How local are local operations in local quantum field theory?

A notion called operational C⁎-separability of local C  *-algebras (A(V1)A(V1) and A(V2)A(V2)) associated with spacelike separated spacetime regions V1 and V2 in a net of local observable algebras satisfying the standard axioms of local, algebraic relativistic quantum field theory is defined in terms of operations (completely positive unit preserving linear maps) on the local algebras A(V1)A(V1) and A(V2)A(V2). Operational C*-separability is interpreted as a “no-signaling” condition formulated for general operations, for which a straightforward no-signaling theorem is shown not to hold. By linking operational C  *-separability of (A(V1),A(V2))(A(V1),A(V2)) to the recently introduced (Rédei & Summers, forthcoming) operational C  *-independence of (A(V1),A(V2))(A(V1),A(V2)) it is shown that operational C  *-separability typically holds for the pair (A(V1),A(V2))(A(V1),A(V2)) if V1 and V2 are strictly spacelike separated double cone regions. The status in local, algebraic relativistic quantum field theory of a natural strengthening of operational C*-separability, i.e. operational W*-separability, is discussed and open problems about the relation of operational separability and operational independence are formulated.