Invariants of Reducible Commuting Squares

Determine the behavior of normalizer factorizations, Weyl-group embeddings, and the associated structural invariants for commuting squares of finite-index II_1 factors that are not strictly irreducible.

Background

The paper’s principal structural results, including the normalizer factorization theorem and the embedding of the Weyl group of the upper inclusion into that of the lower inclusion, rely on strict irreducibility assumptions. The authors explicitly state that the behavior of these invariants is unknown when the relevant commuting squares lack irreducibility. Extending the analysis to this reducible regime is identified as a subject for future investigation.