Idempotents in a complex group algebra, projective modules, and the von Neumann algebra

Abstract.

The complex group algebra ${\Bbb C}G$ of a countable group G can be imbedded in the von Neumann algebra NG of G. If G is torsion-free, and if P is a finitely generated projective module over ${\Bbb C}G$ it is proved that the central-valued trace of $NG\otimes _{{\Bbb C}G}P$, i.e. of an idempotent ${\Bbb C}G$-matrix A defining P is equal to the canonical trace $\kappa (P)$ times identity I. It follows that $\kappa (P)$ characterizes the isomorphism type of $NG\otimes _{{\Bbb C}G}P$.¶If $\kappa (P)$ is an integer, e.g., if the weak Bass conjecture holds for G then $NG\otimes _{{\Bbb C} G}P$ is free. It is also shown that for certain classes of groups geometric arguments can be used to prove the Bass conjecture.