Lifting idempotents in group rings

Abstract

In a celebrated remark, Kaplansky observed that the theory of von Neumann algebras implies that the abstract group algebra over a field of characteristic zero is von Neumann finite. This means that if x and y are elements of k[G] such that xy= 1 then it follows that yx= 1. In what is essentially the only known proof, Kaplansky first argues that the group algebra trace of a nonzero idempotent is a positive real number not exceeding 1 (5, p. 381. Since Iyx is an idempotent with zero trace, it must be zero. The same problem over fields of positive characteristic is open. For one thing, the trace of a nonzero idempotent may be zero. If G is the group of order three generated by c then t2+ t is an idempotent in Z/2Z[G] whose trace is zero. On the other hand, this idempotent can be lifted uniquely to the ‘p-adic group ring’ z2[G] where the resulting idempotent has trace 2/3. The hope is that a lifted trace can substitute for the very effective trace in characteristic zero. This paper is a preliminary study of the trace. We show that the trace of the lifting of an idempotent is independent of lifting. It is a function only of the equivalence class of an idempotent. Finally, and most important, the trace is always a p-adic integer. As a consequence, it is independent of specialization. Out general notation follows [5]. FP will denote the field with p elements and FP will be its algebraic closure. 2, is the ring of p-adic integers; its quotient field is QP. Throughout the paper A will denote a commutative algebra over the field FP.