Under certain additional assumptions, it is proved that a w^∗-closed subalgebra X of L^∞(μ) (more generally, a w^∗-closed module over X) verifies the Grothendieck theorem. The assumptions in question imitate a property of the classical harmonic conjugation operator but are less restrictive than it is usual in similar settings. Specifically, μ may fail to be multiplicative on X, etc.