A Rationality Criterion for Unbounded Operators

Résumé 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 CG in U(G). Thus D(G) is the smallest subring of U(G) containing CG 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. Copyright 2000 Academic Press. Soient G un groupe, U(G) lʹensemble dʹopérateurs non bornés affiliés à lʹalgèbre de von Neumann de groupe de G, et D(G) la clôture de division de CG dans U(G). Ainsi D(G) est le plus petit anneau qui est fermé sous lʹopération dʹinverse. Si G est un group libre, nous donnons un critère pour quʹun élément de U(G) soit dans D(G).