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Relative Positions of Half-sided Modular Inclusions

Ian Koot

Abstract

Let $K_1 \subset H$ and $K_2 \subset H$ be half-sided modular inclusions in a common standard subspace $H$. We prove that the inclusion $K_1 \subset K_2$ holds if and only if we have an inclusion of spectral subspaces of the generators of the positive one-parameter groups associated to the half-sided modular inclusions $K_1 \subset H$ and $K_2 \subset H$. From this we give a characterization of this situation in terms of (operator valued) symmetric inner functions. We illustrate these characterizations with some examples of non-trivial phenomena occurring in this setting.

Relative Positions of Half-sided Modular Inclusions

Abstract

Let and be half-sided modular inclusions in a common standard subspace . We prove that the inclusion holds if and only if we have an inclusion of spectral subspaces of the generators of the positive one-parameter groups associated to the half-sided modular inclusions and . From this we give a characterization of this situation in terms of (operator valued) symmetric inner functions. We illustrate these characterizations with some examples of non-trivial phenomena occurring in this setting.

Paper Structure

This paper contains 9 sections, 18 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Modular theory, standard subspaces and their inclusions
  3. Standard subspaces
  4. Half-sided modular inclusions and standard pairs
  5. Representation theory of standard pairs
  6. Spectral characterization of inclusions of standard pairs
  7. Characterization in terms of inner functions and concrete examples
  8. Distributional Fourier transform of $e^{ise^\theta}$
  9. Vector valued uniform convergence

Key Result

Theorem 2.2

Let $H \subset \mathcal{H}$ be a standard subspace.

Theorems & Definitions (50)

  • Definition 2.1
  • Theorem 2.2: Tomita-Takesaki theorem, Takesaki1970 LongoSibiu
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Proposition 2.6: Borchers1999a, ARAKI2005
  • Example 2.7
  • Definition 2.8
  • Theorem 2.9: Borchers' theorem, Borchers1992 Florig1998
  • Definition 2.10
  • ...and 40 more