On the structure of abstract h*-algebras

Previously we have shown that the topos approach to quantum theory of Doering and Isham can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke. In the monoidal approach to quantum theory H^*-algebras provide an axiomatisation of states and observables. Here we show that H^*-algebras naturally correspond with the notions of states and observables in the generalised topos approach to quantum theory. We then combine these results with the dagger-kernel approach to quantumlogic of Heunen and Jacobs, which we use to prove a structure theorem for H^*-algebras. This structure theorem is a generalisation of the structure theorem of Ambrose for H^*-algebras the category of Hilbert spaces.