Valuations, non-commutative determinants, and quaternionic pluripotential theory.

We present a new construction of translation invariant continuous valuations on convex compact subsets of a quaternionic space H n ≃ R4n. This construction is based on the theory of plurisubharmonic functions of quaternionic variables started by the author in [4] and [5] which is based in turn on the notion of non-commutative determinants. In this paper we also establish some new properties of quaternionic plurisubharmonic functions necessary for the construction of valuations, and which have independent interest.