Von Neumann algebra invariants of Dirac operators

Abstract

In this paper we define and study certain von Neumann algebra invariants associated to the Dirac operator acting onL2spinors on the universal covering space of a compact, Riemannian spin manifold. We first study a Novikov-Shubin type invariant, which is aconformal invariantbut which isnotindependent of the choice of metric. However, we prove results which give evidence that this invariant may always be positive. When the Novikov-Shubin type invariant is positive, we can define the von Neumann algebra determinant of the Dirac Laplacian and the corresponding element in the determinant line of the space ofL2harmonic spinors on the universal covering space, which we also compute for certain locally symmetric spaces. Finally, we study a von Neumann algebra eta invariant associated to the Dirac operator and we show that it is sometimes an obstruction to the existence of metrics of positive scalar curvature. © 1998 Academic Press.