Operational independence and operational separability in algebraic quantum mechanics

Recently, new types of independence of a pair of C (*)- or W (*)-subalgebras (A(1,) A(2)) of a C (*)- or W (*)-algebra have been introduced: operational C (*)- and W (*)-independence (Redei and Summers, http://arxiv.org/abs/0810.5294, 2008) and operational C (*)- and W (*)-separability (Redei and Valente, How local are local operations in local quantum field theory? 2009). In this paper it is shown that operational C (*)-independence is equivalent to operational C (*)-separability and that operational W (*)-independence is equivalent to operational W (*)-separability. Specific further sub-types of both operational C (*)- and W (*)-separability and operational C (*)- and W (*)-independence are defined and the problem of characterization of the logical interdependencies of the independence notions is raised.