Abstract
Let Γ be a discrete group. Following Linnell and Schick one can define a continuous ring c(Γ) associated with Γ. They proved that if the Atiyah Conjecture holds for a torsion-free group Γ, then c(Γ) is a skew field. Also, if Γ has torsion and the Strong Atiyah Conjecture holds for Γ, then c(Γ) is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group Γ = Z2 ≀ Z. It is known that C(Z2 ≀ Z) does not even have a classical ring of quotients. Our main result is that if H is amenable, then c(Z2 ≀H) is isomorphic to a continuous ring constructed by John von Neumann in the 1930 ′ s.
Introduction
Let us consider Mat k×k (C) the algebra of k by k matrices over the complex field. This ring is a unital * -algebra with respect to the complex transposes. For each element A ∈ Mat k×k (C) one can define A * satisfying the following properties.
• (λA) * = λA *
• (A + B) * = A * + B *
• (AB) * = B * A *
• 0 * = 0, 1 * = 1
Also, each element has a normalized rank rk(A) = Rank(A)/k with the following properties.
• rk(0) = 0, rk(1) = 1,
• rk(A + B) ≤ rk(A) + rk(B)
• rk(AB) ≤ min{rk(A), rk(B)}
• rk(A * ) = rk(A)
• If e and f are orthogonal idempotents then rk(e + f ) = rk(e) + rk(f ).
The ring Mat k×k (C) has an algebraic property that von Neumann called regularity: Any principal left-(or right) ideal can be generated by an idempotent. Furthermore, among these generating idempotents there is a unique projection (that is Mat k×k (C) is a * -regular ring). In a von Neumann regular ring any non-zerodivisor is necessarily invertible. One can also observe that the algebra of matrices is proper, that is n i=1 a i a * i = 0 implies that all the matrices a i are zero. One should note that if R is a * -regular ring with a rank function, then the rank extends to Mat k×k (R) [6] , where the extended rank has the same property as rk except that the rank of the identity is k. One can immediately see that the rank function defines a metric d(A, B) ≡ rk(A − B) on any algebra with a rank, and the matrix algebra is complete with respect to this metric. These complete * -regular algebras are called continuous * -algebras (see [5] for an extensive study of continuous rings). Note that for the matrix algebras the possible values of the rank functions are 0, 1/k, 2/k, . . . , 1. John von Neumann observed that there are some interesting examples of infinite dimensional continuous * -algebras, where the rank function can take any real values in between 0 and 1. His first example was purely algebraic. Example 1. Let us consider the following sequence of diagonal embeddings.
One can observe that all the embeddings are preserving the rank and the * -operation. Hence the direct limit lim − → Mat 2 k ×2 k (C) is a * -regular ring with a proper rank function. The addition, multiplication, the * -operation and the rank function can be extended to the metric completion M of the direct limit ring. The resulting algebra M is a simple, proper, continuous * -algebra, where the rank function can take all the values on the unit interval.
Example 2. Consider a finite, tracial von Neumann algebra N with trace function tr N . Then N is a * -algebra equipped with a rank function. If P is a projection, then rk N (P ) = tr N (P ). For a general element A ∈ N , rk N (A) = 1−lim t→∞ t 0 tr N (E λ )dλ, where ∞ 0 E λ dλ is the spectral decomposition of A * A. In general, N is not regular, but it has the Ore property with respect to its zero divisors. The Ore localization of N with respect to its non-zerodivisors is called the algebra of affiliated operators and denoted by U (N ). These algebras are also proper continuous * -algebras [1] . The rank of an element A ∈ U (N ) is given by the trace of the projection generating the principal ideal U (N )A. It is important to note, that U (N ) is the rank completion of N (Lemma 2.2 ( [12] ). Linnell and Schick observed [9] that if X is a subset of a proper * -regular algebra R, then there exists a smallest * -regular subalgebra containing X, the * -regular closure. Now let Γ be a countable group and CΓ be its complex group algebra. Then one can consider the natural embedding of the group algebra to its group von Neumann algebra CΓ → N Γ. Let U (Γ) denote the Ore localization of N (Γ) and the embedding CΓ → U (Γ). Since U (Γ) is a proper * -regular ring, one can consider the smallest * -algebra A(Γ) in U (Γ) containing C(Γ). Let c(Γ) be the completion of the algebra A above. It is a continuous * -algebra [5] . Of course, if the rank function has only finitely many values in A, then c(Γ) equals to A(Γ). Note that if CΓ is embedded into a continuous * -algebra T , then one can still define c T (Γ) as the smallest continuous ring containing CΓ. In [3] we proved that if Γ is amenable, c(Γ) = c T (Γ) for any embedding CΓ → T associated to sofic representations of Γ, hence c(Γ) can be viewed as a canonical object. Linnell and Schick calculated the algebra c(Γ) for several groups, where the rank function has only finitely many values on A. They proved that the following results:
• If Γ is torsion-free and the Atiyah Conjecture holds for Γ, then c(Γ) is a skew-field. This is the case, when Γ is amenable and CΓ is a domain. Then c(Γ) is the Ore localization of CΓ. If Γ is the free group of k generators, then c(Γ) is the Cohen-Amitsur free skew field of k generators. The Atiyah Conjecture for a torsion-free group means that the rank of an element in Mat k×k (CΓ) ⊂ Mat k×k (U (N (Γ))) is an integer.
• If the orders of the finite subgroups of Γ are bounded and the Strong Atiyah Conjecture holds for Γ, then R(Γ) is a finite dimensional matrix ring over some skew field. In this case the Strong Atiyah Conjecture means that the ranks of an element in Mat k×k (CΓ) ⊂ Mat k×k (U (N (Γ))) is in the abelian group 1 lcm(Γ) Z, where lcm(Γ) indicates the least common multiple of the orders of the finite subgroups of Γ.
The lamplighter group Γ = Z 2 ≀Z has finite subgroups of arbitrarily large orders. Also, although Γ is amenable, CΓ does not satisfy the Ore condition with respect to its non-zerodivisors [8] . In other words, it has no classical ring of quotients. The goal of this paper is to calculate c(Z 2 ≀ Z) and even c(Z 2 ≀ H), where H is a countably infinite amenable group. Theorem 1. If H is a countably infinite amenable group, then c(Z 2 ≀ H) is the simple continuous ring M of von Neumann.
Crossed Product Algebras
In this section we recall the notion of crossed product algebras and the groupmeasure space construction of Murray and von Neumann. Let A be a unital, commutative * -algebra and φ : Γ → Aut(A) be a representation of the countable group Γ by * -automorphisms. The associated crossed product algebra A ⋊ Γ is defined the following way. The elements of A ⋊ Γ are the finite formal sums
where a γ ∈ A. The multiplicative structure is given by
The * -structure is defined by γ
Now let (X, µ) be a probability measure space and τ : Γ X be a measure preserving action of a countable group Γ on X. Then we have a * -representation
is the commutative * -algebra of bounded measurable functions on X (module zero measure perturbations).
That is, each element of H can be written in the form of
is the von Neumann algebra N (τ ) associated to the action. Here L ∞ c (X, µ) denotes the subspace of functions in L ∞ (X, µ) having only countable many values. Note that one can extend Tr to Tr N (τ ) on the von Neumann algebra to make it a tracial von Neumann algebra. We will denote by c(τ ) the smallest continuous algebra in
is the same as the weak closure of L ∞ (X, µ) ⋊ Γ. Hence our definition for the von Neumann algebra of an action coincides with the classical definition. On the other hand, c(L
The Bernoulli Algebra
Let H be a countable group. Consider the Bernoulli shift space B H := h∈H {0, 1} with the usual product measure ν H . The probability measure preserving action τ H :
where x ∈ B H , δ, h ∈ H. Let A H be the commutative * -algebra of functions that depend only on finitely many coordinates of the shift space. It is well-known that the Rademacher functions {R S } S⊂H, |S|<∞ form a basis in A H , where
The Rademacher functions with respect to the pointwise multiplication form an Abelian group isomorphic to ⊕ h∈H Z 2 the Pontrjagin dual of the compact group B H satisfying
The group H acts on A H bŷ
Hence,τ
Therefore, the elements of A H ⋊ H can be uniquely written as in the form of the finite sums
where δ · R S = R δS · δ . Now let us turn our attention to the group algebra C(Z 2 ≀ H). For δ ∈ H, let t δ be the generator in h∈H Z 2 belonging to the δ-component. Any element of C(Z 2 ≀ H) can be written in a unique way as a finite sum δ S c δ,S t S · δ ,
Hence we have the following proposition.
Proposition 3.1. There exists a trace preserving * -isomorphism
Recall that if A ⊂ N 1 , B ⊂ N 1 are weakly dense *-subalgebras in finite tracial von Neumann algebras N 1 and N 2 and κ : A → B is a trace preserving * -homomorphism, then κ extends to a trace preserving isomorphism between the von Neumann algebras themselves (see e.g. [7] Corollary 7.1.9.). Therefore, κ : C(Z 2 ≀ H) → A H ⋊ H extends to a trace (and hence rank) preserving isomorphism between the von Neumann algebras N (Z 2 ≀ H) and N (τ H ).
Proposition 3.2. For any countable group H,
Proof. The rank preserving isomorphism κ : N (Z 2 ≀ H) → N (τ H ) extends to a rank preserving isomorphism between the rank completions, that is, the algebras of affiliated operators. It is enough to prove that the rank closure of
Proof. By definition,
where E λ is the spectral projection of f * f corresponding to λ.
Therefore our proposition follows from the lemma below.
By (1) 
The Odometer Algebra
The Odometer Algebra is constructed via the odometer action using the algebraic crossed product construction. Let us consider the compact group of 2-adic integersẐ (2) . Recall thatẐ (2) is the completion of the integers with respect to the dyadic metric
where k is the power of two in the prime factor decomposition of |m − n|. The groupẐ (2) can be identified with the compact group of one way infinite sequences with respect to the binary addition. The Haar-measure µ haar onẐ (2) is defined by µ haar (U l n ) = 1/2 n , where 0 ≤ l ≤ 2 n − 1 and U l n is the clopen subset of elements inẐ (2) having residue l modulo 2 n . Let T be the addition map x → x + 1 inẐ (2) . The map T defines an action ρ : Z (Ẑ (2) , µ haar ) The dynamical system (T,Ẑ (2) , µ haar ) is called the odometer action. As in Section 3, we consider the * -subalgebra of function
, µ haar ) that depend only on finitely many coordinates ofẐ (2) . We consider a basis for A M . For n ≥ 0 and 0 ≤ l ≤ 2 n − 1 let
with respect to the pointwise multiplication. The discrete group Z (2) is the Pontrjagin dual of the compact Abelian groupẐ (2) . The element F 1 n is the generator of the cyclic subgroup Z 2 n . Note that
Hence we have the following lemma. 
Periodic operators
Definition 5.1. A function Z × Z → C is a periodic operator if there exists some n ≥ 1 such that
Observe that the periodic operators form a * -algebra, where
• AB(x, y) = z∈Z A(x, z)B(z, y)
• A * (x, y) = A(y, x) Proposition 5.1. The algebra of periodic operators P is * -isomorphic to a dense subalgebra of M.
Proof. We call A ∈ P an element of type-n if
Clearly, the elements of type-n form an algebra P n isomorphic to Mat 2 n ×2 n (C) and P n → P n+1 is the diagonal embedding. Hence, we can identify the algebra of finite type elements P f = ∪ ∞ n=1 P n with lim − → Mat 2 n ×2 n (C). For A ∈ P, if n ≥ 1 is large enough, let A n ∈ P n be defined the following way.
• A n (x, y) = A(x, y) if 2 n l ≤ x, y ≤ 2 n l + 2 n − 1 for some l ∈ Z.
• Otherwise, A(x, y) = 0.
is a Cauchy-sequence in M.
( Proof. First observe that for any Q ∈ P n
Suppose that A(x, y) = A(x + 2 k , y + 2 k ) and k < n < m. Then
Hence by the previous observation, {A n } ∞ n=1 is a Cauchy-sequence. Note that (iii) and (iv) can be proved similarly, the proof of (ii) is straightforward. In order to prove (v) let us suppose that A(x, y) = 0 whenever |x − y| ≥ 2 k . Let n > k and 0 ≤ y ≤ 2 k − 1 such that A(x, y) = 0 for some −2
. Thus (v) follows. Let us define φ : P → M by φ(A) = lim n→∞ A n . By the previous lemma, φ is an injective * -homomorphism. Proof. For n ≥ 1 and 0 ≤ l ≤ 2 n − 1 let E l n ∈ D be defined by
It is easy to see that E 2l n+1 = E l n and the multiplicative group generated by E 1 n is isomorphic to Z 2 n . Observe that the set {E l n } n,l,(l,n)=1 form a basis in the space of n-type diagonal operators. Therefore, D ∼ = ∪ ∞ n=1 C(Z 2 n ) = C(Z (2) ). Let J ∈ P be the following element.
• J(x, y) = 1, if y = x + 1.
• Otherwise, J(x, y) = 0.
Also, any periodic operator A can be written in a unique way as a finite sum
where D k is a diagonal operator in the form
Thus, by (2) and (3), we have the following corollary.
is a * -isomorphism of algebras.
Lück's Approximation Theorem revisited
The goal of this section is to prove the following proposition.
Proposition 6.1. We have c(ρ) ∼ = M where ρ is the odometer action.
Proof. Let us define the linear map t : P → C by
where A ∈ P and A(x + 2 n , y + 2 n ) for all x, y ∈ Z.
Lemma 6.1. Tr N (ρ) (ψ(A)) = t(A) , where ψ is the * -isomorphism of Corollary 5.1.
Proof. Recall that Tr N (ρ) (F l n ) = 0 , except, when l = 0, n = 0, F l n = 1. If n = 0 and l = 0, then t(E l n ) is the sum of all k-th roots of unity for a certain k, hence t(E l n ) = 0. Also, t(1) = 1. Thus, the lemma follows.
It is enough to prove that
Indeed by (4), ψ is a rank-preserving * -isomorphism between P and A M ⋊ Z.
Hence the isomorphism ψ extends to a metric isomorphism
where P is the closure of P in M and
, since the * -ring operations are continuous with respect to the rank metric. Therefore A M ⋊ Z is a continuous algebra isomorphic to M. Observe that the rank closure A M ⋊ Z is isomorphic to the rank closure of L ∞ c (Ẑ (2) , µ haar ) ⋊ Z by the argument of Lemma 3.2. Therefore, c(ρ) ∼ = M. Thus from now on, our only goal is to prove (4). Lemma 6.2. Let A ∈ P and A n ∈ Mat 2 n ×2 n (C) as in Section 5. Then the matrices {A n } ∞ n=1 have uniformly bounded norms.
Proof. Let M, N be chosen in such a way that
• |A n (x, y)| ≤ M for any x, y ∈ Z, n ≥ 1.
Therefore, for any n ≥ 1, A n ≤ M N .
Proof. Let m ≥ 1, l ≥ 1, q ≥ 1 be integers such that
• A(x, y) = A(x + 2 m , y + 2 m ) for any x, y ∈ Z.
• A(x, y) = 0, if |x − y| ≥ l.
•
By definition,
Observe that if 2lk < x, 2 n − 2lk, then
Hence,
2 n . Thus our lemma follows. Now, we follow the idea of Lück [10] . Let µ be the spectral measure of ψ(A) ∈ N (ρ). That is
or all f ∈ C[0, K]. As in [10] , we can see that the measures {µ n } ∞ n=1 converge weakly to µ. Indeed by Lemma 6.3,
for any real polynomial P , therefore
, in order to prove (4) it is enough to see that
Observe that rk M (A * n A n ) = 1 − µ n (0) and
Hence, our proposition follows from the lemma below (an analogue of Lück's Approximation Theorem). 
Hence, by Lemma 3.5 [2] ,
Therefore, {F n } ∞ n=1 converges uniformly.
Orbit Equivalence
First let us recall the notion of orbit equivalence. Let
(Y, ν) be essentially free probability measure preserving actions of the countably infinite groups Γ 1 resp. Γ 2 . The two actions are called orbit equivalent if there exists a measure preserving bijection Ψ : (X, µ) → (Y, ν) such that for almost all x ∈ X and γ ∈ Γ 1 there exists γ x ∈ Γ 2 such that
Feldman and Moore [4] proved that if τ 1 and τ 2 are orbit equivalent then N (τ 1 ) ∼ = N (τ 2 ) . The goal of this section is to prove the following proposition. By the famous theorem of Ornstein and Weiss [11] , the odometer action and the Bernoulli shift action of a countably infinite amenable group are orbit equivalent. Hence M ∼ = c(Z 2 ≀ H) .
Proof. We build the proof of our proposition on the original proof of Feldman and Moore. Let
Observe that Ψ(N (δ, γ)) = M (γ, δ) . following Feldman and Moore ( [4] , Proposition 2.1) for any
are well-defined. That is, k n=1 h n · 1 M(hn,γ) converges weakly to κ(γ) ∈ N (τ 2 ) as k → ∞ and k n=1 g n · 1 N (gn,δ) converges weakly to λ(δ) ∈ N (τ 1 ) as k → ∞, where {γ n } ∞ n=1 resp. {δ n } ∞ n=1 are enumerations of the elements of Γ 1 resp. Γ 2 . Furthermore, one can extend κ resp. λ to maps
and
The maps κ ′ resp. λ ′ are injective trace-preserving * -homomorphisms with weakly dense ranges. Hence they extend to isomorphisms of von Neumann algebrasκ : N (τ 1 ) → N (τ 2 ),λ : N (τ 2 ) → N (τ 1 ) ,
whereκ andλ are, in fact, the inverses of each other.
Lemma 7.1. 

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