  We consider weakly positive semidefinite kernels valued in ordered $*$-spaces
with or without certain topological properties, and investigate their
linearisations (Kolmogorov decompositions) as well as their reproducing kernel
spaces. The spaces of realisations are of VE (Vector Euclidean) or VH (Vector
Hilbert) type, more precisely, vector spaces that possess gramians (vector
valued inner products). The main results refer to the case when the kernels are
invariant under certain actions of $*$-semigroups and show under which
conditions $*$-representations on VE-spaces, or VH-spaces in the topological
case, can be obtained. Finally we show that these results unify most of
dilation type results for invariant positive semidefinite kernels with operator
values as well as recent results on positive semidefinite maps on
$*$-semigroups with values operators from a locally bounded topological vector
space to its conjugate $Z$-dual space, for $Z$ an ordered $*$-space.
