}^{(2)}(M) = (r − 1) − ∑_{j=1}^{r} \frac{1}{| π_{1}(M_{j}) |} − χ(M) + | \{C ∈ π_{0}(∂M) s.t. C ≅ S^{2}\} |

b_{2}^{(2)}(M) = (r − 1) − ∑_{j=1}^{r} \frac{1}{| π_{1}(M_{j}) |} + | \{C ∈ π_{0}(∂M) s.t. C ≅ S^{2}\} |
b_{3}^{(2)}(M) = 0.

Two of the outstanding conjectures on L^{2}-cohomology are Singer’s conjecture that the L^{2}-Betti numbers of a closed aspherical manifold vanish possible except in the middle dimension and Atiyah’s conjecture that the L^{2}-Betti numbers of a compact manifold are rational and, provided that the fundamental group is torsionfree, are integers. Atiyah’s conjecture implies Kaplanski’s conjecture that the rational group ring of a group has no zero-divisors if and only if the group is torsionfree.

A group π is called residually finite if for any g ∈ π there is a finite group G and a homomorphism ϕ : π ⟶ G satisfying ϕ(g) ≠ 1. If a group π is countable and residually finite, then there is a nested sequences of normal subgroups … ⊂ Γ_{m+1} ⊂ Γ_{m} ⊂ … ⊂ π such that Γ_{m} ⊂ π is of finite index and ∩_{m=0}^{∞}Γ_{m} = {1}. We list some basic facts about residually finite groups, for more information we refer to the survey article [27].

• The free product of two residually finite groups is again residually finite [16], [6, page 27].

• A finitely generated residually finite group has a solvable word problem [29].

• The automorphism group of a finitely generated residually finite group is residually finite [3].

• A finitely generated residually finite group is hopfian, i.e, any surjective endomorphism is an automorphism [26], [30, Corollary 41.44].

• Let π be a finitely generated group possessing a faithful representation into GL(n, F) for F a field. Then π is residually finite [26], [32, Theorem 4.2].

• Let π be a finitely generated group. Let π^{rf} be the quotient of π by the normal subgroup which is the intersection of all normal subgroups of π of finite index. The group π^{rf} is residually finite and any finite-dimensional representation of π in a field factorizes over the canonical projection π ⟶ π^{rf}.

• The fundamental group of a compact 3-manifold whose prime decomposition consists of non-exceptional manifolds (i.e., which are finitely covered by a manifold which is homotopy equivalent to a Haken, Seifert or hyperbolic manifold) is residually finite [17, page 380] .

• There is an infinite group with four generators and four relations which has no finite quotient except the trivial one [18].

2. Proof of Kazhdan’s equality

For the remainder of this section we fix the following

Data 2.1

• a countable residually finite group π.

• a nested sequence … ⊂ Γ_{m+1} ⊂ Γ_{m} ⊂ … ⊂ π such that Γ_{m} ⊂ π is a normal subgroup of finite index [π : Γ_{m}] and ∩_{m=0}^{∞}Γ_{m} = {1}.

• a ℤπ-linear map f : ⊕_{i=1}^{a}ℤπ ⟶ ⊕_{i=1}^{b}ℤπ ( of left ℤπ-modules).

We introduce the following notation. We have explained N (π), tr_{N (π)} and dim_{N (π)} in section 1.

Notation 2.2

• Let B ∈ M(a, b, ℤπ) be the matrix describing f, i.e. f(x) is given by xB.

• Let f_{m} : ⊕_{i=1}^{a}ℂ[Γ_{m}\π] ⟶ ⊕_{i=1}^{b}ℂ[Γ_{m}\π] be the ℂ-linear map induced by f. Denote by f^{(2)} : ⊕_{i=1}^{a}l^{2}(π) ⟶ ⊕_{j=1}^{b}l^{2}(π) the bounded π-equivariant operator induced by f.

• Let {P (λ) | λ ∈ [0, ∞)} be the right continuous spectral family of the positive operator (f^{(2)})^{∗}f^{(2)}. The spectral density function of f^{(2)} is defined by

F : [0, ∞) ⟶ [0, a] λ ↦ dim_{N (π)} (im(P (λ))) = tr_{N (π)} (P (λ)) .

• The L^{2}-Betti number of f is defined by

b^{(2)}(f) = dim_{N (π)}(ker(f^{(2)})) = F (0).

• Let E_{m}(λ) be the ordered set of eigenvalues μ of f_{m}^{∗}f_{m} satisfying μ ≤ λ listed with multiplicity.

• Define

F_{m}^{rf}(λ) = \frac{|E_{m}(λ)|}{[π : Γ_{m}]}
\overline{F^{rf}}(λ) = lim sup_{m→∞} F_{m}^{rf}(λ)
\underline{F^{rf}}(λ) = lim inf_{m→∞} F_{m}^{rf}(λ)
\overline{F^{rf}}^{+}(λ) = lim_{δ→0+} \overline{F^{rf}}(λ + δ)
\underline{F^{rf}}^{+}(λ) = lim_{δ→0+} \underline{F^{rf}}(λ + δ).

Notice that all functions appearing in Notation 2.2 are monotone increasing and, possibly except \underline{F^{rf}} and \overline{F^{rf}}, are right continuous. The main technical result of this paper is

Theorem 2.3 1. F (λ) = \overline{F^{rf}}^{+}(λ) = \underline{F^{rf}}^{+}(λ).

2. The functions \overline{F^{rf}} and \underline{F^{rf}} are right continuous at zero. We have

\overline{F^{rf}}(0) = \overline{F^{rf}}^{+}(0) = F (0) = \underline{F^{rf}}(0) = \underline{F^{rf}}^{+}(0) = lim_{m→∞} F_{m}^{rf}(0) = lim_{m→∞} \frac{|E_{m}(0)|}{[π : Γ_{m}]}.

3. We have for 0 < λ < 1 if K is the constant introduced in 2.4

F (λ) − F (0) ≤ \frac{a · ln(K^{2})}{− ln(λ)}.

Before we give its proof, we explain how it implies Theorem 0.1, Theorem 0.2 and Theorem 0.3.

We begin with Theorem 0.1. Let c_{p+1} and c_{p} be the differentials of C(X̃, Ã) Denote by n_{p} the number of p-cells in X − A. Then we get for any finite-dimensional complex π-representation V if c_{p}^{V} is the linar map obtained by tensoring with V

b_{p}(X, A; V) = dim_{ℂ} (ker(c_{p}^{V})/im(c_{p+1}^{V}))
= dim_{ℂ} (ker(c_{p}^{V})) − dim_{ℂ} (im(c_{p+1}^{V}))
= dim_{ℂ} (ker(c_{p}^{V})) + dim_{ℂ} (ker(c_{p+1}^{V})) − n_{p+1} · dim_{ℂ}(V)
= dim_{ℂ} (ker((c_{p}⊕_{p+1})^{V})) − n_{p+1} · dim_{ℂ}(V).

Similiarly we get

b_{p}^{(2)}(X, A) = b^{(2)} ((c_{p} ⊕ c_{p+1})^{(2)}) − n_{p+1}.

Now Theorem 0.1 follows from Theorem 2.3.2 applied to f = c_{p} ⊕ c_{p+1} since

\frac{b_{p}(X_{m}, A_{m})}{[π : Γ_{m}]} = \frac{b_{p}(X, A; ℂ[Γ_{m}\π])}{[π : Γ_{m}]} = F_{m}^{rf}(0) − n_{p+1}

and

b_{p}^{(2)}(X, A) = F (0) − n_{p+1}.

Next we prove Theorem 0.2 and 0.3. Let f : C_{p}(M̃) ⟶ C_{p}(M̃) be the ℂπ-linear map c_{p+1}c_{p}^{∗} + c_{p}^{∗}c_{p} for a given non-negative integer p. Then F and the analytic spectral density function F^{p} defined in the introduction are dilitationally equivalent [9],[15] and [22] , i.e., there are constants ϵ > 0 and D > 1 such that for all λ ∈ (0, ϵ)

F (D^{−1} · λ) ≤ F^{p}(λ) ≤ F (D · λ).

Hence it suffices to prove the claims in Theorems 0.2 and 0.3 for F instead of F^{p}. These follow from Theorem 2.3.

The proof of Theorem 2.3 needs some preparations.

For u = ∑_{g∈π} λ_{g} · g ∈ ℂπ define |u|_{1} = ∑_{g∈π} |λ_{g}|. Recall that

B ∈ M(a, b, ℤπ) describes f. For the sequel fix a real number K satisfying

2.4 K ≥ a · ∑_{j=1}^{b} max {|B_{i,j}|_{1} | i = 1, 2 … a}.

Lemma 2.5 The number K is greater or equal to the operator norm of f^{(2)} and f_{m}.

\underline{\underline{Proof}}: We give the proof only for f^{(2)}, the one for f_{m} is completely analogous. Consider x ∈ l^{2}(π) and u = ∑_{g∈π} λ_{g} · g ∈ ℂπ. Then

|ux| = |∑_{g∈π} λ_{g} · g · x| ≤ ∑_{g∈π} |λ_{g}| · |g · x| ≤ ∑_{g∈π} |λ_{g}| · |x| ≤ |u|_{1} · |x| .

We conclude for x = (x_{1}, x_{2}, … x_{a}) ∈ ⊕_{i=1}^{a}l^{2}(π)

|f^{(2)}(x)|^{2} = ∑_{j=1}^{b} |∑_{i=1}^{a} x_{i} · B_{i,j}|^{2}

≤ ∑_{j=1}^{b} (∑_{i=1}^{a} |x_{i} · B_{i,j}|)^{2}

≤ ∑_{j=1}^{b} (∑_{i=1}^{a} |x_{i}| · |B_{i,j}|_{1})^{2}

≤ ∑_{j=1}^{b} (∑_{i=1}^{a} |x_{i}| · max{|B_{i,j}|_{1} | 1 ≤ i ≤ a})^{2}

≤ ∑_{j=1}^{b} (max{|B_{i,j}|_{1} | 1 ≤ i ≤ a})^{2} · (∑_{i=1}^{a} |x_{i}|)^{2}

≤ ∑_{j=1}^{b} (max{|B_{i,j}|_{1} | 1 ≤ i ≤ a})^{2} · a^{2} · ∑_{i=1}^{a} |x_{i}|^{2}

≤ (∑_{j=1}^{b} max{|B_{i,j}|_{1} | 1 ≤ i ≤ a} · a)^{2} · |x|^{2}

This finishes the proof of Lemma 2.5.

For ∑_{g∈π} λ_{g} · g ∈ ℤπ define

tr_{ℤπ} (∑_{g∈π} λ_{g} · g) = λ_{e}

where e ∈ π is the unit element. Recall that B ∈ M(a, b, ℤπ) is the matrix describing f. Put

tr_{ℤπ} (f) = ∑_{i=1}^{a} tr_{ℤπ} (B_{i,i})

provided a = b. Denote by B^{∗} the matrix (\overline{B_{j,i}}) in M(b, a, ℤπ) if ¯ : ℤπ ⟶ ℤπ is the involution sending ∑_{g∈π} λ_{g} · g to ∑_{g∈π} λ_{g} · g^{−1}. Let f^{∗} : ⊕_{j=1}^{b}ℤπ ⟶ ⊕_{i=1}^{a}ℤπ be the ℤπlinear map described by B^{∗}. Notice that (f^{(2)})^{∗} = (f^{∗})^{(2)}. One easily checks that f^{∗}f is described by BB^{∗} and

tr_{N (π)} ((f^{(2)})^{∗}f^{(2)}) = tr_{ℤπ} (f^{∗}f) .

The proof of the following lemma is the only place where we need the assumption that Γ_{m} ⊂ π is normal.

Lemma 2.6 Let p(μ) be a polynomial. There is a number m_{0} ((Γ_{m})_{m≥0}, p) such that for all m ≥ m_{0} ((Γ_{m})_{m≥0}, p)

tr_{ℤπ} (p(f^{∗}f)) = \frac{1}{[π : Γ_{m}]} · tr_{ℂ} (p(f_{m}^{∗}f_{m})) .

\underline{\underline{Proof}}: Recall that the matrix B ∈ M(a, b, ℤπ) describes f. Fix elements g_{0}, g_{1}, …, g_{r} ∈ π and λ_{0}, λ_{1}, …, λ_{r} ∈ ℂ such that g_{0} = e, g_{i} ≠ e and λ_{i} ≠ 0 for 1 ≤ i ≤ r and

∑_{j=1}^{a} (p(BB^{∗}))_{j,j} = ∑_{i=0}^{r} λ_{i}g_{i}.

We get:

tr_{ℤπ} (p(f^{∗}f)) = λ_{0}

and

tr_{ℂ} (p(f_{m}^{∗}f_{m})) = tr_{ℂ} (∑_{i=1}^{r} λ_{i} · r(g_{i}) : ℂ[Γ_{m}\π] ⟶ ℂ[Γ_{m}\π])

where r(g_{i}) is right multiplication with g_{i}. Since the intersection of the Γ_{m}-s is trivial, there is a number m_{0} such that for m ≥ m_{0} none of the elements g_{i} for 1 ≤ i ≤ r lies in Γ_{m}. Since Γ_{m} ⊂ π is normal, we conclude for m ≥ m_{0} and 1 ≤ i ≤ r

tr_{ℂ}(r(g_{i}) : ℂ[Γ_{m}\π] ⟶ ℂ[Γ_{m}\π]) = 0.

This implies for m ≥ m_{0}

tr_{ℤπ} (p(f^{∗}f)) = \frac{1}{[π : Γ_{m}]} · tr_{ℂ} (p(f_{m}^{∗}f_{m})) .

Lemma 2.7 Let p_{n}(μ) be a sequence of polynomials such that for the characteristic function χ_{[0,λ]}(μ) of the interval [0, λ] and an appropriate real number L

lim_{n→∞} p_{n}(μ) = χ_{[0,λ]}(μ) and |p_{n}(μ)| ≤ L

holds for each μ ∈ [0, |f^{(2)}|^{2}]. Then

lim_{n→∞} tr_{ℤπ} (p_{n}(f^{∗}f)) = F (λ).

\underline{\underline{Proof}}: Let \{P (λ) | λ ∈ [0, |f^{(2)}|^{2}\} be the right continuous spectral family of (f^{(2)})^{∗}f^{(2)}. We get using Lebesgues Theorem of Majorized Convergence and the fact that the von Neumann trace is linear, monotone and ultraweakly continuous

lim_{n→∞} tr_{ℤπ} (p_{n}(f^{∗}f)) = lim_{n→∞} tr_{N (π)} (p_{n}((f^{(2)})^{∗}f^{(2)}))
= lim_{n→∞} tr_{N (π)} (∫_{0}^{||f||^{2}} p_{n}(λ) dP (λ))

= lim_{n→∞} ∫_{0}^{||f||^{2}} p_{n}(λ) dF (λ)
= ∫_{0}^{||f||^{2}} ( lim_{n→∞} p_{n}(λ)) dF (λ)
= ∫_{0}^{||f||^{2}} χ_{[0,λ]} dF (λ)
= F (λ)

This finishes the proof of Lemma 2.7.

Lemma 2.8 Let g : V ⟶ W be a linear map of finite-dimensional Hilbert spaces V and W. Let p(t) = det(tid − g^{∗}g) be the characteristic polynomial of g^{∗}g. Write p(t) = t^{k} · q(t) for a polynomial q satisfying q(0) ≠ 0. Let K be a positive real number such that K ≥ 1 and K ≥ ||g|| and let C be a positive real number such that C ≤ |q(0)|. Let E(λ) be the ordered set of eigenvalues μ of g^{∗}g satisfying μ ≤ λ listed with multiplicity. Then we get for 0 < λ < 1:

\frac{|E(λ)| − |E(0)|}{dim_{ℂ}(V)} ≤ \frac{− ln(C)}{dim_{ℂ}(V) · (− ln(λ))} + \frac{ln(K^{2})}{− ln(λ)}.

\underline{\underline{Proof}}: Let 0 ≤ μ_{0} ≤ μ_{1} ≤ … ≤ μ_{dim_{ℂ}(V)} be the eigenvalues of g^{∗}g listed with multiplicity. Let r be the integer for which μ_{i} = 0 for i ≤ r and μ_{i} > 0 for i > r holds. Let s be the integer for which μ_{i} ≤ λ for i ≤ s and μ_{i} > λ for i > s is valid. Then

q(t) = ∏_{i=r+1}^{dim_{ℂ}(V)}(t − μ_{i})

and

|E(λ)| − |E(0)| = s − r.

We conclude for λ < 1 ( and hence ln(λ) < 0) since μ_{i} ≤ K^{2} for all i

∏_{i=r+1}^{dim_{ℂ}(V)}(−μ_{i}) = q(0)

∏_{i=r+1}^{s} μ_{i} = |q(0)| · ∏_{i=s+1}^{dim_{ℂ}(V)} μ_{i}^{−1}

∏_{i=r+1}^{s} λ ≥ C · ∏_{i=s+1}^{dim_{ℂ}(V)} K^{−2}
λ^{|E(λ)|−|E(0)|} ≥ C · K^{−2 dim_{ℂ}(V)}
(|E(λ)| − |E(0)|) · ln(λ) ge ln(C) − dim_{ℂ}(V) · ln(K^{2})
\frac{|E(λ)| − |E(0)|}{dim_{ℂ}(V)} ≤ \frac{− ln(C)}{dim_{ℂ}(V) · (− ln(λ))} + \frac{ln(K^{2})}{− ln(λ)}.

This finishes the proof of Lemma 2.8.

Next we proof Theorem 2.3.1. Fix λ ≥ 0. Define for n ≥ 1 a continuous function

f_{n} : ℝ ⟶ ℝ μ ↦ \{ \begin{matrix}1 + 1/n & μ ≤ λ \\ 1 + 1/n − n · (μ − λ) & λ ≤ μ ≤ λ + \frac{1}{n} \\ 1/n & λ + \frac{1}{n} ≤ μ \end{matrix}

Obviously χ_{[0,λ]}(μ) < f_{n+1}(μ) < f_{n}(μ) and f_{n}(μ) converges for n → ∞ to χ_{[0,λ]}(μ) for all μ ∈ [0, ∞). For each n choose a polynomial p_{n} such that χ_{0,λ]}(μ) < p_{n}(μ) < f_{n}(μ) holds for all μ ∈ [0, K^{2}]. Such polynomial can be found by approximating f_{n+1} sufficiently close. Hence

χ_{[0,λ]}(μ) < p_{n}(μ) < 2 and lim_{n→∞} p_{n}(μ) = χ_{[0,λ]}(μ) for μ ∈ [0, K^{2}].

Recall that E_{m}(λ) is the ordered set of eigenvalues μ of f_{m}^{∗}f_{m} satisfying μ ≤ λ listed with multiplicity. We conclude since K^{2} ≥ ||f_{m}^{∗}f_{m}|| by Lemma 2.5

\frac{1}{[π : Γ_{m}]} · tr_{ℂ} (p_{n}(f_{m}^{∗}f_{m}))

= \frac{1}{[π : Γ_{m}]} · ∑_{μ∈E_{m}(K^{2})} p_{n}(μ)

= \frac{|E_{m}(λ)|}{[π : Γ_{m}]} + \frac{1}{[π : Γ_{m}]} ·

( ∑_{μ∈E_{m}(λ)}(p_{n}(μ) − 1) + ∑_{μ∈E_{m}(λ+1/n)−E_{m}(λ)} p_{n}(μ) + ∑_{μ∈E_{m}(K^{2})−E_{m}(λ+1/n)} p_{n}(μ))

This implies

F_{m}^{rf}(λ) = \frac{|E_{m}(λ)|}{[π : Γ_{m}]} ≤ \frac{1}{[π : Γ_{m}]} · tr_{ℂ} (p_{n}(f_{m}^{∗}f_{m}))

We conclude further

\frac{1}{[π : Γ_{m}]} · tr_{ℂ} (p_{n}(f_{m}^{∗}f_{m}))

≤ \frac{|E_{m}(λ)|}{[π : Γ_{m}]} + \frac{sup\{p_{n}(μ) − 1 | μ ∈ [0, λ]\} · |E_{m}(λ)|}{[π : Γ_{m}]}

+\frac{sup\{p_{n}(μ) | μ ∈ [λ, λ + 1/n]\} · |E_{m}(λ + 1/n) − E_{m}(λ)|}{[π : Γ_{m}]}

+\frac{sup\{p_{n}(μ) | μ ∈ [λ + 1/n, K^{2}]\} · |E_{m}(K^{2}) − E_{m}(λ + 1/n)|}{[π : Γ_{m}]}

≤ \frac{|E_{m}(λ)|}{[π : Γ_{m}]} + \frac{1/n · |E_{m}(λ)|}{[π : Γ_{m}]} + \frac{(1 + 1/n) · |E_{m}(λ + 1/n) − E_{m}(λ)|}{[π : Γ_{m}]} +

\frac{1/n · |E_{m}(K^{2}) − E_{m}(λ + 1/n)|}{[π : Γ_{m}]}

≤ \frac{|E_{m}(λ + 1/n)|}{[π : Γ_{m}]} + \frac{1}{n} · \frac{|E_{m}(K^{2})|}{[π : Γ_{m}]}
≤ F_{m}^{rf}(λ + 1/n) + \frac{1}{n} · a

Because of Lemma 2.6 there is for each n ∈ ℕ a number m(n) such that for all m ≥ m(n)

\frac{1}{[π : Γ_{m}]} · tr_{ℂ} (p_{n}(f_{m}^{∗}f_{m})) = tr_{ℤπ} (p_{n}(f^{∗}f)) .

Hence we get for m ≥ m(n):

F_{m}^{rf}(λ) ≤ tr_{ℤπ} (p_{n}(f^{∗}f)) ≤ F_{m}^{rf}(λ + 1/n) + \frac{1}{n} · a

Taking for fixed n the limit superior respectively limit inferior for m → ∞ gives

\overline{F^{rf}}(λ) ≤ tr_{ℤπ} (p_{n}(f^{∗}f)) ≤ \underline{F^{rf}}(λ + 1/n) + \frac{a}{n}

Now we take the limit for n → ∞ and obtain because of Lemma 2.7

\overline{F^{rf}}(λ) ≤ F (λ) ≤ \underline{F^{rf}}^{+}(λ)

We have for all ϵ > 0

F (λ) ≤ \underline{F^{rf}}^{+}(λ) ≤ \underline{F^{rf}}(λ + ϵ) ≤ \overline{F^{rf}}(λ + ϵ) ≤ F (λ + ϵ)

Since lim_{ϵ→0+} F (λ + ϵ) = F (λ) we get

F (λ) = \overline{F^{rf}}^{+}(λ) = \underline{F^{rf}}^{+}(λ).

This finishes the proof of Theorem 2.3.1.

Next we show Theorem 2.3.2 and 2.3.3. We want to apply Lemma 2.8 to the functions f_{m} simultaneously. Let p_{m} be the characteristic polynomial of f_{m} and p_{m}(t) = t^{r_{m}} · q_{m}(t) for appropriate r_{m} and q_{m}(t) with q_{m}(0) ≠ 0. Since f is a ℤπ-linear map, the matrix describing f_{m} has integers as entries. Hence p_{m} is an integer polynomial and |q_{m}(0)| ≥ 1. We get for 0 < λ < 1 from Lemma 2.8 and Lemma 2.5 with the constants K and C = 1 which are independent of m

\frac{F_{m}^{rf}(λ) − F_{m}^{rf}(0)}{a} ≤ \frac{− ln(1)}{a · [π : Γ_{m}] · (− ln(λ))} + \frac{ln(K^{2})}{− ln(λ)}

and hence

F_{m}^{rf}(λ) ≤ F_{m}^{rf}(0) + \frac{a · ln(K^{2})}{− ln(λ)}.

Taking the limit inferior and limit superior for m → ∞ gives:

\underline{F^{rf}}(λ) ≤ \underline{F^{rf}}(0) + \frac{a · ln(K^{2})}{− ln(λ)}

and

\overline{F^{rf}}(λ) ≤ \overline{F^{rf}}(0) + \frac{a · ln(K^{2})}{− ln(λ)}.

Taking the limit for λ → 0 gives

\underline{F^{rf}}(0) = \underline{F^{rf}}^{+}(0)

and

\overline{F^{rf}}(0) = \overline{F^{rf}}^{+}(0).

We already know \overline{F^{rf}}^{+}(0) = F (0) = \underline{F^{rf}}(0) from Theorem 2.3.1. This proves Theorem 2.3.2. Since \frac{a·ln(K^{2})}{− ln(λ)} is right continuous, we conclude:

\overline{F^{rf}}^{+}(λ) ≤ F (0) + \frac{a · ln(K^{2})}{− ln(λ)}.

Now Theorem 2.3.3 follows from Theorem 2.3.1. This finishes the proof of Theorem 2.3.

3. Density functions and their invariants

We consider density functions and their invariants like Betti numbers, Novikov-Shubin invariants and determinants. We show that these invariants are not changed if the density function in question is made right continuous. This is useful in connection with Theorem 2.3.1. We apply this to the spectral density function of f (given in Data 2.1) and will express these invariants for f^{(2)} in terms of the f_{m}-s. We will prove Theorem 0.5 and Theorem 0.6.

Definition 3.1 A density function is a monotone increasing function F : [0, ∞) ⟶ [0, ∞). Put

b^{(2)}(F) = lim_{λ→0+} F (λ).

Define

α(F) = lim inf_{λ→0^{+}} \frac{ln(F (λ) − b^{(2)}(F))}{ln(λ)} ∈ [0, ∞],

provided that F (λ) > b^{(2)}(F) holds for all λ > 0. Otherwise, we put α(F) = ∞^{+}. Suppose there is a K satisfying F (λ) = F (K^{2}) for all λ ≥ K^{2}. If the integral

∫_{0+}^{K^{2}} \frac{F (λ) − b^{(2)}(F)}{λ}dλ

exists as real number, put

det(F) = exp (\frac{1}{2} · (∫_{0+}^{K^{2}} \frac{F (λ) − b^{(2)}(F)}{λ}dλ))

If the integral does not exist, define det(F) = 0.

It is not hard to check that the definition of det(F) is independent of the choice of K and that the integral exists as real number if α(F) > 0. If g : ⊕_{i=1}^{r}l^{2}(π) ⟶ ⊕_{i=1}^{r}l^{2}(π) is an invertible bounded π-equivariant operator its Fuglede Kadison determinant is defined by [11]

det(g) = exp (\frac{1}{2} · tr_{N (π)} (ln (g^{∗}g))) ∈ (0, ∞).

It coincides with the determinant of Definition 3.1 applied to the spectral density function of g by [23, Lemma 4.2].

Lemma 3.2 Let F : [0, ∞) ⟶ [0, ∞) be a density function. Define the right continuous density function

F^{+} : [0, ∞) ⟶ [0, ∞) λ ↦ lim_{δ→0+} F (λ + δ).

Then

b^{(2)}(F) = b^{(2)}(F^{+}) = F^{+}(0), α(F) = α(F^{+}) and det(F) = det(F^{+}).

\underline{\underline{Proof}}: Obviously we have b^{(2)}(F) = b^{(2)}(F^{+}) = F^{+}(0). We have for all λ > 0

F (λ) ≤ F^{+}(λ) ≤ F (2 · λ)

and hence for 0 < λ < 1

\frac{F (λ) − b^{(2)}(F)}{ln(λ)} ≥ \frac{F^{+}(λ) − b^{(2)}(F^{+})}{ln(λ)} ≥ \frac{F (2 · λ) − b^{(2)}(F)}{ln(λ)} = \frac{F (2 · λ) − b^{(2)}(F)}{ln(2 · λ)}·(1 + \frac{ln(2)}{ln(λ)})

Now the we get by taking the limit inferior for λ → 0+

α(F) ≥ α(F^{+}) ≥ α(F)

and thus the claim

α(F) = α(F^{+}).

For ϵ > 0 and δ > 0 we estimate

0 ≤ ∫_{ϵ}^{K^{2}} \frac{F (λ + δ) − b^{(2)}(F)}{λ}dλ − ∫_{ϵ}^{K^{2}} \frac{F (λ) − b^{(2)}(F)}{λ}dλ
= ∫_{ϵ}^{K^{2}} \frac{F (λ + δ) − b^{(2)}(F)}{λ + δ} · \frac{λ + δ}{λ} dλ − ∫_{ϵ}^{K^{2}} \frac{F (λ) − b^{(2)}(F)}{λ}dλ
= ∫_{ϵ}^{K^{2}} \frac{F (λ + δ) − b^{(2)}(F)}{λ + δ}dλ + δ · ∫_{ϵ}^{K^{2}} \frac{F (λ + δ) − b^{(2)}(F)}{(λ + δ) · λ}dλ − ∫_{ϵ}^{K^{2}} \frac{F (λ) − b^{(2)}(F)}{λ}dλ
= ∫_{ϵ+δ}^{K^{2}+δ} \frac{F (λ) − b^{(2)}(F)}{λ}dλ − ∫_{ϵ}^{K^{2}} \frac{F (λ) − b^{(2)}(F)}{λ}dλ + δ · ∫_{ϵ}^{K^{2}} \frac{F (λ + δ) − b^{(2)}(F)}{(λ + δ) · λ}dλ
= ∫_{K^{2}}^{K^{2}+δ} \frac{F (λ) − b^{(2)}(F)}{λ}dλ − ∫_{ϵ}^{ϵ+δ} \frac{F (λ) − b^{(2)}(F)}{λ}dλ + δ · ∫_{ϵ}^{K^{2}} \frac{F (λ + δ) − b^{(2)}(F)}{(λ + δ) · λ}dλ
≤ δ · \frac{F (K^{2})}{K^{2}} + δ · K^{2} · \frac{F (K^{2})}{ϵ^{2}}
= δ · (\frac{F (K^{2})}{K^{2}} + \frac{K^{2} · F (K^{2})}{ϵ^{2}})

We conclude from Lebesgue’s Theorem of Majorized Convergence

∫_{ϵ}^{K^{2}} \frac{F^{+}(λ) − b^{(2)}(F^{+})}{λ}dλ

= ∫_{ϵ}^{K^{2}} lim_{δ→0} \frac{F (λ + δ) − b^{(2)}(F)}{λ}dλ
= lim_{δ→0} ∫_{ϵ}^{K^{2}} \frac{F (λ + δ) − b^{(2)}(F)}{λ}dλ
= ∫_{ϵ}^{K^{2}} \frac{F (λ) − b^{(2)}(F)}{λ}dλ

From Levi’s Theorem of Monotone Convergence we conclude

∫_{0+}^{K^{2}} \frac{F^{+}(λ) − b^{(2)}(F^{+})}{λ}dλ

= lim_{ϵ→∞} ∫_{ϵ}^{K^{2}} \frac{F^{+}(λ) − b^{(2)}(F^{+})}{λ}dλ
= lim_{ϵ→∞} ∫_{ϵ}^{K^{2}} \frac{F (λ) − b^{(2)}(F)}{λ}dλ
= ∫_{0+}^{K^{2}} \frac{F (λ) − b^{(2)}(F)}{λ}dλ

This finishes the proof of Lemma 3.2. Now we consider Data 2.1 and use Notation 2.2

Lemma 3.3

1.

∫_{0+}^{K^{2}} \frac{F (λ) − F (0)}{λ} dλ ≤ lim inf_{m→∞} ∫_{0+}^{K^{2}} \frac{F_{m}(λ) − F_{m}(0)}{λ} dλ

2. If

∫_{0+}^{K^{2}} sup \{ \frac{F_{m}(λ) − F_{m}(0)}{λ}| m ≥ 0\} dλ < ∞

then

∫_{0+}^{K^{2}} \frac{F (λ) − F (0)}{λ} dλ = lim_{m→∞} ∫_{0+}^{K^{2}} \frac{F_{m}(λ) − F_{m}(0)}{λ} dλ

∫_{0+}^{K^{2}} \frac{F (λ) − F (0)}{λ} dλ = lim_{ϵ→0} lim_{m→∞} ∫_{ϵ}^{K^{2}} \frac{F_{m}(λ) − F_{m}(0)}{λ} dλ

\underline{\underline{Proof}}: 1.) We get from Theorem 2.3, Lemma 3.2 and Levi’s Theorem of Monotone Convergence

∫_{0+}^{K^{2}} \frac{F (λ) − F (0)}{λ} dλ

= ∫_{0+}^{K^{2}} \frac{\underline{F^{rf}}^{+}(λ) − \underline{F^{rf}}^{+}(0)}{λ} dλ
= ∫_{0+}^{K^{2}} \frac{\underline{F^{rf}}(λ) − \underline{F^{rf}}(0)}{λ} dλ
= ∫_{0+}^{K^{2}} \frac{lim inf_{m→∞} F_{m}(λ) − lim_{m→∞} F_{m}(0)}{λ} dλ
= ∫_{0+}^{K^{2}} lim inf_{m→∞} \frac{F_{m}(λ) − F_{m}(0)}{λ} dλ
= ∫_{0+}^{K^{2}} lim_{m→∞} (inf \{ \frac{F_{n}(λ) − F_{n}(0)}{λ}| n ≥ m\}) dλ
= lim_{m→∞} ∫_{0+}^{K^{2}} inf \{ \frac{F_{n}(λ) − F_{n}(0)}{λ}| n ≥ m\} dλ

≤ lim_{m→∞} (inf \{∫_{0+}^{K^{2}} \frac{F_{n}(λ) − F_{n}(0)}{λ} dλ| n ≥ m\})

= lim inf_{m→∞} ∫_{0+}^{K^{2}} \frac{F_{m}(λ) − F_{m}(0)}{λ} dλ

2.) We get from Theorem 2.3, Lemma 3.2 and Lebegues Theorem of Majorized Convergence

lim sup_{m→∞} ∫_{0+}^{K^{2}} \frac{F_{m}(λ) − F_{m}(0)}{λ} dλ

= lim_{m→∞} (sup \{∫_{0+}^{K^{2}} \frac{F_{n}(λ) − F_{n}(0)}{λ} dλ| n ≥ m\})

≤ lim_{m→∞} ∫_{0+}^{K^{2}} sup \{ \frac{F_{n}(λ) − F_{n}(0)}{λ}| n ≥ m\} dλ

= ∫_{0+}^{K^{2}} lim_{m→∞} (sup \{ \frac{F_{n}(λ) − F_{n}(0)}{λ}| n ≥ m\}) dλ
= ∫_{0+}^{K^{2}} lim sup_{m→∞} \frac{F_{m}(λ) − F_{m}(0)}{λ} dλ
= ∫_{0+}^{K^{2}} \frac{lim sup_{m→∞} F_{m}(λ) − lim_{m→∞} F_{m}(0)}{λ} dλ
= ∫_{0+}^{K^{2}} \frac{\overline{F^{rf}}(λ) − \overline{F^{rf}}(0)}{λ} dλ

= ∫_{0+}^{K^{2}} \frac{\overline{F^{rf}}^{+}(λ) − \overline{F^{rf}}^{+}(0)}{λ} dλ

∫_{0+}^{K^{2}} \frac{F (λ) − F (0)}{λ} dλ

Now assertion 2.) follows using assertion 1.)

3.) If we substitute the lower bound 0+ in the calculations of assertion 1.) and assertion 2.) by ϵ > 0, they remain true without the additional assumption in assertion 2.) that a certain integral is smaller than ∞. Hence we get

∫_{ϵ}^{K^{2}} \frac{F (λ) − F (0)}{λ} dλ = lim_{m→∞} ∫_{ϵ}^{K^{2}} \frac{F_{m}(λ) − F_{m}(0)}{λ} dλ

Now we derive from Levi’s Theorem of Monotone Convergence

∫_{0+}^{K^{2}} \frac{F (λ) − F (0)}{λ} dλ = lim_{ϵ→0} ∫_{ϵ}^{K^{2}} \frac{F (λ) − F (0)}{λ} dλ.

This finishes the proof of Lemma 3.3.

The main technical result of this section is

Theorem 3.4 1. Let p(t) = det_{ℂ}(t − f_{m}^{∗}f_{m}) be the characteristic polynomial of the ℂlinear endomorphism f_{m}^{∗}f_{m}. Write p(t) = t^{r} · q(t) for appropriate r and q satisfying

q(0) ≠ 0. Then

det(F_{m}) = ( ∏_{μ∈E_{m}(K^{2}),μ>0} μ)^{\frac{1}{2·[π:Γm])}} = |q(0)|^{\frac{1}{2·[π:Γm])}} ≥ 1

where E_{m}(K^{2}) is the ordered set of eigenvalues of f_{m}^{∗}f_{m} counted with multiplicity.

det(F) ≥ lim sup_{m→∞} det(F_{m}) ≥ 1.

3. Suppose there are constants α > 0, ϵ > 0 and C > 0 such that for all λ ∈ (0, ϵ) and

m ≥ 0

F_{m}(λ) − F_{m}(0) ≤ C · λ^{α}.

Then

α(F) ≥ α
det(F) = lim_{m→∞} det(F_{m}).

\underline{\underline{Proof}}: 1.) follows from the definition of det(F_{m}) by an elementary calculation since F_{m}(λ) is a step function and

p(t) = ∏_{μ∈E_{m}(K^{2})} (t − μ).

Since f_{m} is described by a matrix with integral coefficients, p(t) is an integer polynomial. Hence q(0) an integer and |q(0)| ≥ 1.

2.) From Theorem 2.3.2 and Lemma 3.3.1

2 · ln(det(F)) =
= ln(K^{2}) · (F (K^{2}) − b^{(2)}(F)) − ∫_{0+}^{K^{2}} \frac{F (λ) − b^{(2)}(F)}{λ}dλ
≥ ln(K^{2}) · (F_{m}(K^{2}) − lim_{m→∞} F_{m}(0)) − lim inf_{m→∞} ∫_{0+}^{K^{2}} \frac{F_{m}(λ) − F_{m}(0)}{λ} dλ

= lim sup_{m→∞} (ln(K^{2}) · (F_{m}(K^{2}) − F_{m}(0)) − ∫_{0+}^{K^{2}} \frac{F_{m}(λ) − F_{m}(0)}{λ} dλ)
= lim sup_{m→∞} 2 · ln(det(F_{m}))

and the claim follows.

3.) We get from the assumption and Theorem 2.3 for small 0 < λ < 0.

\overline{F^{rf}}(λ) − b^{(2)}(\overline{F^{rf}}) = lim sup_{m→∞} F_{m}(λ) − lim_{m→∞} F_{m}(0) ≤ sup \{F_{m}(λ) − F_{m}(0)| m ≥ 0\} ≤ C · λ^{α}

Hence α(\overline{F^{rf}}) ≥ α Since α(F) = α(\overline{F^{rf}}^{+}) = α(\overline{F^{rf}}) by Theorem 2.3.1 and Lemma 3.2 we get α(F) ≥ α. Because

∫_{0+}^{K^{2}} sup \{ \frac{F_{m}(λ) − F_{m}(0)}{λ}| m ≥ 0\} dλ ≤ C · ∫_{0+}^{K^{2}} \frac{λ^{α}}{λ} dλ < ∞

holds, det(F) = lim_{m→∞} det(F_{m}) follows from Lemma 3.3.2. This finishes the proof of Theorem 3.4

Now we can prove Theorem 0.5. We have already mentioned in the introduction that assertions 1.) and 2.) are equivalent so that we only have to prove assertion 2.). Suppose that f : ⊕_{i=1}^{a}ℤπ ⟶ ⊕_{i=1}^{a}ℤπ is a ℤπ-automorphism. Choose K such big that the inequality 2.4 holds for both f and f^{−1}. Then the operator norms of f^{(2)}, (f^{(2)})^{−1}, f_{m} and f_{m}^{−1} for all m ≥ 0 are less or equal to K. In particular F_{m}(λ) = F_{m}(0) = 0 holds for λ < K^{−1}. From Theorem 3.4.3 we get for the Fuglede Kadison determinant of f^{(2)} which is det(F) if det_{ℂ} denotes the determinant of a linear automorphism of a finite-dimensional ℂ-vector space

det(F) = lim_{m→∞} det_{ℂ}(f_{m}^{∗}f_{m})^{\frac{1}{2·[π:Γm]}} .

Since f is described by a matrix with coefficients in ℤπ, the matrix describing f_{m} has integer entries. Hence det_{ℂ}(f_{m}) = ±1 and therefore det_{ℂ}(f_{m}^{∗}f_{m}) = 1 for all m. This implies det(F) = 1 and finishes the proof of Theorem 0.5.

Finally we prove Theorem 0.6. The proof of [23, Theorem 2.4] shows that for an appropriate ℤπ-endomorphism f : ⊕_{i=1}^{a}ℤπ ⟶ ⊕_{i=1}^{a}ℤπ

ρ(M) = det(F)^{−1}

holds. Now apply Theorem 3.4.2

4. Extending Swan’s theorem from finite to residually finite groups

This section is devoted to the proof of Theorem 0.4. We mention that it is not hard to prove Theorem 0.4 using Bass [2, remark 6.11]. We begin with explaining the meaning of assertion 1.) and why it does imply assertion 2.)

Denote by tr_{N (π)}^{c} the center valued trace of the finite von Neumann algebra N (π) [19, Theorem 8.2.8]. Two idempotents p ∈ M(n, n, N (π)) and q ∈ M(m, m, N (π)) are called stably equivalent if p ⊕ 0 = xy and q ⊕ 0 = yx for some l ≥ m, n and x, y ∈ M(l, l, N (π)) holds.

This is equivalent to the statement that im(p) and im(q) are isometrically π-isomorphic Hilbert N (π)-modules. Moreover, p and q are stably e quivalent if and only if they have the same center valued trace [19, Theorem 8.4.3]. Hence the first assertion in Theorem 0.4 means that for any idempotent p ∈ M(n, n, ℤπ) the idempotent p^{(2)} is stably equivalent to the idempotent given by the identity matrix I_{r} ∈ M(r, r, N (π)) for r the rank of the abelian group ℤ ⊗_{ℤπ} im(p). Equivalently, im(p^{(2)}) is isometrically π-isomorphic to ⊕_{i=1}^{r}l^{2}(π)). Hence assertion 1.) implies assertion 2.) and it remains to prove assertion 1.)

We have to show for an idempotent p ∈ M(n, n, ℤπ) that tr_{N (π)}^{c}(p^{(2)}) = rI_{1} for I_{1} the unit element in N (π). Consider an element u in the center Z(ℤπ) of ℤπ. Denote by uI_{n} the diagonal (n, n)-matrix with all diagonal entries equal to u. From Lemma 2.6 applied to f = uI_{n}p and f = uI_{1} we obtain the existence of a positive integer m satisfying (in the notation 2.2)

tr_{N (π)} (uI_{n} · p)) = \frac{1}{[π : Γ_{m}]} · tr_{ℂ} ((uI_{n} · p)_{m})

and

tr_{N (π)}(uI_{1}) = \frac{1}{[π : Γ_{m}]} · tr_{ℂ} (u_{m}I_{1}) .

By Swan’s result [31, Theorem 8.1] applied to the finite group Γ_{m}\π the image of p_{m} is a finitely generated free ℂ[Γ_{m}\π]-module of rank r. Hence we get

tr_{ℂ} ((uI_{n} · p)_{m}) = tr_{ℂ} ((uI_{r})_{m}) = ru_{m}I_{1}.

This implies

tr_{N (π)}(uI_{n} · p) = tr_{N (π)}(ruI_{1}).

From the universal property of the center valued trace [19, Proposition 8.3.10],

tr_{N (π)} (uI_{n} · p) = tr_{N (π)} (uI_{1} · tr_{N (π)}^{c}(p)) .

Hence we get for all u ∈ Z(ℤπ)

tr_{N (π)} (uI_{1} · (tr_{N (π)}^{c}(p) − rI_{1})) = 0.

For g ∈ π denote by (g) the set of elements of π which are conjugated to g. The center valued trace applied to gI_{1} gives 0 if (g) is infinite and \frac{1}{|(g)|} · (∑_{h∈(g)} h) I_{1} otherwise. Hence tr_{N (π)}^{c}(p) − rI_{1} = vI_{1} for an appropriate element v in Z(ℤπ). We conclude for u = v^{∗}

tr_{N (π)} ((tr_{N (π)}^{c}(p) − rI_{1})^{∗} · (tr_{N (π)}^{c}(p) − rI_{1})) = 0.

Since tr_{N (π)} is positive,

(tr_{N (π)}^{c}(p) − rI_{1})^{∗} · (tr_{N (π)}^{c}(p) − rI_{1}) = 0

and hence

tr_{N (π)}^{c}(p) − rI_{1} = 0.

This finishes the proof of Theorem 0.4.

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Wolfgang Lück Fachbereich Mathematik Johannes Gutenberg-Universität 55099 Mainz Bundesrepublik Deutschland email: lueck@topologie.mathematik.uni-mainz.de FAX: 06131 393867

Version of November 8, 1996