On the predual of non-commutative $H^{\mathrm{\infty }}$

The unique predual $M_{\star }\mathrm{/}{A}_{\perp }$ of a non-commutative $H^{\mathrm{\infty }}$-algebra $A=H^{\mathrm{\infty }}(M,\tau )$ is investigated. In particular, we will prove the liftability property of weakly relatively compact subsets in $M_{\star }\mathrm{/}{A}_{\perp }$ to $M_{\star }$.