A correspondence between maximal abelian sub-algebras and linear logic fragments

We show a correspondence between a classification of maximal abelian sub-algebras (MASAs) proposed by Jacques Dixmier (Dixmier 1954. Annals of Mathematics 59 (2) 279–286) and fragments of linear logic. We expose for this purpose a modified construction of Girard's hyperfinite geometry of interaction (Girard 2011. Theoretical Computer Science 412 (20) 1860–1883). The expressivity of the logic soundly interpreted in this model is dependent on properties of a MASA which is a parameter of the interpretation. We also unveil the essential role played by MASAs in previous geometry of interaction constructions.