Noncommutativity as a Colimit

We give substance to the motto “every partial algebra is the colimit of its total subalgebras” by proving it for partial Boolean algebras (including orthomodular lattices), the new notion of partial C*-algebras (including noncommutative C*-algebras), and variations such as partial complete Boolean algebras and partial AW*-algebras. Both pairs of results are related by taking projections. As corollaries we find extensions of Stone duality and Gelfand duality. Finally, we investigate the extent to which the Bohrification construction (Heunen et al. 2010), that works on partial C*-algebras, is functorial.