The Atiyah Conjecture and Artinian Rings

Let G be a group such that its finite subgroups have bounded order, let d denote the lowest common multiple of the orders of the finite subgroups of G, and let K be a subfield of C that is closed under complex conjugation. Let U(G) denote the algebra of unbounded operators affiliated to the group von Neumann algebra N(G), and let D(KG,U(G)) denote the division closure of K GinU(G); thus D(KG,U(G)) is the smallest subring of U(G) containing KG that is closed under taking inverses. Suppose n is a positive integer, andα∈Mn(KG). Thenαinduces a bounded linear mapα:`2(G)n→`2(G)n, and ker α has a well-defined von Neumann dimension dimN(G)(kerα). This is a nonnegative real number, and one version of the Atiyah conjecture states that ddimN(G)(kerα)∈Z. Assuming this conjecture, we shall prove that if G has no nontrivial finite normal subgroup, then D(KG,U(G)) is ad×d matrix ring over a skew field. We shall also consider the case when G has a nontrivial finite normal subgroup, and other subrings of U(G) that contain KG.