L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-Betti numbers arising from the lamplighter group

We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^2$$\end{document}-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^2$$\end{document}-Betti numbers arising from the lamplighter group algebra Q[Z2≀Z]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb Q}[{\mathbb Z}_2 \wr {\mathbb Z}]$$\end{document}. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^2$$\end{document}-Betti numbers from the algebras Q[Zn≀Z]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb Q}[{\mathbb Z}_n \wr {\mathbb Z}]$$\end{document}, where n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 2$$\end{document} is a natural number. We also apply the techniques developed to the generalized odometer algebra O(n¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}({\overline{n}})$$\end{document}, where n¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{n}}$$\end{document} is a supernatural number. We compute its ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-regular closure, and this allows us to fully characterize the set of O(n¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}({\overline{n}})$$\end{document}-Betti numbers.


Introduction and the Atiyah problem
During the last 40 years, the importance of both the SAC and BSAC has increased due to the wide variety of their consequences in several branches of mathematics. For instance, in differential geometry and topology the SAC has connections with the Hopf conjecture on the possible sign of the Euler characteristic of a Riemannian manifold ( [12], see also [33,Chapters 10 and 11]). In group theory, the SAC has implications relating group-theoretic properties of a group and its homological dimension. In particular, it is known that if a group has homological dimension one and satisfies the SAC, then it must be locally free [25].
One of the main problems (following these lines) is to actually compute the whole set C(G, K ) of 2 -Betti numbers arising from G with coefficients in K . In this paper, we uncover a portion of this set for the lamplighter group Γ with coefficients in Q. The group Γ is defined to be the semidirect product of Z copies of the finite group Z 2 by Z, i.e., Γ = i∈Z Z 2 ρ Z, whose automorphism ρ implementing the semidirect product is the well-known Bernoulli shift. A precursor of our work here can be found in the paper [4] by Ara and Goodearl. In that article, the authors attack this problem algebraically, by trying to uncover the structure of what is called the * -regular closure R K Γ of the lamplighter group algebra K Γ inside U(Γ ), the algebra of unbounded operators affiliated to the group von Neumann algebra N (Γ ) or, more algebraically, the classical ring of quotients of N (Γ ). The precise connection between the * -regular closure R K Γ and the set C(Γ , K ) has been provided recently by a result of Jaikin-Zapirain [22], which states that the rank function on R K Γ , obtained by restricting the canonical rank function on U(Γ ), is completely determined by its values on matrices over K Γ . Since R K Γ is * -regular, this can be rephrased in the form where φ is the state on K 0 (R K Γ ) induced by the restriction of the rank function on K Γ , and G(Γ , K ) is the subgroup of R generated by C(Γ , K ) [3, Proposition 4.1] (see, e.g., [37] for the definition of the K 0 -group of a ring). The algebra R K Γ , together with tight connections with C(Γ , K ), has been recently studied by the authors in [3].
A key observation in light of the development of the work presented in this paper is to realize the lamplighter group algebra as a Z-crossed product * -algebra K Γ ∼ = C K (X ) T Z through the Fourier transform. This method was first (somewhat implicitly) used in [13], and very explicitly in [7]. Here X = {0, 1} Z is the Pontryagin dual of the group i∈Z Z 2 , topologically identified with the Cantor set, C K (X ) is the set of locally constant functions f : X → K , and T : X → X is the homeomorphism of X implemented by the Bernoulli shift. There is a natural measure μ on X , namely the usual product measure, having taken the 1 2 , 1 2 -measure on each component {0, 1}.
This measure is ergodic, full and T -invariant, so we can apply the techniques developed in [2] to study the Z-crossed product algebra A := C K (X ) T Z by giving 'μapproximations' of the space X , which at the level of the algebra A correspond to certain 'approximating' * -subalgebras A n ⊆ A (see [2,Section 4.1], also [3,Section 6]). By using [2, Theorem 4.7 and Proposition 4.8], we obtain a canonical faithful Sylvester matrix rank function rk A on A which coincides, in case K is a subfield of C closed under complex conjugation, with the rank function rk K Γ on the group algebra K Γ naturally inherited from the canonical rank function in the * -regular ring U(Γ ) [3,Proposition 5.10]. In light of this, one can define 'generalized' 2 -Betti numbers in this more general setting, that is, arising from the Z-crossed product algebra A = C K (X ) T Z, for K an arbitrary field and T an arbitrary homeomorphism on a Cantor set X . Turning back to the lamplighter group algebra K Γ , Graboswki has shown in a recent paper [18] the existence of irrational (in fact transcendental) 2 -Betti numbers arising from Γ , exhibiting a concrete example in [18,Theorem 2]. With this result, the lamplighter group has become the simplest known example which gives rise to irrational 2 -Betti numbers. Very roughly, his idea is to compute 2 -Betti numbers by means of decomposing them as an infinite sum of (normalized) dimensions of kernels of finite-dimensional operators (i.e., matrices). He then realizes these matrices as adjacency-labeled matrices of certain graphs in order to determine the global behavior of the dimensions of their kernels. We use these ideas in this work, but applied to our construction. Our main result is the following: Here b (2) (A) stands for the 2 -Betti number of A (see Definition 2.2). Thus, we obtain a whole family of irrational and even transcendental 2 -Betti numbers arising from Γ . Another source of irrational 2 -Betti numbers arising from Γ comes from the fact that the * -regular closure R K Γ contains a copy of the algebra of non-commutative rational power series K rat X in infinitely many indeterminates, see [ We also apply our machinery to study a particular crossed product algebra known as the odometer algebra. It is defined as the Z-crossed product algebra O := C K (X ) T Z where X = {0, 1} N is the one-sided shift space, and the automorphism T : X → X implementing the crossed product is given by addition of the element (1, 0, 0, . . . ) with carry over. Although it is not possible to realize the odometer algebra as a group algebra, this example is interesting in its own right because we are able to fully determine the structure of its * -regular closure R O (see Theorem 5.4), thus giving a complete description of the set of O-Betti numbers (Theorem 5.6). The algebra O has also been studied by Elek in [15], although the author does not exactly compute the * -regular closure R O ; instead, the author computes its rank completion, showing that it must be isomorphic to the von Neumann continuous factor M K , which is by definition the rank completion of lim − →i M 2 i (K ) with respect to its unique rank metric (cf. [3,Proposition 4.2]). In fact, we study a general version of the classical odometer algebra, namely the dynamical system generated by 'addition of 1' for arbitrary profinite completions of Z.
This work is structured as follows. In Sect. 2, we provide essential background and preliminary concepts about Sylvester matrix rank functions and 2 -Betti numbers for general group algebras. We summarize, in Sect. 3, a general approximation construction for Z-crossed product algebras using measure-theoretic tools, which enables us to construct a canonical Sylvester matrix rank function over the crossed product algebra. Using this construction, we derive a formula for computing generalized 2 -Betti numbers (see Definition 3.3) over this large class of algebras (Formula (3.4)). In Sect. 4, we focus on the particular case of the lamplighter group algebra. We explicitly realize it as a Z-crossed product algebra, thus enabling us to apply the whole theory developed in [2,3]. Our main result (Theorem 4.5) provides a vast family of irrational, and even transcendental, 2 -Betti numbers arising from the lamplighter group.
We also explore another method to find 2 -Betti numbers arising from the lamplighter group, using the algebra of non-commutative rational series. In this direction, we obtain in Theorem 4.13 that the irrational algebraic number 1 4 2 7 belongs to G(Γ , Q).
Finally we study, in Sect. 5, the generalized odometer algebra O(n) in great detail, and we are able to completely determine the algebraic structure of its * -regular closure (Theorem 5.4). We use this characterization in Theorem 5.6, where we explicitly compute the whole set of O(n)-Betti numbers.

* -Regular rings and rank functions
A * -regular ring is a regular ring R endowed with a proper involution * , that is, x * x = 0 if and only if x = 0. In a * -regular ring R, for every x ∈ R there exist unique projections e, f ∈ R such that x R = eR and Rx = R f . It is common to denote them by e = LP(x) and f = RP(x), and are termed the left and right projections of x, respectively. We refer the reader to [1,8] for further information on * -regular rings.
For any subset S ⊆ R of a unital * -regular ring, there exists a smallest unital *regular subring of R containing S ([4, Proposition 6.2], see also [32,Proposition 3.1] and [22,Proposition 3.4]). This * -regular ring is denoted by R(S, R), and called the * -regular closure of S in R. Let us denote by M(R) the set of finite matrices over R of arbitrary size, i.e., n≥1 M n (R).

Definition 2.1
A Sylvester matrix rank function on a unital ring R is a map rk : M(R) → R + satisfying the following conditions: (c) rk for any matrices M 1 and M 2 ; for any matrices M 1 , M 2 and M 3 of appropriate sizes.
The notion of Sylvester matrix rank function was first introduced by Malcolmson in [35] on a question of characterizing homomorphisms from a fixed ring to division rings. Equivalent definitions exist, introduced by Malcolmson itself and Schofield [39]. For more theory and properties about Sylvester matrix rank functions, we refer the reader to [22] and [39,Part I,Chapter 7].
In a regular ring R, any Sylvester matrix rank function rk is uniquely determined by its values on elements of R, see, e.g., [16,Corollary 16.10]. This is no longer true if R is not regular.
Any Sylvester matrix rank function rk on a unital ring R defines a pseudo-metric by the rule d(x, y) = rk(x − y). The rank function is called faithful if the only element with zero rank is the zero element. In this case, d becomes a metric on R. The ring operations are continuous with respect to d, and rk extends uniquely to a Sylvester matrix rank function rk on the completion R of R with respect to d.
For a * -subring S of a * -regular ring R, there are tight connections between the structure of projections of the * -regular closure R(S, R) and possible values of a Sylvester matrix rank function defined on R, see for instance [3,22].

2 -Betti numbers for group algebras
In this subsection, we define 2 -Betti numbers arising from a group G with coefficients in a subfield K ⊆ C closed under complex conjugation.
Let G be a discrete, countable group. For any subring R ⊆ C closed under complex conjugation, let RG denote the group * -algebra of G with coefficients in R, consisting of formal finite sums γ ∈G a γ γ with a γ ∈ R. The sum operation is defined pointwise, the product is induced by the group product and the * -operation is defined by linearity and according to the rule (a γ γ ) * = a γ γ −1 , for a γ ∈ R and γ ∈ G. Let also 2 (G) denote the Hilbert space of all square-summable functions f : G → C with obvious addition and scalar multiplication, and inner product defined by The space 2 (G) has an orthonormal basis, naturally identified with G, consisting of indicator functions ξ γ ∈ 2 (G) for each γ ∈ G. Here ξ γ is defined to be 1 over the element γ and 0 otherwise.
We denote by N (G) the weak completion of CG ⊆ B( 2 (G)), which is commonly known as the group von Neumann algebra of G. An equivalent algebraic definition can be given: It consists exactly of those bounded operators T : 2 (G) → 2 (G) that are G-equivariant, i.e., the relation ρ γ • T = T • ρ γ is satisfied for every γ ∈ G. The algebra N (G) is endowed with a normal, positive and faithful trace, defined as Note that for an element T = γ ∈G a γ γ ∈ CG, its trace is simply the coefficient a e .
All the above constructions can be extended to k×k matrices: The ring M k (RG) acts faithfully on 2 (G) k by left (resp. right) multiplication. We denote the extended actions by λ k and ρ k , respectively. We identify M k (RG) with its image M k (RG) ⊆ B( 2 (G) k ) under λ k . We denote by N k (G) the weak completion of M k (CG) inside B( 2 (G) k ), which is easily seen to be equal to M k (N (G)). The previous trace can be extended to an unnormalized trace over M k (N (G)) by setting, for a matrix T = (T i j ) ∈ M k (N (G)), A finitely generated Hilbert (right) G-module is any closed subspace V of 2 (G) k , invariant with respect to the right action ρ ⊕k := ρ⊕ k · · · ⊕ρ. For V ≤ 2 (G) k a finitely generated Hilbert G-module, the corresponding orthogonal projection operator p V : 2 (G) k → 2 (G) k onto V belongs to N k (G). One then defines the von Neumann dimension of V as the trace of p V : In the particular case of matrix group rings M k (K G), being K ⊆ C a subfield closed under complex conjugation, every matrix operator A ∈ M k (K G) gives rise to an 2 -Betti number, in the following way. Consider A as an operator A : 2 (G) k → 2 (G) k acting on the left, and take p A ∈ N k (G) to be the projection onto ker A, which is a finitely generated Hilbert G-module. One can then consider the von Neumann dimension of ker A, which is simply the trace of the projection p A .

Definition 2.2 Let
A be a matrix operator in M k (K G) for some integer k ≥ 1. We define the 2 -Betti number of A by b (2) The set of all 2 -Betti numbers of operators A ∈ M k (K G) will be denoted by C(G, K ), and will be referred to as the set of all 2 -Betti numbers arising from G with coefficients in K . It should be noted that this set is always a subsemigroup of (R + , +). We also write G(G, K ) for the subgroup of (R, +) generated by C(G, K ).
It is also possible to define the von Neumann dimension by means of a Sylvester matrix rank function, as follows. Let U(G) be the algebra of unbounded operators affiliated to N (G); equivalently, the classical ring of quotients of N (G). It is a *regular ring possessing a Sylvester matrix rank function rk U (G) defined by for any matrix U ∈ M k (U(G)), where LP(U ) and RP(U ) are the left and right projections of U inside the * -regular algebra M k (U(G)), respectively. Notice that these projections actually belong to N k (G). In particular, we obtain by restriction a Sylvester matrix rank function rk K G over K G. So for a matrix operator A ∈ M k (K G) we have p A = 1 k − RP(A) and we get the equality b (2) (2.1) Here 1 k stands for the identity matrix in k dimensions.

Approximating crossed product algebras through a dynamical perspective
We recall the general construction used in [2] on approximating Z-crossed product algebras.
Let T : X → X be a homeomorphism of a totally disconnected, compact metrizable space X , which we also assume to be infinite (e.g., one can take X to be the Cantor space). Let also K be an arbitrary field endowed with a positive definite involution, that is, an involution * such that for all n ≥ 1 and a 1 , . . . , a n ∈ K , we have n i=1 a * i a i = 0 ⇒ a i = 0 for each i = 1, . . . , n.
The algebras of interest are Z-crossed product algebras of the form where C K (X ) denotes the algebra of locally constant functions f : X → K ; equivalently, the algebra of continuous functions f : X → K when K is endowed with the discrete topology. For the approximation process, we choose a T -invariant, ergodic and full probability measure μ on X . We refer the reader to [2, Section 3] for a detailed exposition of the construction. For any clopen subset ∅ = E ⊆ X and any (finite) partition P of the complement X \E into clopen subsets, let B be the unital * -subalgebra of A generated by the partial isometries {χ Z t | Z ∈ P}. Here t denotes the generator of the copy of Z inside A, and χ A denotes the characteristic function of the set A. There exists a quasi-partition of X (i.e., a countable family of non-empty, pairwise disjoint clopen subsets whose union has full measure) given by the T -translates of clopen subsets W of the form for k ≥ 1 and Z i ∈ P, whenever these are non-empty. In fact, if we write |W | := k (the length of W ) and V := {W = ∅ as above}, then for a fixed W ∈ V and 0 ≤ i < |W | the element χ T i (W ) belongs to B, and moreover, the set of elements In this way, one obtains an injective * -representation π : B → W ∈V M |W | (K ) =: Proposition 3.13]. From now on, the * -algebra B corresponding to (E, P) as above will be denoted by A(E, P).
Take now {E n } n≥1 to be a decreasing sequence of clopen sets of X together with a family {P n } n≥1 consisting of (finite) partitions into clopen sets of the corresponding complements X \E n , satisfying: (a) the intersection of all the E n consists of a single point y ∈ X ; (b) P n+1 ∪ {E n+1 } is a partition of X finer than P n ∪ {E n }; (c) n≥1 (P n ∪ {E n }) generates the topology of X .
By writing V n for the set of all the non-empty subsets W of the form (3.1) corresponding to the pair (E n , P n ), and setting A n := A(E n , P n ) and R n := W ∈V n M |W | (K ), we get injective * -representations π n : A n → R n , in such a way that the diagrams Here ι n is the natural embedding ι n (χ Z t) = Z χ Z t where the sum is taken with respect to all the Z ∈ P n+1 satisfying Z ⊆ Z , the maps j n : R n → R n+1 are the embeddings given in [2,Proposition 4.2], and A ∞ , R ∞ are the inductive limits of the direct systems (A n , ι n ), (R n , j n ), respectively.

Remark 3.1
The algebra A ∞ can be explicitly described in terms of the crossed product, as follows. For U ⊆ X an open set, denote by C c,K (U ) the ideal of C K (X ) generated by the characteristic functions χ V , where V ranges over the clopen subsets One can define a Sylvester matrix rank function on each R n by the rule being Rk the usual rank of matrices. These Sylvester matrix rank functions are compatible with respect to the embeddings j n , so they give rise to a well-defined Sylvester matrix rank function rk ∞ on R ∞ .
Theorem 3.2 [2, Theorem 4.7 and Proposition 4.8] Following the above notation, if R rk := R ∞ denotes the rank completion of R ∞ with respect to its Sylvester matrix rank function rk ∞ (see Sect. 2.1), then there exists an injective * -homomorphism π rk : A → R rk making the diagram commutative, and sending the element t to π rk (t) = lim n π n (χ X \E n t). Moreover, we obtain a Sylvester matrix rank function rk A on A by restriction of rk ∞ (the extension of rk ∞ to R rk ) on A, which is extremal and unique with respect to the following property: Finally, the rank completion of A with respect to rk A gives back R rk , that is In particular, since R rk has the structure of a * -regular algebra, we can consider the * -regular closure of A inside R rk , which we denote by R A := R(A, R rk ), see [3] for more details. The * -regular closure R A is extensively studied in [3]. For our purposes, it will become important when computing O(n)-Betti numbers arising from the generalized odometer algebra O(n) in Sect. 5, and also in Sect. 4.3.

A formula for computing A-Betti numbers inside B
In this subsection, we give a formula for computing A-Betti numbers of elements from matrix algebras over the approximation algebra B = A(E, P). We will use ideas from [18], although applied to our construction.
By Theorem 3.2, the algebra A = C K (X ) T Z possesses a 'canonical' Sylvester matrix rank function rk A , unique with respect to Property (3.3). It is then natural to ask which is the set of positive real numbers reached by such a Sylvester matrix rank function.

Definition 3.3 Let
The set consisting of all A-Betti numbers of elements A ∈ M k (A) will be denoted by C(A). This set has the structure of a semigroup of (R + , +) (cf. Definition 2.2). The subgroup of (R, +) generated by C(A) will be denoted by G(A). It is shown in Sect. 4 that the definition of A-Betti number coincides with Definition (2.2) in case A is the lamplighter group algebra Γ , and thus, C(A) = C(Γ , K ).
We now focus on the approximation algebra B = A(E, P), where recall that E is any non-empty clopen subset of X , and P a (finite) partition of the complement X \E into clopen subsets. Let π : B → R B be the faithful * -representation on R B = W ∈V M |W | (K ) given by π(a) = (h W · a) W . We extend π to a faithful * -representation, also denoted by π , over matrix algebras in a canonical way. Hence, rk A can be computed over elements where Rk is the usual rank of matrices. Note that π(A) W ∈ M k (K ) ⊗ M |W | (K ) = M k|W | (K ). We thus obtain the following proposition.

Proposition 3.4 With the above notation, for a given element A
Here dim K (·) denotes the usual K-dimension of finite-dimensional K-vector spaces.
Proof It is just a matter of computation, using that Rk Let ·, · be the scalar product on K k ⊗ K |W | rendering the basis B orthonormal. Then for an element A ∈ M k (B), the entry of the matrix π(A) W corresponding to the e i j ⊗ e i j (W ) component is given by One can think of the matrix π(A) W as the adjacency-labeled matrix of an edge-labeled graph E A (W ) defined as follows: In this case, we label the arrow as Here adjacency-labeled matrix means that the coefficient of There is an example of such a graph in Fig. 1, corresponding to the matrix If we denote by Gr A (W ) the set consisting of the connected components of the graph E A (W ) and by A C , for C ∈ Gr A (W ), the adjacency-labeled matrix of the corresponding connected component C, then the matrix π(A) W is similar to the block-diagonal As a consequence, and by using Proposition 3.4, we obtain the formula which will be used in Sect. 4 for explicit computations of 2 -Betti numbers.
The following lemma, whose proof is trivial, is an adaptation of [18, Lemma 20] in our notation, and provides a graphical way for computing dimensions of kernels of finite matrices.
One can think of this system of equations in a graphical way: If we think of the variable λ (i,i ) as the label of the vertex (e i , e i (W )), then the flow lemma at this vertex is telling us that all arrows having range (i,i ) (label of the arrow) · (label of the source of the arrow) = 0.
To see how exactly the flow lemma works explicitly, we refer the reader to the appendix given in [18], where some applications of it in concrete examples are described.

The lamplighter group algebra
In this section, we apply the construction given in Sect. 3 to the lamplighter group algebra. This algebra is of great relevance because, among other things, it gave the first counterexample to the strong Atiyah conjecture (SAC), see for example [13,19].
We refer the reader to [3,Section 5], where a further application of the construction from Sect. 3 to a wider class of group algebras is given.

Definition 4.1
The lamplighter group Γ is defined to be the wreath product of the finite group of two elements, Z 2 , by Z. In other words, where the action implementing the semidirect product is the well-known Bernoulli shift σ defined by If we denote by t the generator corresponding to Z and by a i the generator corresponding to the i th copy of Z 2 , we have the following presentation for Γ : Let now K ⊆ C be a subfield of C closed under complex conjugation, which will be the involution on K . As in [3, Section 6], the Fourier transform F gives a and U i is the clopen set consisting of all points x ∈ X having a 0 at the i th component.
The collection of all the cylinder sets constitutes a basis for the topology of X .
We take μ to be the product measure on X , where the 1 2 , 1 2 -measure on each component {0, 1} is considered. It is well known (see, e.g., [24,Example 3.1]) that μ is an ergodic, full and shift-invariant probability measure on X . Therefore, by Theorem 3.2 we can construct a Sylvester matrix rank function rk A on A = C K (X ) T Z, as exposed in Sect. 3.

Proposition 4.2
As in Sect. 2.2, let rk K Γ be the Sylvester matrix rank function on K Γ inherited from U(Γ ). Then rk K Γ and rk A coincide through the Fourier transform F : (4.1) Proof Both rk K Γ and rk A coincide by Theorem 3.2, as observed in [3, Subsection 6.1]. The result follows from Formula (2.1).
We now recall the definition of the m-step Fibonacci sequence, as in [3, Definition 6.2].  We thus have injective * -representations π n : A n → R n defined by the . We will use the sequence {A n } n≥0 in the next subsection for computing rational values of 2 -Betti numbers arising from Γ .

Some computations of 2 -Betti numbers for the lamplighter group algebra
Although it does not appear in our approximating sequence, we now introduce the algebra A 1/2 , which gives the first non-trivial approximating algebra exhibiting irrational behavior of 2 -Betti numbers. Throughout this subsection, K will be any subfield of C closed under complex conjugation, which will be the involution on K .
The set V 1/2 consists of the non-empty W of the form: (a) either W = [111] of length 1, or (b) W = [110 k 1 10 k 2 1 · · · 0 k r 11] of length k = k 1 + · · · + k r + (r + 1), with r , k i ≥ 1, where 0 k i denotes 0 k i · · · 0. It is easily checked that ..,k r ≥1 k 1 + · · · + k r + (r + 1) 2 k 1 +···+k r +(r +3)  There exists then an element A inside some matrix algebra over A 1/2 with rational coefficients such that b (2) where q 0 , q 1 are nonzero rational numbers which can be explicitly computed. Therefore, we get a collection of irrational and even transcendental 2 -Betti numbers arising from the lamplighter group Γ . If moreover a 0,0 = 0, then Proof Since the proof of the theorem is quite technical, we will start by giving some examples of elements whose 2 -Betti number behave as stated, first by just considering a single polynomial p(k) and then by considering p(k)d k . After that, we will present an explicit element having 2 -Betti number as in (4.2). We start by giving a concrete example, p(x) = 2 + x + x 2 . Consider the element from M 11 (A 1/2 ) given by , we observe that π(A) W equals the zero 11 × 11 matrix, so its kernel has dimension 11. Figure 2 gives the prototypical graph E A (W ) that appears in case one takes a W of the second form [110 k 1 10 k 2 1 · · · 0 k r 11] with length k = k 1 +· · ·+k r +(r +1). By having in mind Formula (3.4), we study the different types of connected components of the graph E A (W ). From now on, in the diagrams each straight line should be labeled with a +1, and each dotted line with a −1. Nevertheless, we will explicitly write down some of the labels when necessary in the diagrams, and we will use straight lines in such cases.
(d) Finally C 4 , given by the graphs We have r − 1 connected components of this kind.
(d) Finally C 4 , given by the graphs We have r − 1 connected components of this kind.
In this case, we compute b (2) ..,k r ≥1 1 2 k 1 +···+k r +(r +3) (3n + 7) +(3n + 5)r + 3(k 1 + · · · + k r ) (3n + 7) + (3n + 11)r 2 r which is again an irrational number, and even transcendental (see, e.g., [40]). Just as before, the rational number accompanying α = k≥1 After these examples, one can derive the pattern in order to obtain an exponent of the form p 0 (k) + p 1 (k)d k 1 + · · · + p n (k)d k n by simply adding more levels, i.e., by considering matrices of higher dimension, and gluing the corresponding graphs in an appropriate way. We write down the corresponding element that gives rise to such a pattern. If we let N = m 0 + · · · + m n to be the sum of the degrees of the polynomials p 0 , . . . , p n respectively, with p i (x) = m i j=0 a j,i x j for 0 ≤ i ≤ n, then the element realizing the preceding pattern belongs to M 3N +n+5 (A 1/2 ), and is given explicitly by The elements in between connect the different polynomials p i , and the contributions (monomials) of the polynomials (that is, the sum p 0 (k) + p 1 (k)d k 1 + · · · + p n (k)d k n ) are accumulated in the χ [01] · e 0,0 component. A simplified schematic of a prototypical graph appearing here is as follows.
Using the same procedure as before, a straightforward (but quite tedious) computation allows us to conclude the proof. We only write down the contribution corresponding to the components C 4 of the graph E A (W ) where W is of the second type, which are the ones that contribute to the irrationality of the 2 -Betti number. We get We have r − 1 connected components of this kind. Thus, the contribution to b (2) (A) coming from these graphs is We leave the rest of the details to the reader. Note, again, that in the special case a 0,0 = 0 the irrational number α = k≥1 is accompanied by a rational number of the form 1 2 m . For the second part of the theorem, note that under the assumption a 0,0 = 0 we have always obtained irrational numbers of the form q 0 + q 1 α, being α the irrational number and q 0 , q 1 nonzero rational numbers with q 1 of the form 1 2 m for some m ≥ 1. In particular, q 0 +q 1 α belongs to G(Γ , Q). By a result of Ara and Goodearl [4, Corollary 6.14], the group G(Γ , Q) contains the rational numbers, and hence, the element q 1 α belongs to G(Γ , Q) too. Since q 1 = 1 2 m , we get This completes the proof of the theorem.

Remark 4.6
By making use of the ideas and techniques developed in Theorem 4.5, one can construct elements whose associated 2 -Betti number has a binary expansion following different kinds of exotic patterns. For example, by gluing in an appropriate way two graphs corresponding to different polynomials p 1 (x) and p 2 (x), that is, by constructing graphs of the form C 3 but substituting the bottom right graph (the one contributing to k i+1 ) by another graph corresponding to the polynomial p 2 (x), we can obtain terms in the computation of the associated 2 -Betti number of the form and more generally of the form being p i (x), q j (x) polynomials satisfying the hypotheses of the theorem, and d i , l j ≥ 2 integers.
To conclude this section, it may be instructive to compute some rational values of 2 -Betti numbers. In [19] (cf. [4,13,17]) the authors compute the 2 -Betti number of the element a 0 = e 0 t + t −1 e 0 = χ X \E 0 t + t −1 χ X \E 0 , which belongs to the first of our approximating algebras, A 0 . We will compute, in general, the 2 -Betti number of the element a n = χ X \E n t + t −1 χ X \E n ∈ QΓ belonging to the * -subalgebra A n . Under π n -and recalling that R n = Q × k≥1 M m+k (Q) Fib m (k) with m = 2n + 1-it gives π n (a n ) = (0, (t m+k + t * m+k , Fib m (k) . . . , where t r is the r × r lower triangular matrix given by It is then straightforward to show that This sum can be computed by using the summation rule k≥1 Fib m (2k) 2 2k = 2 m−1 2 m+1 − 1 2 m+2 + 1 , whose proof can be found in [10, Lemma 3.2.11]. We then have b (2) (a n ) = 3 1 + 2 2n+3 .
Note that b (2) (a 0 ) = 1 3 , and we recover the result from [13,19]. Also, as n → ∞, this value tends to zero, as expected since a n → t + t −1 in rank, which is invertible inside R rk .

Rational series and 2 -Betti numbers
In this section, we let again K ⊆ C be any field closed under complex conjugation. A more algebraic perspective to attack the problem of computing values from G(A 1/2 ) ⊆ G(Γ , K ) is through the determination of the * -regular closure R 1/2 of the algebra A 1/2 seen inside R 1/2 = K × k≥1 M k+2 (K ) Fib 2 (k) through the embedding π 1/2 : A 1/2 → R 1/2 , and taking advantage of [3,Proposition 4.1] which states that the values of ranks of elements from R 1/2 are all included in G (Γ , K ).
In order to clarify the exact relationship of this approach with Atiyah's problem, it is convenient to introduce the following definitions for a general unital ring.

Definition 4.7
Let R be a unital ring and let rk be a Sylvester matrix rank function on R. We denote by C rk (R) the set of real numbers of the form k − rk(A) for A ∈ M k (R), k ≥ 1. We denote by C rk (R) the set of real numbers of the form rk(A) for A ∈ M k (R), k ≥ 1. Both C rk (R) and C rk (R) are subsemigroups of (R + , +). We denote by G rk (R) the subgroup of R generated by C rk (R). Clearly Remark 4.8 With this notation, [22,Corollary 6.2] says that, if U is a * -regular ring and R is a * -subring of U such that U is the * -regular closure of R in U, then we have for every Sylvester matrix rank function rk on U. Thus, the main object of study within this approach is the group G rk (R). Of course, C(G, K ) = C rk (K G) for any discrete group G, where rk is the canonical rank. It would be interesting to know conditions under which C rk (R) = C rk (R) and under which C rk (R) = G rk (R) ∩ R + , or C rk (R) = G rk (R) ∩ R + . Note that ∀g ∈ G rk (R) ∃t ∈ Z + : g + t ∈ C rk (R) ⇐⇒ ∀g ∈ G rk (R) ∃t ∈ Z + : g + t ∈ C rk (R).
In case R is von Neumann regular, we have that C rk (R) = C rk (R) and the above equivalent conditions hold automatically, for every Sylvester matrix rank function rk.
Next, consider the subset such that each b i t i ∈ A 1/2,i t i belongs to the K -linear span of the special terms of degree i. We denote by S 1/2 [t; T ] the subspace of S 1/2 [[t; T ]] consisting of elements with finite support. Let X = {x 1 , x 2 , . . . } be an infinite countable set, and consider the algebras K X and K X of non-commutative polynomials and non-commutative formal power series, respectively, with coefficients in K . We consider the degree in K X given by d(x i ) = i + 1 for all i ≥ 1 and d(1) = 0. With this degree function, K X is a graded K -algebra, where K has the trivial degree.
The following result is shown in [3, Lemma 6.5 and Proposition 6.6]. Note that [3, Propositions 6.6 and 6.7] are stated only for n ≥ 1, but they hold, with the same proof, for n = 1/2.

Proposition 4.9 The linear subspaces
There is an isomorphism of unital graded algebras K X ∼ = S 1/2 [t; T ] sending the element x i r · · · x i 1 to χ [10 i 1 10 i 2 1···10 ir 1] t i , where i = i 1 + · · · + i r + r , and 1 to χ [1] . This isomorphism extends naturally to an isomorphism between The special terms of the form χ [10 i 1] t i+1 , for i ≥ 1, which are the algebra generators of S 1/2 [t; T ], are called pure special terms, see the proof of [3,Proposition 6.6].
Let X * denote the free monoid on X . We now recall the concept of the Hadamard product on K X , see [9] for more details. If a = w∈X * a w w and b = w∈X * b w w are elements in K X , set We will denote by K X • the algebra of non-commutative formal power series endowed with the Hadamard product. Note that K X • is a * -regular ring with respect to the involution given by w∈X * a w w = w∈X * a w w. The algebra of noncommutative rational series, denoted by K rat X is defined as the division closure of K X in K X , that is, K rat X is the smallest subalgebra of K X containing K X and closed under inversion, see [9] for details. It is well known that K rat X is closed under the Hadamard product [9, Theorems 1.5.5 and 1.7.1].
Let p 1/2 := p E 1/2 = π 1/2 (χ E 1/2 ) be the projection in T ]] from Proposition 4.9 (which respects the respective Hadamard products and involutions) with this isomorphism, we obtain a * -isomorphism is the coefficient of f corresponding to the monomial x i r · · · x i 1 .
For each n ≥ 1, set X n = {x 1 , . . . , x n }. Let L be a language over X n , i.e., a subset of the free monoid X * n on X n . We now give a formula for the rank of the element (L), where L := w∈L w ∈ K X n is the characteristic series of L ( [9, p.8]). Note that, for W = [110 i 1 10 i 2 1 · · · 10 i r 11] ∈ V 1/2 , we have Moreover, we have μ(W ) = 2 −(i 1 +···+i r +r +3) = 2 −3 2 −d(w) , where w = x i r · · · x i 2 x i 1 . Let s(L) = j≥0 α j x j be the series in Z + [[x]] such that α j is the number of words w ∈ L such that d(w) = j. Then we have Here we make use of (3.2) for the first equality, of the above observations for the second, and of the definition of s(L) for the third. We say that s(L) ∈ Z + [[x]] is the generating function of L with respect to d, and we denote by α L the real number rk( (L)) described in (4.3). We now recall the notion of a rational language (see [9,Chapter 3]).

Definition 4.10
The set of rational languages over a finite set F is the smallest set of subsets of the free monoid F * containing all the finite subsets of F * and closed under the following operations: (a) union L 1 ∪ L 2 ; (b) product L 1 L 2 := {w 1 w 2 ∈ F * | w 1 ∈ L 1 , w 2 ∈ L 2 }; (c) * -product L * := k≥0 L k .
A characterization of the rational languages is the following ([9, Lemma 3.1.4]): a language over a finite alphabet F is rational if and only if it is the support of some rational series z ∈ Z + F . In fact, if L is a rational language over F, its characteristic series L = w∈L w belongs to R rat F • for any semiring R ([9, Proposition 3.2.1]).
It turns out that the set of rational languages over F, which we will denote by L(F), forms a Boolean subalgebra of P(F * ), the power set of F * ([9, Corollary 3.1.5]). However, the converse may not be true. Indeed, for a subfield K of C closed under complex conjugation, given a rational series z = w∈F * λ w w ∈ K rat F it may be the case that its support L = supp(z) is not a rational language. We put and we let B K rat (F) be the Boolean algebra of subsets of F * generated by K(F). Then by [3,Proposition 6.10] the * -regular closure of K rat F • in K F • is contained in the set of formal power series whose support belongs to B K rat (F), and for each set L ∈ B K rat (F), the characteristic series L belongs to the * -regular closure. We gather some of these facts in the following.

Proposition 4.11
Let n ≥ 1 and let X n = {x 1 , . . . , x n }, endowed with the degree function d(x i ) = i + 1. With the above notation, we have L(X n ) ⊆ K(X n ) ⊆ B K rat (X n ). The sets L(X n ) and B K rat (X n ) are Boolean subalgebras of P(X * n ), but this is not the case in general for K(X n ). Moreover, if L ∈ B K rat (X n ), then α L ∈ G(Γ , K ).
We now show that rational languages give rise to rational 2 -Betti numbers.

Lemma 4.12
If a series j≥0 α j x j ∈ Z + [[x]] is the generating function of a rational language L ∈ L(X n ) with respect to d, then it is a rational series with constant term either 0 or 1. Consequently, α L ∈ Q.
We obtain now a concrete irrational algebraic number of the form α L for a suitable L ∈ B Q rat (X 2 ).

Theorem 4.13
There is L ∈ B Q rat (X 2 ) such that α L = 1 4 2 7 . In particular, 1 Proof Let L be the language on X 2 = {x 1 , x 2 } given by Here |w| x i is the number of appearances of x i in w, for i = 1, 2. Then by [9, Example 3.4.1], L ∈ B Q rat (X 2 ) (although L is not a rational language). If w ∈ L, then since d(x 1 ) = 2 and d(x 2 ) = 3, we get that d(w) = 5l, where l = |w| x 1 = |w| x 2 . Therefore, we get from Proposition 4.11 that Taking different choices of variables x r , x s , with r = s, we indeed obtain that G(Γ , Q) contains all the algebraic numbers Similar formulas can be obtained to compute the ranks of elements coming from the copy of the algebra Q rat X in the * -regular closure R n of A n in R n , where A n are the approximations of QΓ described in Sect. 4.1. The only differences are that we have different degree functions on X , depending on n, and that the factor 1/8 must be substituted by the factor 2 −2n−2 . is itself an 2 -Betti number, and in that case, to determine a concrete matrix over QΓ witnessing this fact. In particular, this would solve a question posed by Grabowski in [17, p. 32], see also [18,Question 3]. We do not know any example of a group G for which C(G, Q) = G(G, Q) ∩ R + .
We close this section with the following problem: Question 4. 15 With the notation established before, is α L always an algebraic number when L ∈ B Q rat (X n )?

The odometer algebra
In this last section, we focus on computing the whole set of values that the Sylvester matrix rank function constructed from Theorem 3.2 can achieve in the case of the generalized odometer algebra. We first recall its definition. Fix a sequence of natural numbers n = (n i ) i∈N with n i ≥ 2 for all i ∈ N, and consider X i to be the finite space {0, 1, . . . , n i −1} endowed with the discrete topology. From these we form the topological space X = i∈N X i endowed with the product topology, which is in fact a Cantor space. Let T be the homeomorphism on X given by the odometer, namely for x = (x i ) ∈ X , T is given by Note that the odometer action is just addition of (1, 0, . . . ) by carryover. Let (K , −) be any field with a positive definite involution −. The generalized odometer algebra is defined as the crossed product * -algebra O(n) := C K (X ) T Z. We obtain a measure μ on X by taking the usual product measure, where we consider the measure on each component X i which assigns mass 1 n i on each point in X i . It is well known (e.g., [11,Section VIII.4]) that μ is an ergodic, full and T -invariant probability measure on X , which in turn coincides with the Haar measure μ on X if one considers X as an abelian group with addition by carryover. We denote by rk O(n) the Sylvester matrix rank function on O(n) given by Theorem 3.2.
We denote by O(n) m the unital * -subalgebra of O(n) generated by the partial isometries {χ Z t | Z ∈ P m }. The quasi-partition P m is really simple in this case: write Z m,l = T l (E m ) for 1 ≤ l < p m . Note that these clopen sets form exactly the partition P m , and that T (Z m, p m −1 ) = E m . Therefore, there is only one possible W ∈ V m , which has length p m and is given by We can completely characterize these * -subalgebras. In particular, since lim − →m M p m (K (t p m )) has a unique Sylvester matrix rank function, it coincides with the one given by Theorem 3.2 under the previous * -isomorphism.

Characterizing the set C (O(n))
We characterize the set C (O(n)) of all positive real values that the Sylvester matrix rank function rk O(n) can achieve, and the subgroup G(O(n)) it generates. First, let where ¶ is the set of prime numbers ordered with respect to the natural ordering, and each ε q (n) ∈ {0} ∪ N ∪ {∞}. In more detail, if n i = q∈ ¶ q ε q (n i ) is the prime decomposition of n i , then, for each q ∈ ¶, ε q (n) is defined by ε q (n) = ∞ i=1 ε q (n i ) ∈ {0} ∪ N ∪ {∞}, and n is defined as the formal product q∈ ¶ q ε q (n) .
As in [26,Definition 7.4.2], from any supernatural number n one can construct an additive subgroup of Q containing 1, denoted by Z(n), consisting of those fractions a b with a ∈ Z, and b ∈ Z\{0} being of the form where ε q (b) ≤ ε q (n) for all q ∈ ¶, and ε q (b) = 0 for all but finitely many q's. If n comes from a sequence n = (n 1 , n 2 , . . . ) as above, Z(n) is exactly the additive subgroup of Q consisting of those fractions of the form a n 1 · · · n r , with a ∈ Z and r ≥ 1.
Each group Z(n) gives rise to a subgroup G(n) = Z(n)/Z of the group Q/Z. Indeed the groups G(n) describe all the subgroups of Q/Z. These groups are called Prüfer groups.

Proof
The argument is similar to the one given in the proof of [3, Proposition 4.1].
Since R O(n) is a * -regular ring with positive definite involution, each matrix algebra M k (R O(n) ) is also * -regular. Hence, for each A ∈ M k (R O(n) ) there exists a projection P ∈ M k (R O(n) ) such that rk R O(n) (A) = rk R O(n) (P). We conclude that C (R O(n) ) equals the set of positive real numbers of the form rk R O(n) (P), where P ranges over matrix projections with coefficients in R O(n) . Now each such projection P is equivalent to a diagonal projection [16,Proposition 2.10], that is of the form diag( p 1 , p 2 , . . . , p r ) for some projections p 1 , . . . , p r ∈ R O(n) , so that rk R O(n) (P) = rk R O(n) ( p 1 ) + · · · + rk R O(n) ( p r ). But since R O(n) ∼ = lim − →m M p m (K (t p m )) by Theorem 5.4, the set of ranks of elements in R O(n) is contained in Z(n) + ∩ [0, 1]. Therefore, rk R O(n) (P) ∈ Z(n) + . This proves the inclusion C (R O(n) ) ⊆ Z(n) + . The inclusion Z(n) + ⊆ C (O(n)) is straightforward, since for 0 ≤ i < p m we have rk O(n) (e (m) ii ) = 1 p m . The equality C(O(n)) = Z(n) + also follows from the above. The last part of the theorem is immediate.
In a sense, Theorem 5.6 confirms the SAC for a class of crossed product algebras, as follows:

Remark 5.7
Let n be a supernatural number and let n = i∈N n i a decomposition of n, as above. Consider the presentation of G(n) given by generators {g m } m∈N and relations g n 1 1 = 1 and g n m+1 m+1 = g m for all m ≥ 1. The topological group X (n) = i∈N X i is naturally isomorphic with the Pontryagin dual G(n) of the Prüfer group G(n). Indeed, let ξ p m denote the canonical primitive p m -root of unity in C, where p m = n 1 · · · n m , and observe that ξ m n+1 p m+1 = ξ p m for each m ∈ N. The map φ : X (n) → G(n) defined by φ x (g m ) = ξ a 1 +a 2 n 1 +···+a m n m−1 p m , where x = (a 1 , a 2 , . . . ) ∈ X (n), is a group isomorphism and a homeomorphism. Therefore, if K is a subfield of C closed under complex conjugation containing all p th m roots of unity, Fourier transform defines an isomorphism F : K [G(n)] −→ C K (X (n)), and we can pull back the automorphism induced by T on C K (X (n)) to an automorphism ρ on K [G(n)] (which is not induced by an automorphism of G(n)). Note that ρ(g m ) = ξ p m g m ∈ K · G(n) for all m ≥ 1. Therefore, we can interpret Theorem 5.6 as giving a positive answer to the SAC for the crossed product A = K [G(n)] ρ Z ∼ = O(n): the set of A-Betti numbers is exactly the semigroup Z(n) + generated by the inverses of the orders of the elements of G(n). Note finally that the groups X (n), n a supernatural, give all the profinite completions of Z. With this view, the dynamical system considered on X (n) is generated by addition by 1.
Data availability Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.