Von Neumann Algebra Invariants of Dirac Operators

✍ Scribed by Varghese Mathai

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 407 KB
• Volume: 152
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3179

✦ Synopsis

In this paper we define and study certain von Neumann algebra invariants associated to the Dirac operator acting on L 2 spinors on the universal covering space of a compact, Riemannian spin manifold. We first study a Novikov Shubin type invariant, which is a conformal invariant but which is not independent 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 of L 2 harmonic 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 * (see Definition 1.2), then we prove results which give evidence that :( g) may always be positive. The results in this section are inspired by the papers of Gromov and Shubin [12], Lott [16] and the author [18], where the invariants associated to the Laplacian of the deRham complex were studied. When :( g) is positive, we define the von Neumann algebra determinant of the Dirac Laplacian and the corresponding element in the article no. FU973179