  We consider positive semidefinite kernels valued in the $*$-algebra of
continuous and continuously adjointable operators on a VH-space (Vector Hilbert
space in the sense of Loynes) and that are invariant under actions of
$*$-semigroups. For such a kernel we obtain two necessary and sufficient
boundedness conditions in order for there to exist $*$-representations of the
underlying $*$-semigroup on a VH-space linearisation, equivalently, on a
reproducing kernel VH-space. We exhibit several situations when the latter
boundedness condition is automatically fulfilled. For example, when
specialising to the case of Hilbert modules over locally $C^*$-algebras, we
show that both boundedness conditions are automatically fulfilled and,
consequently, this general approach provides a rather direct proof of the
general Stinespring-Kasparov type dilation theorem for completely positive maps
on locally $C^*$-algebras and with values adjointable operators on Hilbert
modules over locally $C^*$-algebras.
