Classification questions for reflection positivity via Krein space analysis

The purpose of our paper is to address the natural classification questions in the context of reflection positivity systems. The latter are defined axiomatically. While the context of reflection positivity has been extensively and widely studied in both mathematical physics (there referred to as Osterwalder–Schrader positivity), and in the theory of representations of Lie group, so far, the question of classification has not been addressed systematically. With the use of Krein space analysis, we offer here classification results for reflection positivity systems. Our context takes the form of triples, (U,J,M+) as follows: A Hilbert space H is fixed, and for the triple (U,J,M+), the first U will be a strongly continuous unitary one-parameter group, J will be a reflection operator, and M+ a closed subspace in H. The three parts in (U,J,M+) are intertwined via the axioms, referred to here as the reflection positivity axioms. After stating our reflection positivity axioms in this context of such triples, (U,J,M+), we then give a solution to the natural classification questions which are entailed by the axioms.