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

Aurelian Gheondea

TL;DR

The paper develops an operator-theoretic framework for Hilbert locally $C^*$-modules by representing them through locally bounded operators on strictly inductive limits of Hilbert spaces. Its central result is a general dilation theorem for positive semidefinite kernels with values in locally bounded operators, invariant under a left $*$-semigroup, which yields a locally Hilbert space linearisation and a $*$-representation, providing a concrete operator model for all Hilbert locally $C^*$-modules. This framework leads to a direct construction of the exterior tensor product of Hilbert locally $C^*$-modules and yields Stinespring-type dilation theorems for completely positive maps on locally $C^*$-algebras with locally bounded values. The combination of Kolmogorov-type decompositions and reproducing kernel space methods in the locally convex setting extends classical dilation theory to a broader noncommutative context, with potential applications in noncommutative geometry and operator algebraic models of bundles.

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.

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

TL;DR

The paper develops an operator-theoretic framework for Hilbert locally -modules by representing them through locally bounded operators on strictly inductive limits of Hilbert spaces. Its central result is a general dilation theorem for positive semidefinite kernels with values in locally bounded operators, invariant under a left -semigroup, which yields a locally Hilbert space linearisation and a -representation, providing a concrete operator model for all Hilbert locally -modules. This framework leads to a direct construction of the exterior tensor product of Hilbert locally -modules and yields Stinespring-type dilation theorems for completely positive maps on locally -algebras with locally bounded values. The combination of Kolmogorov-type decompositions and reproducing kernel space methods in the locally convex setting extends classical dilation theory to a broader noncommutative context, with potential applications in noncommutative geometry and operator algebraic models of bundles.

Abstract

We single out the concept of concrete Hilbert module over a locally -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 -modules and, as an application, we obtain a direct construction of the exterior tensor product of Hilbert locally -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 -algebras and with values locally bounded operators.

Paper Structure

This paper contains 15 sections, 9 theorems, 86 equations.

Table of Contents

  1. Preliminaries
  2. Projective Limits of Locally Convex Spaces.
  3. Inductive Limits of Locally Convex Spaces.
  4. Locally Hilbert Spaces.
  5. Locally Bounded Operators.
  6. Tensor Products of Locally Hilbert Spaces.
  7. Locally $C^*$-Algebras.
  8. The Spatial Tensor Product of Locally $C^*$-Algebras.
  9. Dilations
  10. Positive Semidefinite Kernels.
  11. The General Dilation Theorem.
  12. Completely Positive Maps.
  13. Applications to Hilbert Locally $C^*$-Modules
  14. Hilbert Locally $C^*$-Modules.
  15. The Exterior Tensor Product of Hilbert Locally $C^*$-Modules.

Key Result

Lemma 1.1

For any strictly inductive system of Hilbert spaces $\{{\mathcal{H}}_\lambda\}_{\lambda\in\Lambda}$, its inductive limit ${\mathcal{H}}=\varinjlim_{\lambda\in\Lambda}{\mathcal{H}}_\lambda$ is a Hausdorff locally convex space.

Theorems & Definitions (16)

  • Lemma 1.1
  • proof
  • Proposition 1.3
  • proof
  • Remark 1.5
  • Theorem 1.7
  • Remark 2.1
  • Proposition 2.2
  • Theorem 2.3
  • proof
  • ...and 6 more