Title: Von Neumann dimension, Hodge index theorem and geometric applications,
Abstract: This note contains a reformulation of the Hodge index theorem within the
framework of Atiyah's $L^2$-index theory. More precisely, given a compact
Kähler manifold $(M,h)$ of even complex dimension $2m$, we prove that
$$\sigma(M)=\sum_{p,q=0}^{2m}(-1)^ph_{(2),\Gamma}^{p,q}(M)$$ where $\sigma(M)$
is the signature of $M$ and $h_{(2),\Gamma}^{p,q}(M)$ are the $L^2$-Hodge
numbers of $M$ with respect to a Galois covering having $\Gamma$ as group of
Deck transformations. Likewise we also prove an $L^2$-version of the
Frölicher index theorem. Afterwards we give some applications of these two
theorems and finally we conclude this paper by collecting other properties of
the $L^2$-Hodge numbers.