  For a given C*-algebra $\mathcal{A}$, we establish the existence of maximal
and minimal operator $\mathcal{A}$-system structures on an AOU
$\mathcal{A}$-space. In the case $\mathcal{A}$ is a W*-algebra, we provide an
abstract characterisation of dual operator $\mathcal{A}$-systems, and study the
maximal and minimal dual operator $\mathcal{A}$-system structures on a dual AOU
$\mathcal{A}$-space. We introduce operator-valued Schur multipliers, and
provide a Grothendieck-type characterisation. We study the positive extension
problem for a partially defined operator-valued Schur multiplier $\varphi$ and,
under some richness conditions, characterise its affirmative solution in terms
of the equality between the canonical and the maximal dual operator
$\mathcal{A}$-system structures on an operator system naturally associated with
the domain of $\varphi$.
