Let \(l^2(G)\) be the Hilbert space over the complex numbers \(\mathbb C\) with orthonormal basis the elements of a group \(G\). So the elements of \(l^2(G)\) have the form \(\sum_{g\in G}a_gg\), where \(a_g\in\mathbb C\) and \(\sum_{g\in G}|a_g|^2<\infty\). Let \(\mathbf B\) denote the bounded linear operators on \(l^2(G)\). Then \(\mathbb CG\) acts faithfully on the left on \(l^2(G)\) as bounded linear operators via the left regular representation and we may consider \(\mathbb CG\) as a subalgebra of \(\mathbf B\). The weak closure of \(\mathbb CG\) in \(\mathbf B\) is the group von Neumann algebra \(N(G)\) of \(G\). Suppose that \(n\) is a positive integer and \(\alpha\in M_n(\mathbb CG)\). Then \(\alpha\) induces a bounded linear map \(\alpha\colon l^2(G)^n\to l^2(G)^n\), and \(\ker\alpha\) has a well-defined von Neumann dimension \(\dim_{N(G)}(\ker\alpha)\) that is a nonnegative real number.
Let \(K\) be a subfield of \(\mathbb C\) that is closed under the complex conjugation. Assume that \(G\) is a group such that its finite subgroups have bounded orders and let \(\text{lcm}(G)\) be the lowest common multiple of the orders of the finite subgroups of \(G\). We say that the strong Atiyah conjecture holds for \(G\) over \(K\) if \(\text{lcm}(G)\dim_{N(G)}(\ker\alpha)\) is an integer for all \(\alpha\in M_n(KG)\).
Let \(U(G)\) be the algebra of unbounded operators on \(l^2(G)\) affiliated to \(N(G)\). Let \(D(KG)\) be the division closure of \(KG\) in \(U(G)\), i.e. \(D(KG)\) is the smallest subring of \(U(G)\) that is closed under taking inverses. If the maximal FC-subgroup of \(G\) is torsion-free and \(d=\text{lcm}(G)<\infty\), then the main result asserts that \(G\) satisfies the strong Atiyah conjecture over \(K\) if and only if \(D(KG)\) is a \((d\times d)\)-matrix ring over a skew field.
Let \(E(KG)\) be the extended division closure of \(KG\) in \(U(G)\), i.e. \(E(KG)\) is the smallest subring of \(U(G)\) containing \(KG\) with the properties:
(a) \(x\in E(KG)\) and \(x^{-1}\in U(G)\) implies \(x^{-1}\in E(KG)\);
(b) \(x\in E(KG)\) and \(xU(G)=eU(G)\), where \(e\) is a central idempotent of \(U(G)\), implies \(e\in E(KG)\).
The second main result asserts that if \(G\) satisfies the strong Atiyah conjecture over \(K\), then \(E(KG)\) is a semisimple Artinian ring. Observe that this result follows immediately from a more general result of the authors.