A Rationality Criterion For Unbounded Operators

Peter A. Linnell

Let G be a group, let U(G) denote the set of unbounded operators on L²(G) which are affiliated to the group von Neumann algebra W(G) of G, and let D(G) denote the division closure of $\mathbb {C}$G in U(G). Thus D(G) is the smallest subring of U(G) containing $\mathbb {C}$G which is closed under taking inverses. If G is a free group then D(G) is a division ring, and in this case we shall give a criterion for an element of U(G) to be in D(G). This extends a result of Duchamp and Reutenauer, which was concerned with proving a conjecture of Connes.