On some dilation theorems for positive definite operator valued functions

The aim of this paper is to prove dilation theorems for operators from a linear complex space to its Z-anti-dual space. The main result is that a bounded positive definite function from a *-semigroup Gamma into the space of all continuous linear maps from a topological vector space X to its Z-anti-dual can be dilated to a *-representation of Gamma on a Z-Loynes space. There is also an algebraic counterpart of this result.