Under consideration for publication in Math. Struct. in Comp. Science A Correspondence between Maximal Abelian Sub-Algebras and Linear Logic Fragments THOMAS SEILLER† 1 † I.H.E.S., Le Bois-Marie, 35, Route de Chartres, 91440 Bures-sur-Yvette, France
[email protected] Received December 15th, 2014; Revised September 23rd, 2015 We show a correspondence between a classification of maximal abelian sub-algebras (MASAs) proposed by Jacques Dixmier (Dix54) and fragments of linear logic. We expose for this purpose a modified construction of Girard’s hyperfinite geometry of interaction (Gir11). 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. Contents 1 Introduction 1 2 von Neumann Algebras and MASAs 4 3 Geometry of Interaction 14 4 Subjective Truth and Matricial GoI 29 5 Dixmier’s Classification and Linear Logic 46 6 Conclusion 58 References 59 arXiv:1408.2125v2 [math.LO] 23 Sep 2015 1. Introduction 1.1. Geometry of Interaction. Geometry of Interaction is a research program initiated by Girard (Gir89b) a year after his seminal paper on linear logic (Gir87a). Its aim is twofold: define a semantics of proofs that accounts for the dynamics of cut-elimination, and then construct realisability models for linear logic around this semantics of cut-elimination. The first step for defining a GoI model, i.e. a construction that fulfills the geometry of 1 This work was partly supported by the ANR-10-BLAN-0213 Logoi.