In systems with infinitely many degrees of freedom, fundamental results from quantum information theory can fail. An important example is the uniqueness of purifications: Even when two subsystems, described by commuting von Neumann algebras A and B, are tomographically complete, purifications of a state on A need not be related by unitaries in B. It was recently shown that this occurs precisely when Haag duality fails, i.e., when the commutant B′
Nearby in the stack
is strictly larger than
A
. This raises the question of which fundamental entanglement properties survive in such a setting. We show that, for a pure global state, the ability to steer any ensemble decomposition of the marginal state on
A
by measurements on
B
is equivalent to the existence of a state-preserving conditional expectation from
B′
onto
A
. This establishes a direct connection between quantum steering and subfactor theory. The key observation is that steering is equivalent to the existence of extensions of ensemble decompositions from
A
to
B′
. Working with general Jordan algebras, we prove that unital positive maps have state-preserving left inverses if and only if ensemble decompositions can be lifted. For the inclusion
A↪B′
, a left inverse is precisely a conditional expectation, yielding the characterization above.
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