Abstract
Denote by Mn the algebra of nn matrices. We consider the dyadic paraproducts b associated with Mn valued functions b, and show that the L 1 NMnO norm of b does not dominatekbk L2N' 2O!L2N' 2O uniformly overn. We also consider paraproducts associated with noncommutative martingales and prove that their boundedness on bounded noncommutative L p - martingale spaces implies their boundedness on bounded non- commutativeL q -martingale spaces for all 1<p<q<1 .
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