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L2-torsion, the measure-theoretic determinant conjecture, and uniform measure equivalence

We show an invariance result for the L2-torsion of groups under uniform measure equivalence provided a measure-theoretic version of the determinant conjecture holds. The measure-theoretic determinant conjecture is discussed and, for instance, proved for Bernoulli actions of residually amenable groups.

If one attempts to solve the conjecture by eventually reducing it to homotopy invariance, which is done in [13] for L 2 -Betti numbers (see [23] for another approach), then one encounters at least two difficulties that do not appear for L 2 -Betti numbers: one is the need of a measure-theoretic version of the determinant conjecture (Definition 1.6 below), the other is that the Fuglede-Kadison determinant lacks a continuity property that is obvious for the trace (discussed in Section 5). The latter difficulty disappears if we restrict to uniformly measure equivalent groups in Conjecture 1.2.
The proof of the invariance of the L 2 -torsion under uniform measure equivalence, which is the main topic of this paper, is nevertheless much more involved than the one of the invariance of L 2 -Betti numbers under uniform measure equivalence. (1) for every g ∈ G there is a finite subset H(g) ⊂ H such that gY ⊂ H(g)Y up to µ-null sets, and (2) for every h ∈ H there is a finite subset G(h) ⊂ G such that hX ⊂ G(h)X up to µ-null sets.
Uniform measure equivalence was introduced by Shalom and studied in the context of quasi-isometry of amenable groups by Shalom and the second author [24,26]. Uniform measure equivalence is much more restrictive and geometric than measure equivalence. Here are important examples: (1) Two uniform lattices in the same locally compact, second countable Hausdorff group are uniformly measure equivalent. Next we introduce a measure-theoretic version of the determinant conjecture. Some definitions are in order: Let R be a standard equivalence relation on a standard probability space (X, µ) with an invariant measure µ in the sense of [8,Section 2]. We call an equivalence relation with these properties just a measured equivalence relation. We call an action of a countable group G on a standard probability space (X, µ) standard if it is measurable and µ is G-invariant. Every measured equivalence relation on (X, µ) is the orbit equivalence relation of some standard action of a countable group on (X, µ) [ endowed with the following multiplication, involution, and trace, respectively: (1) (f g)(x, y) = z∼x f (x, z)g(z, y), (2) f * (x, y) = f (y, x), The integral groupoid ring is ZR = CR ∩ L ∞ (R, Z). The finite von Neumann algebra of R [9, Section 2], which contains CR as a weakly dense subset and to which tr N (R) extends as a finite trace, is denoted by N (R). If R is the orbit equivalence relation of a standard action G X, we will also use the notation N (G X) instead of N (R).
Definition 1.6. Let R be a standard equivalence relation and G be a countable group. We say that R satisfies the measure-theoretic determinant conjecture (abbreviated as MDC ) if the (generalized) Fuglede-Kadison determinant of every matrix A ∈ M (m × n, ZR) satisfies det N (R) (A) ≥ 1.
A standard action of G satisfies MDC if its orbit equivalence relation satisfies MDC. The group G satisfies MDC if every essentially free, standard action of G satisfies MDC. The following theorem, actually a stronger version thereof, is proved in Section 3.
Theorem 1.8. Let G be a countable group.
(2) If G satisfies MDC, then every subgroup of G satisfies MDC.
(3) If G satisfies MDC, then every amenable extension of G satisfies MDC.
(4) Let 1 → K → G → Q → 1 be a group extension such that K is finite. If G satisfies MDC, then Q satisfies MDC. (5) Assume that G = colim i∈I G i where (G i ) i∈I is a directed system of countable groups (whose structure maps are not necessarily injective). If every G i satisfies MDC, then G satisfies MDC. (6) Bernoulli actions of countable residually amenable groups satisfy MDC. Theorem 1.9. Let G and H be measure equivalent groups. If G satisfies MDC, then H satisfies the determinant conjecture.
The preceding theorem is proved in Section 4. In the same section we also prove the following theorem as a first step towards Conjecture 1.2. The conclusion about the vanishing of the L 2 -Betti numbers of H below is of course due to the corresponding theorem of Gaboriau [13], which we built in for a clean formulation. Theorem 1.10. Let G and H be groups that admit finite CW-models for their classifying spaces. Assume that all the L 2 -Betti numbers of G vanish and G satisfies MDC. If G and H are uniformly measure equivalent with index c = I(G, H) > 0, then all the L 2 -Betti numbers of H vanish, H satisfies the determinant conjecture, and Example 1.4 yields the following corollary: Corollary 1.11. Let G and H be amenable groups that admit finite CW-models for their classifying spaces. If G and H are quasi-isometric, then We emphasize that there is the conjecture that the L 2 -torsion vanishes for all infinite amenable groups. In [28] it is shown that ρ (2) (G) = 0 if G contains an infinite elementary amenable normal subgroup and has a finite model of its classifying space.
2. Background in L 2 -invariants and orbit equivalence relations 2.1. Uniform orbit equivalence. The notion of orbit equivalence has its roots in the pioneering work of Dye [5,6]. We recall it here: If µ X (A) = µ Y (B) = 1, then we call the actions orbit equivalent. The map f is called a weak orbit equivalence or orbit equivalence, respectively.
The following modification of orbit equivalence was introduced in [22] (under the name bounded orbit equivalence).
The following theorem is proved in [11,Theorem 3.3] (see also [22,Theorem 2.33] for the uniform version). Theorem 2.3. Two countable groups G and H are (uniformly) measure equivalent with respect to a measure coupling of index c > 0 if and only if there exist essentially free, standard actions G and H that are (uniformly) weakly orbit equivalent of index c.

2.2.
Algebras associated to groups and equivalence relations. Definition 2.4. Let G (X, µ) be a standard action. The crossed product ring L ∞ (X, C) * G is the free L ∞ (X, C)-module with G as L ∞ (X, C)-basis. Here L ∞ (X, C) denotes the ring of equivalence classes of essentially bounded measurable functions X → C. The multiplication is given by The crossed product ring L ∞ (X, Z) * G is defined analogously using the ring of equivalence classes of essentially bounded measurable functions X → Z. . Let G (X, µ) be an essentially free, standard action and R be its orbit equivalence relation. The rings L ∞ (X, Z) * G and L ∞ (X, C) * G embed as subrings into ZR or CR, respectively. Let i g : X → R be the map i g (x) = (x, g −1 x). We obtain a (multiplicative) isomorphism . There is an analogous statement for integral (instead of complex) coefficients. In particular, we obtain a trace-preserving inclusion CG ֒→ CR; this inclusion extends to the von Neumann algebras N (G) of G and N (R) of R.
The following lemmas about the crossed product of a standard action G X are rather easy to verify.
Proof. This follows from the fact that L ∞ (X, Z) is torsionfree and the isomorphism Recall that that an idempotent p in a ring R is called full if the additive group generated by elements of the form rpr ′ , r, r ′ ∈ R, is R. If p ∈ R is full, then the rings pRp and R are Morita equivalent. This implies e.g. that P is a finitely generated projective R-module if and only if pP is a finitely generated projective pRp-module [17, (18.30B) on p. 490]. Note also that if R ⊂ S is a unital subring and p ∈ R is a full idempotent in R, then p is also a full idempotent in S.
Remark 2.7. Let R be the orbit equivalence relation of a standard action of G on (X, µ). Let A ⊂ X be a subset. We denote the restricted equivalence relation by . Let A ⊂ X be a measurable subset such that finitely many G-translates of A cover X up to null sets. Then the characteristic function χ A is a full idempotent in χ A L ∞ (X, Z) * Gχ A . Lemma 2.9 ([22, Lemma 4.23]). Let f : A → B be an orbit equivalence between essentially free, standard actions of G on (X, µ X ) and H on (Y, µ Y ). Let R 1 and R 2 be the orbit equivalence relations of G X and H Y , respectively.
Thus it extends to an isomorphism N ( (2) If f is uniform, the isomorphism f * restricts to an isomorphism Convention 2.10. Let G X be a standard action, A ⊂ X be a measurable subset, and R be the orbit equivalence relation of the action. We introduce some equivalent notations: Instead of N (R) and N (R|A) we also write N (G X) and N (G X| A ), respectively. For the GNS-construction l 2 (N (R| A )) of N (R| A ) we write L 2 (R| A ) or L 2 (G X| A ).
2.3. The Fuglede-Kadison determinant and L 2 -torsion. In the sequel, let A be a finite von Neumann algebra with trace tr A : A → C.
Our main examples are the group von Neumann algebra N (G) of a group G and the von Neumann algebra N (R) of a standard equivalence relation. A Hilbert space with a (left) A-action that embeds isometrically and equivariantly as a closed subspace into a Hilbert sum of copies of l 2 (A) is a called a Hilbert A-module. A bounded A-equivariant operator between Hilbert A-modules is called a morphism of Hilbert A-modules. The trace tr A extends in a natural way to positive morphisms of Hilbert A-modules. Further, every Hilbert A-module H has a real-valued dimension dim A (H) ∈ [0, ∞]. This dimension satisfies, for example, dim A (l 2 (A)p) = tr A (p) for any projection p ∈ A. We refer to [20,Chapter 6] for more information. The following convention is adopted: Convention 2.11. The n × n-matrices M (n × n, A) over A are again a von Neumann algebra. We equip M (n × n, A) with the unnormalized trace tr M(n×n,A) (A) = n i=1 tr A (A ii ). If the context is clear, we just write tr A instead of tr M(n×n,A) . There is a one-to-one correspondence between A ∈ M (n×n, A) and (left)-A-equivariant bounded operators l 2 (A) n → l 2 (A) n via right matrix multiplication. If we want to stress the point of view of A as a bounded operator, we also use the notation r A for the right multiplication on l 2 (A) n .
We remind the reader of the definition of the spectral density function and the Fuglede- The spectral density function F (f ) is monotonous and right-continuous.
Here the integral is understood to be the Lebesgue-Stieltjes integral with respect to F (f ).
The main properties of this determinant are described in [20, Theorem 3.14].
Remark 2.13 (Induction). Let A ֒→ B be a trace-preserving inclusion of finite von Neumann algebras. Let f : l 2 (A) m → l 2 (A) n be a morphism of Hilbert A-modules. Then f is given by right multiplication r A with an m × n-matrix over A. The morphism of Hilbert B-modules l 2 (B) m → l 2 (B) n defined by right multiplication with the same matrix is denoted by ind B A (f ). It is obvious that tr B P (ind B A (f )) = tr A P (f ) for any complex polynomial P and m = n. Normality of the trace yields that Remark 2.14 (Restriction). Let p ∈ A be a projection. Then pAp is a finite von Neumann algebra with normalized trace tr pAp = 1 trA(p) tr A [3, Prop. 1 on p. 17]. Let f : U → V be a morphism of finitely generated Hilbert A-modules. The morphism f * f decomposes as an orthogonal sum Since spectral calculus respects orthogonal sums, we obtain that im(E The following conjecture is true for all sofic groups [7]; no example of a group that is not sofic is known. Let X be a finite CW-complex with vanishing L 2 -Betti numbers such that G = π 1 (X) satisfies the determinant conjecture. We define the L 2 -torsion of X as . Since the differentials in the cellular chain complex C cell * ( X) are matrices over ZG with respect to cellular bases and thus have positive determinant, this definition is justified. If G is of determinant class, then this definition only depends on the homotopy type of X [20, Lemma 13.6 on p. 456].
Definition 2.17. Let G be a group that admits a finite CW-model X of its classifying space BG. Suppose that the group G satisfies the determinant conjecture and that all its L 2 -Betti numbers vanish. Then we define the L 2 -torsion of G as ρ (2)

The measure-theoretic determinant conjecture
The goal of this section is to prove Theorem 1.8; we actually prove the following slightly stronger formulation: (1) The trivial group satisfies MDC.
(3) Let H ⊂ G be a normal subgroup such that the quotient G/H is amenable. Let G X be a standard action. If H X satisfies MDC, then G X satisfies MDC.
injective structure maps) of countable groups. If every G i satisfies MDC, then G satisfies MDC. (6) Let (X, µ) be a standard probability space. Suppose that G is a directed limit of countable groups G = lim i∈I G i such that the shift action of G i on the product space (X, µ) Gi satisfies MDC. Then the shift action of G on (X, µ) G satisfies MDC.
Actions as in (6) are called Bernoulli actions. The reason that we restrict to Bernoulli actions in (6) is that we do not know how to approximate an arbitrary G-action by actions of the groups G i . The remainder of this section is devoted to the proof of the theorem above.
3.1. The approximation lemma. The proof of the next lemma is essentially the same as in the special case of group von Neumann algebras. Such proofs are given in [ Then lim i∈I dim Ai (ker A i ) = dim A (ker A) and det A (A) ≥ 1.
Remark 3.3. In Connes' pioneering paper [2] the question was raised whether every finite von Neumann algebra embeds into an ultraproduct of the hyperfinite II 1 -factor (nowadays referred to as the Connes embedding problem). If the Connes embedding problem has a positive answer for the finite von Neumann algebra A, then for every self-adjoint ∆ ∈ A there is a sequence of matrices ∆ i ∈ M (d i × d i , C) that satisfies (3) of Lemma 3.2. If the ∆ i have only integral entries, then det A (∆) ≥ 1. In that regard the determinant conjecture is an integral relative of the Connes embedding problem.
The following lemma is often helpful for verifying the second condition in Lemma 3.2. We omit its proof which is essentially the same as in [20, Proof of Lemma 13.33 on p. 466].
3.2. Some reductions used in the proof. Lemma 3.5. Let G (X, µ) be an essentially free, standard action of a countable group G. Let R be its orbit equivalence relation on X.
. Choose an enumeration G = {g 1 , g 2 , . . . }. We define an increasing sequence (X n ) n≥1 of measurable subsets of X by Obviously, µ(X n ) → 1. Set A n = χ Xn A. Then A n ∈ M (d×d ′ ; L ∞ (X, Z) * G) and r AnA * n = r χX n AA * χX n ≤ r AA * . By the continuity of the trace and the trace property, we obtain that The assertion now follows from Lemma 3.2.
Proof. This directly follows from the identity At certain stages in the proof of Theorem 3.1 it is convenient to allow for the flexibility of non-free actions on a probability space. Let G (X, µ) be a, not necessarily free, standard action. The crossed product ring L ∞ (X) * G with its trace can be completed to a von Neumann algebra L ∞ (X) * vN G. This von Neumann algebra is the von Neumann algebra associated to the translation groupoid of the action [27, XIII §2]. If the action is essentially free, then L ∞ (X) * G = N (R) where R ⊂ X × X is the orbit equivalence relation of G X. On the other extreme, if X is just a point, then we have N (R) = C and L ∞ (X) * vN G = N (G).
Lemma 3.7. Assume that G satisfies MDC. Let G X be a (not necessarily free) standard action. Then Proof. Let G Y be an essentially free, standard action; take, for example, Y = [0, 1] G with its shift action. Then the diagonal action of G on the product probability space X ×Y is essentially free. The projection pr : X ×Y → X induces a trace-preserving * -homomorphism which extends to the von Neumann algebras. Since G X ×Y satisfies MDC by hypothesis, the assertion follows (see Remark 2.13).
The remainder of this section is devoted to the proof of Theorem 3.1. Because of Lemma 3.5 it suffices in each case to show MDC only for matrices in the crossed product ring.

The trivial group and transition to subgroups.
Proof of Theorem 3.1 (1). Let (X, µ) be a standard probability space. Let A ∈ M (m × n, L ∞ (X, Z)). We have to show that By [19,Lemma 4.1] there is a unitary matrix U ∈ M (m×m, L ∞ (X)) such that U −1 AA * U is a diagonal matrix whose diagonal entries f 1 , f 2 , . . . , f m ∈ L ∞ (X) are positive functions. We conclude from [20, Theorem 3.14 (1) and (4) and Lemma 3.15 (3), (4) and (7) on p. 128/129] that According to [20, Example 3.13 on p. 128] we have Combining the aforementioned equalities yields Let p(t) be its characteristic polynomial. It can be written as p(t) = t a ·q(t) for a polynomial q(t) with integer coefficients and q(0) = 0. Then q(0) is the product of the positive eigenvalues of Now the assertion follows from q(0) ≥ 1 and (3.1).
Proof of Theorem 3.1 (2). Let i : H → G be the inclusion of a subgroup, and let (X, µ) be a standard probability space endowed with an essentially free standard H-action. Let i ! X be the coinduction of X, i.e. the G-space map H (G, X), on which g ∈ G acts from the left by composition with the G-map r g −1 : G → G, g 0 → g 0 g −1 . By choosing a set theoretic section s : G/H → G of the projection with s(1) = 1, we obtain a bijection We endow i ! X with the structure of a standard probability space (i ! X, ν) by pulling back the product measure on gH∈G/H X. This structure does not depend on the choice of s; the measure ν is G-invariant [14, 3.4]. Let pr : i ! X → X be the map sending φ to φ(1). We obtain a trace preserving, Hequivariant * -homomorphism pr * : L ∞ (X) → L ∞ (i ! X) by composition with pr. Thus we obtain a trace preserving * -algebra homomorphism Let A ∈ M (m × n; L ∞ (X, Z) * H). Let u * A ∈ M (m × n; L ∞ (i ! X, Z) * G) be the matrix obtained from A by applying elementwise the ring homomorphism u. By hypothesis, 3.4. Extensions with amenable quotients.
We have to show that det N (G X) (A) ≥ 1 for every matrix A over the ring L ∞ (X) * G. Taking G ′ ⊂ G to be the subgroup generated by the finitely many elements of G appearing in such A, it is enough to show that G ′ ⊂ X satisfies MDC for every finitely generated subgroup G ′ ⊂ G. By our first remark, we thus may and will assume that G is finitely generated.
Let p : G → G/H be the projection. We choose a left-invariant word-metric d on the finitely generated group G/H. For R > 0 and a subset Z ⊂ G/H we define By amenability (compare [20, Lemma 13.40 on p. 469]) there exists an increasing exhaustion of G/H by finite subsets Z 1 ⊂ Z 2 ⊂ Z 3 ⊂ · · · ⊂ G/H (Følner exhaustion) such that for all R > 0 and ǫ > 0 we find N ∈ N satisfying |N R (Z n )| ≤ ǫ · |Z n | for all n ≥ N . Let S be a transversal for H in G. We set S n = {s ∈ S sH ∈ Z n } ⊂ G. We have |S n | = |Z n |. Let p n : L 2 (G X) → L 2 (G X) be the projection onto the closure of the span of p −1 (Z n ) and L 2 (X), i.e. p n g∈G r g · g = g∈G gH∈Zn r g · g.
The map p n is not L ∞ (X) * G-equivariant in general but L ∞ (X) * H-equivariant.
We remark that in the group case (X = { * }) a more general statement, where H is not necessarily normal, is stated in [25,Section 4] and [20,Proposition 13.93 on p. 469]. However, the proofs of these statements are not correct: the mistakes are related to the equivariance of the map p n above for which normality of H is essential.
Notice that we have an isometric L ∞ (X) * H-equivariant isomorphism In order to show det N (G X) (A) ≥ 1 we may assume that d = d ′ and A is positive (Lemma 3.6). Consider the operator It is easy to see that this operator is given by right multiplication with a positive matrix A n ∈ M (d · |Z n | × d · |Z n |; L ∞ (X, Z) * H). By hypothesis, we have det N (H X) (A n ) ≥ 1 for every n ≥ 1. Since p n ≤ 1 holds for all n ∈ N, we conclude r An ≤ r A . By Lemma 3.2 it suffices to show that This is proven in the case X = { * } in [20, Lemma 13.42 on page 470], and the proof is essentially the same in our setting.

Extensions with finite kernels.
Proof of Theorem 3.1 (4). Let Q X be an essentially free, standard action. Let G act on X via p : G → Q. In the sequel, we write F = l 2 (L ∞ (X) * vN G) for the GNS-construction of the von Neumann algebra L ∞ (X) * vN G.
Let N K ∈ G be the element g∈K g. Consider the Hilbert L ∞ (X) * vN G-morphism Then r |K| −1 ·NK is an orthogonal projection with the K-fixed points F K as image since K ⊆ G is normal and K acts trivially on X. In particular, F K is a finitely generated Hilbert L ∞ (X) * vN G-module. Define an isometric bijective operator v : We also set For every g ∈ G, f ∈ L ∞ (X), and a ∈ L 2 (Q X) we have v(r p(g) (a)) = r g (v(a)), This, in particular, implies that the image of a N (Q Upon choosing an isometric embedding into L 2 (Q X) d and thus into (F K ) d via diag(v), we can equip every finitely generated Hilbert N (Q X)-module V with the structure of a finitely generated Hilbert L ∞ (X) * vN G-module. We denote V endowed with this new structure by res p V . Because of (3.2) these module structures are related by gφ(x) = p(g)x and f φ(x) = f x, where we denote the identity V → res p V by φ for better distinction. From this we also see that the module structure on res p V does not depend on the chosen embedding V ֒→ L 2 (Q X) d and that every morphism f : V → W of finitely generated Hilbert N (Q X)-modules induces a morphism res p f : res p V → res p W of finitely generated Hilbert L ∞ (X) * vN Gmodules by the same map. Next we show that It suffices to treat the case V = L 2 (Q X). Let e G ∈ F and e Q ∈ L 2 (Q H) be the elements given by the unit element in the rings L ∞ (X) * G and L ∞ (X) * Q. Write f (e Q ) = q∈Q λ q · q. Then This implies

Hence (3.3) follows.
If {E λ | λ ∈ R} is the spectral family of f : V → V , then {res p E λ | λ ∈ R} is the spectral family of res p f : res p V → res p V . Hence (3.3) successively yields that We have to show that det N (Q X) (A) ≥ 1. By Lemma 3.6 we may and will assume that d ′ = d and A is positive.
Let n ∈ N. We get a morphism res p r A n : res p L 2 (Q X) d → res p L 2 (Q X) d of finitely generated Hilbert L ∞ (X) * vN G-modules. We have the orthogonal sum decomposition Consider the morphism We conclude from (3.4) and [20, Theorem 3.14 (1) on p. 128 and Lemma 3.15 (7) on p. 130] that where I d is the identity matrix in M (d × d, L ∞ (X, Z) * G). Notice that |K| · I d + B lies in M (d×d, L ∞ (X, Z) * G); thus, by hypothesis and Lemma 3.7, det L ∞ (X) * vN G r |K|·I d +B ≥ 1.
We conclude from [20, Theorem 3.14 (1) on page 128] that We conclude from (3.5) and (3.6) that 3.6. Colimits. Throughout this subsection, we consider a directed system of groups {G i | i ∈ I} over the directed set I. Denote its colimit by G = colim i∈I G i . Let ψ i : G i → G for i ∈ I and ψ i,j : G i → G j for i, j ∈ I, i ≤ j, be the structure maps. We do not require that ψ i or ψ i,j are injective.
Proof of Theorem 3.1 (5). Let G (X, µ) be an essentially free, standard action. Every G i acts on X via ψ i (but not necessarily free). We obtain a trace-preserving ring * -homomorphism Let A ∈ M (m × n, L ∞ (X; Z) * G). Write A = g∈G f g g with f g ∈ M (m × n, L ∞ (X, Z)).
Let i 0 ∈ I be such that for every i ≥ i 0 the implication holds. Let V = {g ∈ G|f g = 0}. For every g ∈ V let g (i) ∈ G i be a preimage of g. Let A i = g∈V f g · g (i) and ∆ i = A i A * i . We have α i (A i ) = A. From Lemma 3.4 it is clear that there is a uniform bound of the operator norms of the ∆ i ∈ M (n × n, L ∞ (X) * vN G i ). By hypothesis and Lemma 3.7 we have Let m ≥ 1. The assertion would follow from Lemma 3.2 provided we show that We can find For i ≥ i 1 the right hand side of (3.7) is stationary and equal to the left hand side.

Bernoulli actions.
Proof of Theorem 3.1 (6). For any countable set A denote by µ A the product measure a∈A µ on X A . The σ-algebra of Borel sets in X G = g∈G X is the σ-algebra S generated by B = g∈G U g U g = X for almost all g ∈ G .
Let A be the algebra generated by B. We say that a measurable function f : Any set M ∈ A can be written as M = ∪ n k=1 M k with disjoint sets M k ∈ B. The sets M ∈ A have the property that there exists a finite subset F ⊆ G with M = pr −1 F (pr F (M )) where pr F : g∈G X → g∈F X is the projection onto the components of F . We denote with F (M ) ⊆ G the smallest subset with this property. Let R be the ring of all bounded, A-measurable functions X G → Z. For f ∈ R, we denote with 1 (z)). Since any non-empty set in A has positive measure, the ring R embeds into L ∞ (X G , Z). Since R is G-invariant, we obtain a subring R * G ⊂ L ∞ (X G , Z) * G that is closed under the involution.
Let ψ i : G → G i , i ∈ I, be the structure maps of the limit. The map ψ i induces a measurable map . Since every f ∈ R is a finite linear combinations of characteristic functions of sets in A, we also obtain that (3.8) We obtain a ring homomorphism which respects the involutions: By applying this homomorphism entry-wise we obtain a ring homomorphism M (d × d ′ , R * G) → M (d × d ′ , L ∞ (X Gi ) * G i ) that we denote by the same name.
Proof of lemma. By considering the matrix entries separately, the assertion reduces to showing that for any set of 2m elements a 1 , . . . , a m , b 1 , . . . , b m ∈ R * G we have Since β i is a ring * -homomorphism, the assertion reduces further to showing that for f ∈ R and g ∈ G we have g)).
Choose i 0 ∈ I such that ψ i | F (f )∪{1,g} is injective for i ≥ i 0 . Then (3.8) yields that for i ≥ i 0 . This concludes the proof of (3.9). By Lemma 3.4 there is an upper bound r ∆ m i that is independent of i. By hypothesis, det N (Gi We need a general fact before we can complete the proof: Let This follows from an iterated application of the corresponding assertion for m = 2. For m = 2 we have: Now we can complete the proof of Theorem 3.1 (6). For any S ∈ S and any ǫ > 0 there exists a set M ∈ A with µ G (S△M ) < ǫ. Since every element in f ∈ L ∞ (X, Z) is a finite linear combination of characteristic functions, there exist functions f (n) ∈ R associated to f such that f (n) Let ∆ = AA * and ∆ n = B (n) (B (n) ) * . For every m ≥ 1 we have (3.11) tr N (G X) (∆ m ) = lim n→∞ tr N (G X) (∆ m n ).
By considering matrix entries separately as in the proof of Lemma 3.8, this easily follows from (3.10) and the G-invariance of the measure. By Lemma 3.4, r ∆ m n has an upper bound independent of n. Lemma 3.8 and Lemma 3.2 complete the proof. 4. Proofs of Theorems 1.9 and 1.10 The following two results are proved in first author's book [20] for group von Neumann algebras. We need them for arbitrary finite von Neumann algebras; the proof translates literally to the general case.
Then the mapping cone cone * (φ * ) is also a finitely generated Hilbert A-chain complex with vanishing L 2 -Betti numbers and positive determinants of his differentials. Moreover, we obtain the equation . Let C * be a finitely generated Hilbert A-chain complexes and γ * a chain contraction. Then b (2) n (C * ) = 0 for all n ∈ Z. If det A (c n ) > 0 for all n ∈ Z then Definition 4.3. Let R be a ring with involution. A finitely generated based free R-module is a finitely generated free R-module together with an isomorphism M ∼ = R n . A finitely generated based projective R-module P is a finitely generated projective R-module P together with an isomorphism P ∼ = R n A where A ∈ M (n × n, R) satisfies A 2 = A and A * = A.
Remark 4.4. Consider a standard action G X and a measurable subset A ⊂ X such that there is a finite subset S ⊂ G with S · A = X up to null sets. We can find (and fix for the following discussion) measurable subsets A g ⊂ A for each g ∈ S such that the sets gA g partition X. We obtain the isomorphism of left χ A L ∞ (X, Z) * G χ A -modules n is a finitely generated based free module, then χ A F becomes a finitely generated based projective module over the ring χ A L ∞ (X, Z) * Gχ A by where is Q is the projection matrix and obtains the structure of a Hilbert N (G X|A)-module from this isomorphism. Let F be a finitely generated based free module over the ring L ∞ (X, Z) * G. Since χ A is full, there is an obvious isomorphism If V is any Hilbert N (G X)-module, then χ A V becomes a Hilbert N (G X| A )-module (see [16, pp. 19-27]). Since L 2 (G X) ⊗ L ∞ (X,Z) * G F is a Hilbert N (G X)-module through the free basis of F , the left hand side in (4.2) is a Hilbert N (G X| A )-module. On the other hand, the right hand side in (4.2) is a Hilbert N (G X| A )-module since χ A F is based projective. The isomorphism (4.2) is an isomorphism of Hilbert N (G X| A )modules.
Lemma 4.5. Let G (X, µ) be an essentially free, standard action and A ⊂ X a measurable subset such there is a finite subset F ⊂ G with F A = X up to null sets. Then the following statements are equivalent: (1) G X satisfies MDC. (2) For every homomorphism of finitely generated based projective L ∞ (X, Z) * G-modules f : P → Q the associated Hilbert N (G X)-morphism L 2 (G X) ⊗ L ∞ (X,Z) * G f has determinant ≥ 1.
(3) For every homomorphism of finitely generated based projective χ A L ∞ (X, Z) * Gχ Amodules f : P → Q the associated Hilbert N (G X| A )-morphism Proof.
(1) implies (2): Every homomorphism f : P → Q between finitely generated based projective L ∞ (X, Z) * G-modules can be extended by zero to an homomorphism of finitely generated based free modules. We have det N (G X) (f ) = det N (G X) (f ⊕ 0), and the latter is ≥ 1 since G satisfies MDC.
(2) implies (3): By a similar argument, it is enough to show that, for every homomorphism of finitely generated based free χ A L ∞ (X, Z) * Gχ A -modules f : P → Q, the associated Hilbert We can view such B also as an element in M (m × n, L ∞ (X, Z) * G). It is obvious (just check on polynomials) that the spectral density function of B with respect to N (G X) is just µ(A) times the spectral density function with respect to N (G X| A ). Assuming (2), we now see that det N (G X|A) (B) ≥ 1 and (3) implies (1): Let f : F → F ′ be a homomorphism of finitely generated based free L ∞ (X, Z) * G-modules. Since χ A is full by Lemma 2.8, χ A F and χ A F ′ are finitely generated (based) projective χ A L ∞ (X, Z) * Gχ A -modules. By (3) and Remark 4.4 the map Thus, by Remark 2.14, the map L 2 (G X) ⊗ f has determinant ≥ 1. By Lemma 3.5 this suffices to show that G X satisfies MDC.
We omit the proof of the following lemma which is essentially the same as that of Lemma 4.5.
Lemma 4.6. Let R be a standard equivalence relation on (X, µ). Let A ⊂ X be a measurable subset such that χ A is a full idempotent in ZR. Then the following statements are equivalent: (1) R satisfies MDC.
(2) For every homomorphism of finitely generated based projective ZR-modules f : P → Q the associated Hilbert N (R)-morphism L 2 (R) ⊗ ZR f has determinant ≥ 1. Proof of Theorem 1.9. By Theorem 2.3 there exist essentially free, standard actions G (X, µ X ) and H (Y, µ Y ) that are weakly orbit equivalent. Let f : B → A be a weak orbit equivalence between measurable subsets A ⊂ X and B ⊂ Y . By [10, Lemma 2.2] we may assume that both actions are ergodic. The map f induces an isomorphism between the restricted orbit equivalence relations R(G X)| A and R(H Y )| B . By hypothesis, R(G X) satisfies MCD. That R(H Y ) satisfies MDC, thus H satisfies the determinant conjecture, follows by applying Lemma 4.6 twice provided that χ A ∈ ZR(G X) and χ B ∈ ZR(H Y ) are full idempotents. For that, we prove in general that χ A ∈ ZR is a full idempotent if R is an ergodic standard equivalence relation and A ⊂ X is a measurable subset of positive measure: Let X = n+1 i=1 A i be a Borel partition of X such that A 1 , . . . , A n all have the same measure as A and µ(A n+1 ) ≤ µ(A n ). By ergodicity and [12,Lemma 2.1] there are measure isomorphisms φ i : X → X for each 1 ≤ i ≤ n + 1 such that (x, φ i (x)) ∈ R for x ∈ X and φ i (A) = A i for 1 ≤ i < n and A n+1 ⊂ φ n+1 (A). The characteristic function χ φi of the graph of each φ i is an element in ZR. From the properties of φ i we obtain that Thus, χ A is a full idempotent.
Proof of Theorem 1.10. Let Z G and Z H be finite CW-models of the classifying spaces BG and BH, respectively. By Theorem 2.3 there exist essentially free, standard actions G (X, µ X ) and H (Y, µ Y ) that are uniformly weakly orbit equivalent. Let f : B → A be a uniform weak orbit equivalence between measurable subsets A ⊂ X and B ⊂ Y . It induces a trace-preserving ring isomorphism f * : . By assumption, G satisfies MDC. Because of the previous ring isomorphism and by applying Lemma 4.5 twice it follows that H Y satisfies MDC; in particular, H satisfies the determinant conjecture.
The cellular chain complex C cell * (Z G ) of the universal coverZ G is a finite based free ZGresolution of Z. By successively applying (2.1) and (2.2) to the differentials of l 2 (G) ⊗ ZG C cell * (Z G ), we obtain that The χ A L ∞ (X, Z) * Gχ A -chain complex C(G) * := χ A L ∞ (X, Z) * G ⊗ ZG C cell * (Z G ) is a finite resolution by based projective modules of χ A L ∞ (X, Z) = L ∞ (A, Z) by Lemma 2.6 and Lemma 2.8. Furthermore, we define: Since χ A is full (Lemma 2.8), By replacing G X by H Y and A by B, we define C(H) * and C(H) (2) * in an analogous fashion. A similar discussion as before applies; so C(H) * is a finite projective resolution of L ∞ (B) over the ring χ B L ∞ (Y, Z) * Hχ B , and It remains to prove that (2) .

A cautionary example
In Gaboriau's proof [13] of the orbit equivalence invariance of L 2 -Betti numbers one encounters the following situation: One obtains a homotopy equivalence between Hilbert N (R)-complexes C * and D * which is not bounded -unless the given orbit equivalence is uniform. To obtain an estimate (and by symmetry an equality) between the L 2 -Betti numbers of these complexes, one constructs increasing sequences of subcomplexes C To be able to drop the uniformity assumption in Theorem 1.10 one would want, among other things, a similar continuity property of the Fuglede-Kadison determinant. The following theorem, whose proof we omit, states such -but with an important, restrictive assumption on the kernels: Theorem 5.1. Let f : U → V be a morphism of finitely generated Hilbert A-modules. Let p k : U → U for k ∈ N be a sequence of projections with ker(f ) ⊆ im(p k ) ⊆ im(p k+1 ) that weakly converges to the identity. Then Next we will show that the condition ker(f ) ⊂ im(p k ) is necessary. In our example, A is L ∞ ([0, 1]), and f is the projection onto the second factor pr 2 : L 2 ([0, 1]) ⊕ L 2 ([0, 1]) → L 2 ([0, 1]). We drop the subscript L ∞ ([0, 1]) in the notation of the trace and the determinant. Obviously, det(pr 2 ) = 1. Let ǫ k for k = 1, 2, . . . be any sequence of positive real numbers. Consider the morphism of finitely generated Hilbert L ∞ ([0, 1])-modules The morphisms u k and v k are isometric L ∞ ([0, 1])-embeddings, and im(u k ) and im(v k ) are othogonal to one another. Then is an increasing sequence of closed subspaces. Since If we choose ǫ k = k 2 k+1 − 1, then we obtain that det(pr 2 •p k ) = 1 k for k = 1, 2 . . . ; det(pr 2 ) = 1.