Operator system structures and extensions of Schur multipliers

For a given C*-algebra A, we establish the existence of maximal and minimal operator A-system structures on an AOU A-space. In the case A is a W*-algebra, we provide an abstract characterisation of dual operator A-systems, and study the maximal and minimal dual operator A-system structures on a dual AOU 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 ϕ and, under some richness conditions, characterise its affirmative solution in terms of the equality between the canonical and the maximal dual operator A-system structures on an operator system naturally associated with the domain of ϕ