Pairs of Subspaces, Split Quaternions and the Modular Operator

TL;DR

This paper explores the structure of pairs of subspaces in real Hilbert spaces using split quaternions and modular operators, relaxing generic position assumptions and providing detailed constructions and examples.

Contribution

It extends the theory of pairs of subspaces by relaxing position assumptions and connects it with split quaternions and modular operator theory.

Findings

Constructs complex Hilbert spaces from real projections

Details the role of split quaternions in the theory

Provides examples with unequal subspace dimensions

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.