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

Representations of *-semigroups associated to invariant kernels with values continuously adjointable operators