Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Representations of *-semigroups associated to invariant kernels with values continuously adjointable operators

Serdar Ay, Aurelian Gheondea

Abstract

We consider positive semidefinite kernels valued in the $*$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $*$-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist $*$-representations of the underlying $*$-semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally $C^*$-algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring-Kasparov type dilation theorem for completely positive maps on locally $C^*$-algebras and with values adjointable operators on Hilbert modules over locally $C^*$-algebras.

Representations of *-semigroups associated to invariant kernels with values continuously adjointable operators

Abstract

We consider positive semidefinite kernels valued in the -algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of -semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist -representations of the underlying -semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally -algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring-Kasparov type dilation theorem for completely positive maps on locally -algebras and with values adjointable operators on Hilbert modules over locally -algebras.

Paper Structure

This paper contains 15 sections, 28 theorems, 134 equations.

Table of Contents

  1. Preliminaries
  2. VE-Spaces and Their Linear Operators.
  3. Admissible Spaces.
  4. Vector Hilbert Spaces and Their Linear Operators.
  5. Positive Semidefinite Kernels with Values Adjointable Operators
  6. Kernels with Values Adjointable Operators.
  7. VE-Module Linearisations.
  8. VH-Space Linearisations and Reproducing Kernels.
  9. Dilation in VH-Spaces
  10. Condition (b2) of Theorem \ref{['t:vhinvkolmo2']}.
  11. Kernels with Values Adjointable Operators on VH-Modules.
  12. $*$-Representations on Hilbert Modules over Locally $C^*$-Algebras
  13. Hilbert Modules over Locally $C^*$-Algebras
  14. Kernels with Values Adjointable Operators in Hilbert Locally $C^*$-Modules.
  15. Completely Positive Maps.

Key Result

Lemma 1.3

Let $Z$ be a topologically ordered $*$-space and ${\mathcal{E}}$ a VE-space over $Z$.

Theorems & Definitions (55)

  • Remark 1.1
  • Lemma 1.3
  • proof : Proof of Lemma \ref{['l:topology']}
  • Corollary 1.4
  • proof
  • Remark 1.6
  • Lemma 1.7
  • proof
  • Remark 1.8
  • Lemma 1.9
  • ...and 45 more