OPERATOR MODELS FOR HILBERT LOCALLY C*-MODULES

We single out the concept of concrete Hilbert module over a locally C*-algebra by means of locally bounded operators on certain strictly inductive limits of Hilbert spaces. Using this concept, we construct an operator model for all Hilbert locally C*-modules and, as an application, we obtain a direct construction of the exterior tensor product of Hilbert locally C*-modules. These are obtained as consequences of a general dilation theorem for positive semidefinite kernels invariant under an action of a *-semigroup with values locally bounded operators. As a by-product, we obtain two Stinespring type theorems for completely positive maps on locally C*-algebras and with values locally bounded operators.