Cambridge University Press, Mathematical Structures in Computer Science, 01(28), p. 77-139
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.