Analytic versions of the zero divisor conjecture

Peter A. Linnell

This is an expanded version of the three lectures I gave at the Durham conference, which took place in July 1994. The material is mainly expository, though there are a few new results, and for those I have given complete proofs. While the subject matter involves analysis, it is written from an algebraic point of view. Thus hopefully algebraists will find the subject matter comprehensible, though analysts may find the analytic part rather elementary.

The topic considered here can be considered as an analytic version of the zero divisor conjecture over $\mathbb {C}$ : recall that this states that if G is a torsion free group and 0$\ne$$\alpha$,$\beta$ $\in$ $\mathbb {C}$G, then $\alpha$$\beta$$\ne$ 0. Here we will study the conjecture that if 0$\ne$$\alpha$ $\in$ $\mathbb {C}$G and 0$\ne$$\beta$ $\in$ L^p(G), then $\alpha$$\beta$$\ne$ 0 (precise definitions of some of the terminology used in this paragraph can be found in later sections). We shall also discuss applications to L^p-cohomology.