This is an announcement for the paper "Conditioned square functions for non-commutative martingales" by Narcisse Randrianantoanina. Abstract: We prove a weak-type (1,1) inequality involving conditioned square functions of martingales in non-commutative $L^p$-spaces associated with finite von Neumann algebras. As application, we determine the optimal orders for the best constants in the non-commutative Burkholder/Rosenthal inequalities from Ann. Proba. 31 (2003), 948-995. We also discuss BMO-norms of sums of non-commuting order independent operators. Archive classification: Functional Analysis; Probability Mathematics Subject Classification: 46L53; 60G42 Remarks: 38 pages The source file(s), condsquaref.tex: 107932 bytes, is(are) stored in gzipped form as 0509226.gz with size 31kb. The corresponding postcript file has gzipped size 136kb. Submitted from: randrin@muohio.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0509226 or http://arXiv.org/abs/math.FA/0509226 or by email in unzipped form by transmitting an empty message with subject line uget 0509226 or in gzipped form by using subject line get 0509226 to: math@arXiv.org.