Notes on matrix valued paraproducts Tao Mei 47B3546L5047B1046L53paraproductsSchatten classesnoncommutative martingalesquare functionsweep functiontriangle projection Denote by $M_n$ the algebra of $n \times n$ matrices. We consider the dyadic paraproducts $\pi_b$ associated with $M_n$ valued functions $b$, and show that the $L^{\infty} (M_n)$ norm of $b$ does not dominate $\|\pi_{b}\|_{L^{2}(\ell_{n}^{2}) \to L^{2}(\ell_{n}^{2})}$ uniformly over $n$. 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 noncommutative $L^{q}$-martingale spaces for all $1 < p < q < \infty$. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2926 10.1512/iumj.2006.55.2926 en Indiana Univ. Math. J. 55 (2006) 747 - 760 state-of-the-art mathematics http://iumj.org/access/