Operator Models for Hilbert Locally $C^{\ast }$-Modules

Abstract

We single out the concept of concrete Hilbert module over a locally $C^{\ast }$-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 $C^{\ast }$-modules and, as an application, we obtain a direct construction of the exterior tensor product of Hilbert locally $C^{\ast }$-modules. These are obtained as consequences of a general dilation theorem for positive semidefinite kernels invariant under an action of a $\ast$-semigroup with values locally bounded operators. As a by-product, we obtain two Stinespring type theorems for completely positive maps on locally $C^{\ast }$-algebras and with values locally bounded operators.

25 pages

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