Regularized and L-Determinants

In the early seventies D. Ray and I. Singer [17] introduced the notion of zeta-regularized determinants. They used it to define the analytic version of Reidemeister torsion as an alternating product of determinants. One way to understand analytic torsion is to consider it as a ”multiplicative index” of an elliptic complex. By the L2-index theorem of M. Atiyah [1] this analogy suggests that one should define ”L2-torsion” via the use of von Neumann traces and one should ask whether a ”multiplicative L2-index theorem” holds, which would say that analytic torsion should coincide with the L2-torsion. This, however, fails to hold. To measure the failure one considers the quotient analytic torsion L2−torsion . For this number there sometimes is a Lefschetz theorem expressing it as (regularized) sum of local geometric contributions (see [7]). Since the torsion is an alternating product of determinants one should more generally consider the quotient regularized determinant L2−determinant . The study of the