Grothendieck theorem for some uniform algebras and modules over them

Under certain additional assumptions, it is proved that a $w^*$-closed subalgebra $X$ of $L^\infty(\mu)$ (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 binding than it is usual in similar settings. Specifically, $\mu$ may fail to be multiplicative on $X$, etc.