We study analogues of the classical (real) H1 and BMO spaces in the noncommutative setting. Let M be a semifinite von Neumann algebra. Let (Tt)t be a semigroup of trace preserving (completely) positive opertors on L p (M). We consider the BMO space and Hardy spaces associated with the subordinated semigroup (Pt)t . An analogue of the classical H1 − BMO duality inequality is obtained if (Tt)t does not increase (decrease) too fast. We also get the inverse relation for some noncommutative H1 and BMO’s with an additional assumption for (Tt)t .