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Pairs of Subspaces, Split Quaternions and the Modular Operator

Jan Naudts, Jun Zhang

TL;DR

This paper extends the Rieffel–van Daele approach to pairs of real subspaces by relaxing the generic-position requirement and developing a real-to-complex framework via a modular-operator construction. It builds a complex Hilbert space from two projections using a split-quaternion symmetry, realized through a central extension and polar decompositions, yielding a Tomita–Takesaki–style modular operator $\Delta$ and conjugation $J$. Two explicit examples, including a non-generic Euclidean-case and a Larmor-precession model, illustrate the geometry and the complexified dynamics. The work highlights potential applications to Hamiltonian flows $H=\log\Delta$ and hints at extensions to manifolds and generalized complex structures, with a companion paper outlining further directions.

Abstract

We revisit the work of Rieffel and van Daele on pairs of subspaces of a real Hilbert space, while relaxing as much as possible the assumption that all the relevant subspaces are in general positions with respect to each other. We work out, in detail, how two real projection operators lead to the construction of a complex Hilbert space where the theory of the modular operator is applicable, with emphasis on the relevance of a central extension of the group of split quaternions. Two examples are given for which the subspaces have unequal dimension and therefore are not in generic position.

Pairs of Subspaces, Split Quaternions and the Modular Operator

TL;DR

This paper extends the Rieffel–van Daele approach to pairs of real subspaces by relaxing the generic-position requirement and developing a real-to-complex framework via a modular-operator construction. It builds a complex Hilbert space from two projections using a split-quaternion symmetry, realized through a central extension and polar decompositions, yielding a Tomita–Takesaki–style modular operator and conjugation . Two explicit examples, including a non-generic Euclidean-case and a Larmor-precession model, illustrate the geometry and the complexified dynamics. The work highlights potential applications to Hamiltonian flows and hints at extensions to manifolds and generalized complex structures, with a companion paper outlining further directions.

Abstract

We revisit the work of Rieffel and van Daele on pairs of subspaces of a real Hilbert space, while relaxing as much as possible the assumption that all the relevant subspaces are in general positions with respect to each other. We work out, in detail, how two real projection operators lead to the construction of a complex Hilbert space where the theory of the modular operator is applicable, with emphasis on the relevance of a central extension of the group of split quaternions. Two examples are given for which the subspaces have unequal dimension and therefore are not in generic position.
Paper Structure (22 sections, 15 theorems, 175 equations, 2 tables)

This paper contains 22 sections, 15 theorems, 175 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Pairs of Subspaces of $\mathscr V$
  3. A pair of orthogonal projections on $\mathscr V$
  4. Identities
  5. Anti-commuting partial isometries
  6. Two subspaces
  7. The corresponding projections
  8. The symmetric subspace
  9. The modular operator
  10. Example: Euclidean space
  11. Complex Structure on $\mathscr V \times \mathscr V$
  12. Split quaternions
  13. Complexification
  14. Complex-linear operators
  15. Antilinear operators
  16. ...and 7 more sections

Key Result

Proposition 2.1

The operator $PQ-QP$ commutes with the positive operators $C$ and $T$ and anti-commutes with the (partial) isometries $J$ and $K$.

Theorems & Definitions (15)

  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Corollary 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Proposition 3.1
  • ...and 5 more