A Rationality Criterion for Unbounded Operators

✍ Scribed by Peter A. Linnell

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 133 KB
• Volume: 171
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1999.3526

✦ Synopsis

Let G be a group, let U(G) denote the set of unbounded operators on L 2 (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.

2000 Academic Press

Soient G un groupe, U(G) l'ensemble d'ope rateurs non borne s affilie s aÁ l'algeÁ bre de von Neumann de groupe de G, et D(G) la clo ^ture de division de CG dans U(G). Ainsi D(G) est le plus petit anneau qui est ferme sous l'ope ration d'inverse. Si G est un group libre, nous donnons un criteÁ re pour qu'un e le ment de U(G) soit dans D(G).