Large time limit and local L2-index theorems for families

We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut - Lott type superconnections in the L2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L2-index formulas. As applications, we prove a local L2-index theorem for families of signature operators and an L2-Bismut - Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber - Tondeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L2-eta forms and L2-torsion forms as transgression forms.