A Correspondence between Maximal Abelian Sub-Algebras and Linear Logic Fragments

Abstract

We show a correspondence between a classification of maximal abelian sub-algebras (MASAs) proposed by Jacques Dixmier and fragments of linear logic. We expose for this purpose a modified construction of Girard's hyperfinite geometry of interaction which interprets proofs as operators in a von Neumann algebra. 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.

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• Characterizing co-NL by a group action
• Logarithmic Space and Permutations
• Values of the Pukanszky Invariant in McDuff Factors
• A Continuous Path of Singular Masas in the Hyperfinite II_1 Factor

Cited by in corpus (1)

• Zeta Functions and the (Linear) Logic of Markov Processes