Operational Independence and Operational Separability in Algebraic Quantum Mechanics

Recently, new types of independence of a pair of C ^*- or W ^*-subalgebras $({\mathcal{A}}_{1},{\mathcal{A}}_{2})$ of a C ^*- or W ^*-algebra have been introduced: operational C ^*- and W ^*-independence (Rédei and Summers, http://arxiv.org/abs/0810.5294, 2008) and operational C ^*- and W ^*-separability (Rédei 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.