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

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We show a correspondence between a classification of maximal abelian sub-algebras (MASAs) proposed by Jacques Dixmier (Dixmier 1954.Annals of Mathematics59(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 Science412(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.