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

Published online by Cambridge University Press:  28 July 2016

THOMAS SEILLER
THOMAS SEILLER*
Affiliation:
I.H.É.S., Le Bois-Marie, 35, Route de Chartres, 91440 Bures-sur-Yvette, France Email: [email protected]
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Abstract

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.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 28 , Issue 1 , January 2018 , pp. 77 - 139
Copyright
Copyright © Cambridge University Press 2016 

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References

Abramsky, S., Haghverdi, E. and Scott, P. (2002). Geometry of interaction and linear combinatory algebras. Mathematical Structures in Computer Science 12 (05) 625–665.CrossRefGoogle Scholar
Aubert, C., Bagnol, M. and Seiller, T. (2016). Unary resolution: Characterizing Ptime. In: Jacobs, B. and Löding, C. (eds.) FOSSACS 2016. Lecture Notes in Computer Science 9634 373–389.CrossRefGoogle Scholar
Aubert, C. and Seiller, T. (2016a). Characterizing co-nl by a group action. Mathematical Structures in Computer Science 26 (4) 606–638.CrossRefGoogle Scholar
Aubert, C. and Seiller, T. (2016b). Logarithmic space and permutations. Information and Computation. Available at: http://dx.doi.org/10.1016/j.ic.2014.01.018 (Available online 6 January 2016).CrossRefGoogle Scholar
Baillot, P. and Pedicini, M. (2001). Elementary complexity and geometry of interaction. Fundamenta Informaticae 45 (1–2) 1–31.Google Scholar
Connes, A., Feldman, J. and Weiss, B. (1981). An amenable equivalence relation is generated by a single transformation. Ergodic Theory and Dynamical Systems 1 (4) 431–450.CrossRefGoogle Scholar
Chifan, I. (2007). On the normalizing algebra of a masa in a II1 factor. Preprint, 2007.Google Scholar
Curien, P.-L. (2006). Introduction to linear logic and ludics. Advances in Mathematics (China) 35 (1) 1–44.Google Scholar
Dixmier, J. (1954). Sous-anneaux abéliens maximaux dans les facteurs de type fini. Annals of Mathematics 59 (2) 279–286.Google Scholar
Danos, V. and Regnier, L. (1993). Local and asynchronous beta-reduction (an analysis of girardís execution formula). In: Proceedings of the 18th Annual Symposium on Logic in Computer Science (LICS '93), Montreal, Canada, June 19–23, 1993, IEEE Computer Society, 296–306.Google Scholar
Danos, V. and Regnier, L. (1999). Reversible, irreversible and optimal λ-machines. Theoretical Computer Science 227 (1–2) 79–97.CrossRefGoogle Scholar
Duchesne, E. (2009). La localisation en logique: géométrie de l'interaction et sémantique dénotationelle. PhD thesis, Université de la Méditerranée, 2009.Google Scholar
Dye, H.A. (1963). On groups of measure preserving transformations. II. American Journal of Mathematics 85 (4) 551–576.CrossRefGoogle Scholar
Feldman, J. and Moore, C.C. (1977a). Ergodic equivalence relations, cohomology, and von neumann algebras. I. Transactions of the American Mathematical Society 234 (2) 289–324.CrossRefGoogle Scholar
Feldman, J. and Moore, C.C. (1977b). Ergodic equivalence relations, cohomology, and von neumann algebras. II. Transactions of the American Mathematical Society 234 (2) 325–359.CrossRefGoogle Scholar
Fuglede, B. and Kadison, R.V. (1952). Determinant theory in finite factors. Annals of Mathematics 56 (3) 520–530.CrossRefGoogle Scholar
Gelfand, I. (1941). Normierte ringe. Matematicheskii Sbornik 51 (1) 3–24.Google Scholar
Girard, J.-Y. (1987a). Linear logic. Theoretical Computer Science 50 (1) 1–101.CrossRefGoogle Scholar
Girard, J.-Y. (1987b). Multiplicatives. In: Lolli, G. (ed.) Logic and Computer Science: New Trends and Applications, Torino, Università di Torino. Rendiconti del seminario matematico dell'università e politecnico di Torino, special issue pp. 11–34.Google Scholar
Girard, J.-Y. (1988). Geometry of interaction II: Deadlock-free algorithms. In: Proceedings of COLOG. Lecture Notes in Computer Science 417, 76–93, Springer.CrossRefGoogle Scholar
Girard, J.-Y. (1989a). Geometry of interaction I: Interpretation of system F. In: Proceedings of the Logic Colloquium '88 221–260.CrossRefGoogle Scholar
Girard, J.-Y. (1989b). Towards a geometry of interaction. In: Proceedings of the AMS Conference on Categories, Logic and Computer Science 69–108.CrossRefGoogle Scholar
Girard, J.-Y. (1995a). Geometry of interaction III: Accommodating the additives. In: Advances in Linear Logic, Lecture Notes Series, volume 222, 329–389, Cambridge University Press.CrossRefGoogle Scholar
Girard, J.-Y. (1995b). Proof nets: The parallel syntax for proof theory. In: Ursini, A. (ed.) Logic and Algebra, Lecture Notes in Pure and Applied Mathematics, volume 180, Marcel Dekker.Google Scholar
Girard, J.-Y. (2001). Locus solum: From the rules of logic to the logic of rules. Mathematical Structures in Computer Science 11 (3) 301–506.CrossRefGoogle Scholar
Girard, J.-Y. (2006). Geometry of interaction IV: The feedback equation. In: Stoltenberg-Hansen, V. and Väänänen, J. (eds.) Logic Colloquium '03, Lecture Notes in Logic 24, Association for Symbolic Logic, La Jolla, CA, USA 76–117.CrossRefGoogle Scholar
Girard, J.-Y. (2007). Le point aveugle, tome 2 : vers l'imperfection. Vision des Sciences.Google Scholar
Girard, J.-Y. (2011). Geometry of interaction V: Logic in the hyperfinite factor. Theoretical Computer Science 412 (20) 1860–1883.CrossRefGoogle Scholar
Gonthier, G., Abadi, A. and Lévy, J.-J. (1992). The geometry of optimal lambda reduction. In: Sethi, R. (ed.) POPL, pp. 15–26, ACM Press.Google Scholar
Haagerup, U. (1975). The standard form of von neumann algebras. Mathematica Scandinavica 37 271–283.CrossRefGoogle Scholar
Haghverdi, E. and Scott, P. (2005). From geometry of interaction to denotational semantics. Proceedings of the 10th Conference on Category Theory in Computer Science (CTCS 2004) Category Theory in Computer Science 2004. Electronic Notes in Theoretical Computer Science 122 67–87.CrossRefGoogle Scholar
Haghverdi, E. and Scott, P. (2006). A categorical model for the geometry of interaction. Theoretical Computer Science 350 (2) 252–274.CrossRefGoogle Scholar
Hyland, J.M.E. and Luke Ong, C.-H. (2000). On full abstraction for PCF: I, II, and III. Information and Computation 163 (2) 285–408.CrossRefGoogle Scholar
Jones, V.F.R. and Popa, S. (1982). Some properties of masas in factors. In: Invariant Subspaces and Other Topics, Operator Theory: Advances and Applications, Volume 6, 89–102, Birkhäuser Basel.CrossRefGoogle Scholar
Krivine, J.-L. (2001). Typed lambda-calculus in classical zermelo-fraenkel set theory. Archive for Mathematical Logic 40 (3) 189–205.CrossRefGoogle Scholar
Krivine, J.-L. (2009). Realizability in classical logic. In: Interactive Models of Computation and Program Behaviour, volume 27, Panoramas et synthèses, Société Mathématique de France, 197–229.Google Scholar
Lago, U.D. (2009). The geometry of linear higher-order recursion. ACM Transactions on Computational Logic 10 (2) 8:1–8:38.CrossRefGoogle Scholar
Laurent, O. (2001). A token machine for full geometry of interaction (extended abstract). In: Abramsky, S. (ed.) Typed Lambda Calculi and Applications '01. Lecture Notes in Computer Science 2044 283–297, Springer.CrossRefGoogle Scholar
Miquel, A. (2011). A survey of classical realizability. In: Luke Ong, C.-H. (ed.) Proceedings of Typed Lambda Calculi and Applications – 10th International Conference, TLCA 2011, Novi Sad, Serbia, June 1–3, 2011. Lecture Notes in Computer Science 6690 1–2, Springer.CrossRefGoogle Scholar
Murray, F. and von Neumann, J. (1936). On rings of operators. Annals of Mathematics 37 (2) 116–229.CrossRefGoogle Scholar
Murray, F. and von Neumann, J. (1937). On rings of operators. II. Transactions of the American Mathematical Society 41 (2) 208–248.CrossRefGoogle Scholar
Murray, F. and von Neumann, J. (1943). On rings of operators. IV. Annals of Mathematics 44 (4) 716–808.CrossRefGoogle Scholar
Naibo, A., Petrolo, M. and Seiller, T. (2016). On the computational meaning of axioms. In: Pombo, O., Nepomuceno, A. and Redmond, J. (eds.) Epistemology, Knowledge and the Impact of Interaction, Springer.Google Scholar
Pukanszky, L. (1960). On maximal abelian subrings of factors of type II1 . Canadian Journal of Mathematics 12 289–296.CrossRefGoogle Scholar
Sakai, S. (1971). C*-Algebras and W*-Algebras, Springer-Verlag.Google Scholar
Seiller, T. (2012). Interaction graphs: Multiplicatives. Annals of Pure and Applied Logic 163 1808–1837.CrossRefGoogle Scholar
Seiller, T. (2013). Interaction graphs: Exponentials. CoRR, abs/1312.1094.Google Scholar
Seiller, T. (2014). Interaction graphs and complexity. Extended Abstract, 2014.Google Scholar
Seiller, S. (2015). Towards a complexity-through-realizability theory. Submitted. ArXiv: 1502.01257.Google Scholar
Seiller, T. (2016a). Interaction graphs: Additives, Annals of Pure and Applied Logic 167 (2) 95–154.CrossRefGoogle Scholar
Seiller, T. (2016b). Interaction graphs: Graphings. Annals of Pure and Applied Logic. To appear.CrossRefGoogle Scholar
Sinclair, A. and Smith, R. (2008). Finite von Neumann algebras and Masas. London Mathematical Society, Lecture Note Series, volume 351, Cambridge University Press.CrossRefGoogle Scholar
Streicher, T. (2013). Krivine's classical realisability from a categorical perspective. Mathematical Structures in Computer Science 23 (6) 1234–1256.CrossRefGoogle Scholar
Takesaki, M. (2001). Theory of Operator Algebras 1, Encyclopedia of Mathematical Sciences, volume 124, Springer.Google Scholar
Takesaki, M. (2003a). Theory of Operator Algebras 2, Encyclopedia of Mathematical Sciences, volume 125, Springer.Google Scholar
Takesaki, M. (2003b). Theory of Operator Algebras 3, Encyclopedia of Mathematical Sciences, volume 127, Springer.Google Scholar
Tomita, M. (1967). Quasi-standard von neumann algebras. Mimeographed Notes (Kyushu University).Google Scholar
von Neumann, J. (1930). Zür algebra der funktionaloperatoren und theorie der normalen operatoren. Mathematische Annalen 102 370–427.CrossRefGoogle Scholar
von Neumann, J. (1938). On infinite direct products. Compositiones Mathematicae 6 1–77.Google Scholar
von Neumann, J. (1940). On rings of operators III. Annals of Mathematics 41 (1) 94–161.CrossRefGoogle Scholar
von Neumann, J. (1943). On some algebraical properties of operator rings. Annals of Mathematics 44 (4) 709–715.CrossRefGoogle Scholar
von Neumann, J. (1949). On rings of operators. reduction theory. Annals of Mathematics 50 (2) 401–485.CrossRefGoogle Scholar
White, S. (2006). Tauer masas in the hyperfinite II1 factor. Quarterly Journal of Mathematics 57 (3) 377–393.CrossRefGoogle Scholar
White, S. (2008). Values of the pukanszky invariant in mcduff factors. Journal of Functional Analysis 254 (3) 612–631.CrossRefGoogle Scholar
White, S. and Sinclair, A. (2007). A continuous path of singular masas in the hyperfinite II1 factor. Journal of the London Mathematical Society 75 (1) 243–254.Google Scholar