This is an announcement for the paper "Compositions of projections in Banach spaces and relations between approximation properties" by M.I. Ostrovskii. Abstract: A necessary and sufficient condition for existence of a Banach space with a finite dimensional decomposition but without the $\pi$-property in terms of norms of compositions of projections is found. Archive classification: math.FA Mathematics Subject Classification: 46B07 Citation: Rocky Mountain Journal of Mathematics, 38 (2008), no. 4, 1253-1262 The source file(s), ostr.tex: 21966 bytes, is(are) stored in gzipped form as 0811.1763.gz with size 7kb. The corresponding postcript file has gzipped size 79kb. Submitted from: ostrovsm@stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0811.1763 or http://arXiv.org/abs/0811.1763 or by email in unzipped form by transmitting an empty message with subject line uget 0811.1763 or in gzipped form by using subject line get 0811.1763 to: math@arXiv.org.