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We consider the geometric and non-geometric faces of closed string vacuaarising by T-duality from principal torus bundles with constant H-flux and payattention to their double phase space description encompassing all toroidalcoordinates, momenta and their dual on equal footing. We construct astar-product algebra on functions in phase space that is manifestly dualityinvariant and substitutes for canonical quantization. The 3-cocycles of theAbelian group of translations in double phase space are seen to account fornon-associativity of the star-product. We also provide alternativecohomological descriptions of non-associativity and draw analogies with thequantization of point-particles in the field of a Dirac monopole or otherdistributions of magnetic charge. The magnetic field analogue of the R-fluxstring model is provided by a constant uniform distribution of magnetic chargein space and non-associativity manifests as breaking of angular symmetry. ThePoincare vector comes to rescue angular symmetry as well as associativity andalso allow for quantization in terms of operators and Hilbert space only in thecase of charged particles moving in the field of a single magnetic monopole.

3-Cocycles, Non-Associative Star-Products and the Magnetic Paradigm of R-Flux String Vacua