This is an announcement for the paper "Geometric, spectral and asymptotic properties of averaged products of projections in Banach spaces" by Catalin Badea and Yuri I. Lyubich. Abstract: According to the von Neumann-Halperin and Lapidus theorems, in a Hilbert space the iterates of products or, respectively, of convex combinations of orthoprojections are strongly convergent. We extend these results to the iterates of convex combinations of products of some projections in a complex Banach space. The latter is assumed uniformly convex or uniformly smooth for the orthoprojections, or reflexive for more special projections, in particular, for the hermitian ones. In all cases the proof of convergence is based on a known criterion in terms of the boundary spectrum. Archive classification: math.FA Remarks: 22 pages Submitted from: catalin.badea@math.univ-lille1.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1006.2052 or http://arXiv.org/abs/1006.2052