Non-commutative BMO space

Abstract.

We consider the non-commutative martingale BMO-version introduced recently by G. Pisier and Q. Xu. Let us denote it by $BMO(\cal M)$, where $\cal M$ is $B(\ell_2)$. We prove that the triangular projection P _T (see [1]) is a bounded linear operator from $B(\ell_2)$ into $BMO(\cal m)$, obtaining a non-commutative analogue of the well-known fact from harmonical analysis that the Riesz projection maps continuously $L_\infty$ into BMO.