Quaternion-valued positive definite functions on locally compact Abelian groups and nuclear spaces

In this paper we study quaternion-valued positive definite functions on locally compact Abelian groups, real countably Hilbertian nuclear spaces and on the space RN=(x1,x2,...):xdϵR endowed with the Tychonoff topology. In particular, we prove a quaternionic version of the Bochner-Minlos theorem. A tool for proving this result is a classical matricial analogue of the Bochner-Minlos theorem, which we believe is new. We will see that in all these various settings the integral representation is with respect to a quaternion-valued measure which has certain symmetry properties.