Operator Models for Hilbert Locally $C^*$-Modules

TL;DR

This paper develops an operator model for Hilbert locally $C^*$-modules using locally bounded operators, enabling constructions like exterior tensor products and establishing dilation theorems and Stinespring-type results for positive maps.

Contribution

It introduces a concrete operator model for Hilbert locally $C^*$-modules and derives new dilation and Stinespring theorems in this setting.

Findings

Constructed an operator model for all Hilbert locally $C^*$-modules.

Provided a direct construction of the exterior tensor product.

Established dilation theorems and Stinespring-type results for positive maps.

Abstract

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.