Non commutative Valuation Theory: from Quantum Algebras and Hopf Algebras to Spectral Families and Observables

The extension of valuations on a central field to a noncommutative extension field is an important ingredient in the theory of orders in finite dimensional simple algebras. It is well known that ramification properties may prevent this "extension" and those are again effective tools in noncommutative arithmetic. Unexpectedly such extension results do hold for quantum algebras like Weyl fields, Sklyanin algebras etc... With an aim to apply such techniques to quantum groups (that are not domains) some modifications are necessary and this makes the introduction of Hopf valuations (on Hopf algebras necessary and useful. Further developments may happen on a very abstract level, indeed valuation filtrations may be defined on lattices or even on noncommutative topologies related to noncommutative geometry. This then ties in with a general approach to spectral families and "observables" that can be extended to this generality, generalizing observables in classical quantum mechanics when the quantum lattice of closed linear subspaces of a Hilbert space is used.