The main theme of this talk is the interplay between the algebra of group rings and the analysis behind \( L^2 \)-Betti numbers.

Let \( \Gamma \) be a discrete group, \( \widetilde X \) a free \( \Gamma \)-CW-complex with finite quotient \( X = \Gamma \backslash \widetilde X \) and chain complex \( C_*(\widetilde X), d_* \).

Fix lifts of the cells of \( X \) in \( \widetilde X \), so that for each \( p \) we can identify \( C_p \cong {\mathbb Z}\Gamma^{n_p} \) and let \( A_p \in M_{n_p \times n_{p-1}}({\mathbb Z}\Gamma) \) represent the differential \( d_p \), which extends to a continuous operator between the Hilbert spaces \( \ell^2(\Gamma)^{n_p} \to \ell^2(\Gamma)^{n_{p-1}} \). Denote by \( P \) the orthogonal projection from \( \ker(d_p) \) onto the closure of \( \mathrm{Im}(d_{p+1}) \). The \( L^2 \)-Betti numbers are then defined by \( b_p^{(2)}(\widetilde X ; \Gamma) = \mathrm{tr}_{\mathcal N(\Gamma)}(P) \).

1. When \( \widetilde X = E\Gamma \) these numbers depend only on the group and are called the \( L^2 \)-Betti numbers of \( \Gamma \).
2. The image of \( P \) is also the kernel of the operator \( \Delta_p = d_p^*d_p + d_{p-1}d_{p-1}^* \in M_{n_p \times n_p}({\mathbb Z}\Gamma) \).

Let \( \mathrm{Bet}(\Gamma) \) be the set of all real numbers appearing as \( L^2 \)-Betti numbers of \( \Gamma \)-complexes \( \widetilde X \) as above, which is also equal (by point 2. just above) to the set of von Neumann dimensions of matrices in \( M_{n \times n}({\mathbb Z}\Gamma) \) for \( n \ge 1 \).

Strong Atiyah Conjecture: Let \( \mathrm{lcm}(\Gamma) = \mathrm{lcm}(|G| : G \le \Gamma, |G| < +\infty) \). Then \( \mathrm{Bet}(\Gamma) \subset \frac 1 {\mathrm{lcm}(\Gamma)} {\mathbb Z} \).

The same statement with \( {\mathbb Z} \) replaced by any number field is equivalent to the statement for \( {\mathbb Z} \). The conjecture is also made for coefficients in \( K[G] \) where \( K \) is any subfield of the complex numbers.

The conjecture is known to hold in many cases. For free groups Linnell proved that the von Neumann dimension of the kernel any matrix with coefficients in a group ring \( K\Gamma, K\subset {\mathbb C} \) is an integer. His statement is in fact more general and rests on an algebraic reformulation of the conjecture. The von Neumann algebra \( \mathcal N\Gamma \) (the von Neumann algebra of bounded operators on \( \ell^2(\Gamma) \) which commute with the right-\( \Gamma \)-action) fits into the following diagram :
\[
\begin{array}{ccc}
K\Gamma & \subset & \mathcal N\Gamma \\
\cap & & \cap \\
D_K\Gamma & \subset & \mathcal U\Gamma \end{array}
\]
where:

• \( \mathcal U\Gamma \) is the algebra of affiliated operators, the Ore localisation of \( \mathcal N\Gamma \) at the multiplicative subset of non-zero-divisors (in general it is not a field, not even an integral domain);
• \( D_K\Gamma \) is the division closure of \( K\Gamma \) in \( \mathcal U\Gamma \), the subset of all \( y^{-1}x \) for \( x\in K\Gamma \) and \( y\in K\Gamma \) invertible in \( \mathcal U\Gamma \).

As an example, in the case \( \Gamma = \langle z \rangle \cong {\mathbb Z} \) these algebras are explicitely described as follows:

• \( \mathcal U\Gamma \) is the algebra of classes of measurable functions on \( \mathbb T^1 = \{z\in{\mathbb C} : |z|=1 \} \);
• \( K\Gamma = K[z, z^{-1}] \) is the subalgebra of Laurent polynomials;
• \( \mathcal N\Gamma = L^\infty(\mathbb T^1) \) is the subalgebra of essentially bounded functions;
• \( D_K\gamma = K(z) \) is the subalgebra of rational functions.

Theorem (Linnell): Let \( \Gamma \) be a torsion-free group. Then the strong Atiyah conjecture with coefficients in \( K \) holds for \( \Gamma \) if and only if \( D_K\Gamma \) is a skew field.

A remarkable feature of this statement is that if one manages to give an intrinsic characterisation of the algebra \( D_K\Gamma \) (i.e. without referring to the von Neumann algebra) then it gives a completely algebraic formulation for the Atiyah conjecture. In special classes of groups this is in fact possible, for instance:

• If \( \Gamma \) is amenable then \( D_K\Gamma \) is the Ore localisation of \( K\Gamma \) at non-zero-divisor elements ;
• If \( \Gamma \) is bi-orderable then \( D_K\Gamma \) is the Mal’cev–von Neumann completion of \( K\Gamma \).

This gives an approach to the Atiyah conjecture which might be valid beyond characteristic 0 fields, in particular for finite fields.

It is desirable to have an extension of Linnell’s approach to the case where \( \Gamma \) is not necessarily torsion-free.

Theorem (Knebusch–Linnell–Schick): Let \( \Gamma \) be a discrete group with \( \mathrm{lcm}(\Gamma) < +\infty \) and let \( K \subset {\mathbb C} \) be a subfield stable under complex conjugation. Then the Atiyah conjecture for center-valued \( L^2 \)-Betti numbers for \( K\Gamma \) holds if and only if \( D_K\Gamma \) is a direct sum of finite matrix algebras over skew fields.

The theorem in fact gives a precise description of the degrees of the matrix algebras involved; the nature of the skew fields remains mysterious (as it is already in general in the torsion-free case).

A particular case where the statement becomes simpler is when \( \Gamma \) does not contain any non-trivial finite normal subgroup. In this setting, if \( d = \mathrm{lcm}(\Gamma) \) the statement is that the center-valued Atiyah conjecture holds for \( K\Gamma \) if and only if \( D_K\Gamma \cong M_{d\times d}(D) \) for some skew field \( D \).

More precisely, in this case « center-valued von Neumann dimensions » are the same as classical von Neumann dimensions, and the result is that the dimensions of kernels of matrices with coefficients in \( K\Gamma \) are dimensions of associated \( M_{d\times d}(D) \)-modules, which are easily seen to lie in \( d^{-1}{\mathbb Z} \) (they are direct sums of the module \( M_{d\times 1}(D) \), which is of dimension \( 1/d \)).

In general the center-valued dimensions lie a priori in the center \( Z(\mathcal N\Gamma) \). The center-valued Atiyah conjecture predicts that they in fact lie in the subalgebra \( Z({\mathbb C}\Gamma) \), which is equal to the elements supported on finite conjugacy classes in \( \Gamma \), and constant along them.