Abstract
Limits of sequences of finite-dimensional (AF)C *-algebras, such as the CAR algebra for the ideal Fermi gas, are a standard mathematical tool to describe quantum statistical systems arising as thermodynamic limits of finite spin systems. Only in the infinite-volume limit one can, for instance, describe phase transitions as singularities in the thermodynamic potentials, and handle the proliferation of physically inequivalent Hilbert space representations of a system with infinitely many degrees of freedom. As is well known, commutative AFC *-algebras correspond to countable Boolean algebras, i.e., algebras of propositions in the classical two-valued calculus. We investigate thenoncommutative logic properties of general AFC *-algebras, and their corresponding systems. We stress the interplay between Gödel incompleteness and quotient structures in the light of the “nature does not have ideals” program, stating that there are no quotient structures in physics. We interpret AFC *-algebras as algebras of the infinite-valued calculus of Lukasiewicz, i.e., algebras of propositions in Ulam's “twenty questions” game with lies.
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References
Behncke, H., and Leptin, H. (1972).C *-algebras with a two-point dual,Journal of Functional Analysis,10, 330–335.
Belluce, P. (1986). Semisimple algebras of infinite-valued logic and bold fuzzy set theory,Canadian Journal of Mathematics,38, 1356–1379.
Berlekamp, E. R. (1968). Block coding for the binary symmetric channel with noiseless, delayless feedback, inError Correcting Codes, Wiley, New York, pp. 61–88.
Birkhoff, G., and von Neumann, J. (1936). The logic of quantum mechanics,Annals of Mathematics,37, 823–843.
Blackadar, B. (1980a). A simple C*-algebra with no nontrivial projections,Proceedings of the American Mathematical Society,78, 504–508.
Blackadar, B. (1980b). Traces on simple AFC *-algebras,Journal of Functional Analysis,38, 156–168.
Blackadar, B. (1987).K-Theory for Operator Algebras, Springer-Verlag, New York.
Bratteli, O. (1972). Inductive limits of finite-dimensionalC *-algebras,Transactions of the American Mathematical Society,171, 195–234.
Bratteli, O., and Robinson, D. W. (1979).Operator Algebras and Quantum Statistical Mechanics I, II, Springer-Verlag, Berlin.
Chang, C. C. (1958). Algebraic analysis of many valued logics,Transactions of the American Mathematical Society,88, 467–490.
Chang, C. C. (1959). A new proof of the completeness of the Lukasiewicz axioms,Transactions of the American Mathematical Society,93, 74–80.
Christensen, E. (1982). Measures on projections and physical states,Communications in Mathematical Physics,86, 529–538.
Cignoli, R., D'Ottaviano, I. M. L., and Mundici, D. (1994). Algebras of many-valued sentential calculi of Lukasiewicz, inColeção CLE, Centro de Lògica, Epistemologia e Historia da Ciência, Universidade Estadual de Campinas, Brazil [in Portuguese].
Cignoli, R., Elliott, G. A., Mundici, D. (n.d.). ReconstructingC *-algebras from their Murray-von Neumann order,Advances in Mathematics, to appear.
Cuntz, J. (1982). The internal structure of simpleC *-algebras,Proceedings Symposia Pure Mathematics I,38, 85–115.
Czyzowicz, J., Mundici, D., and Pelc, A. (1989). Ulam's searching game with lies,Journal of Combinatorial Theory A,52, 62–76.
Di Nola, A. (n.d.). MV algebras in the treatment of uncertainty, in Proceedings IFSA Congress, Bruxelles 1991, R. Lowen, ed., Kluwer, Dordrecht, to appear.
Dixmier, J. (1977).C *-algebras, North-Holland, Amsterdam.
Effros, E. G. (1981). Dimensions andC *-algebras, inCBMS Regional Conference Series in Mathematics, Vol. 46,American Mathematical Society, Providence, Rhode Island.
Effros, E. G., and Rosenberg, J. (1978).C *-algebras with approximately inner flip,Pacific Journal of Mathematics,77, 417–443.
Elliott, G. A. (1976). On the classification of inductive limits of sequences of semisimple finite-dimensional algebras,Journal of Algebra,38, 29–44.
Elliott, G. A. (1979). On totally ordered groups andK 0, inLecture Notes in Mathematics, Vol. 734, pp. 1–49.
Elliott, G. A., and Mundici, D. (1993). A characterization of lattice-ordered Abelian groups,Mathematische Zeitschrift,213, 179–185.
Emch, G. G. (1984).Mathematical and Conceptual Foundations of 20th Century Physics, North-Holland, Amsterdam.
Goodearl, K. R. (1982).Notes on Real and Complex C *-Algebras, Birkhäuser, Boston.
Grigolia, R. (1977). Algebraic analysis of Tarski Lukasiewiczn-valued logical systems, inSelected Papers on Lukasiewicz Sentential Calculi, R. Wòjcicki and G. Malinowski, eds., Polish Academy of Sciences, Ossolineum, Wroclaw, pp. 81–92.
Haag, D., and Kastler, D. (1964). An algebraic approach to quantum field theory,Journal of Mathematical Physics,5, 848–861.
Kastler, D. (1982). Does ergodicity plus locality imply the Gibbs structure?Proceedings Symposia Pure Mathematics II,38, 467–489.
Maeda, S. (1990). Probability measures on projections in von Neumann algebras,Reviews in Mathematical Physics,1, 235–290.
Mundici, D. (1986). Interpretation of AFC *-algebras in Lukasiewicz sentential calculus,Journal of Functional Analysis,65, 15–63.
Mundici, D. (1987a). The Turing complexity of AFC *-algebras with lattice-orderedK 0,Lecture Notes in Computer Science,270, 256–264.
Mundici, D. (1987b). Satisfiability in many-valued sentential logic is NP-complete,Theoretical Computer Science,52, 145–153.
Mundici, D. (1988a). Farey stellar subdivisions, ultrasimplicial groups, andK 0 of AFC *-algebras,Advances in Mathematics,68, 23–39.
Mundici, D. (1988b). Free products in the category of Abelianl-groups with strong unit,Journal of Algebra,113, 89–109.
Mundici, D. (1988c). The derivative of truth in Lukasiewicz sentential calculus, in Methods and Applications of Mathematical Logic,Contemporary Mathematics,69, 209–227.
Mundici, D. (1989). TheC *-algebras of three-valued logic, inProceedings Logic Colloquium 1988, North-Holland, Amsterdam, pp. 61–77.
Mundici, D. (1991). The complexity of adaptive error-correcting codes,Lecture Notes in Computer Science,533, 300–307.
Mundici, D. (1992a). Turing complexity of Behncke-LeptinC *-algebras with a two-point dual,Annals of Mathematics and Artificial Intelligence,6, 287–294.
Mundici, D. (1992b), The logic of Ulam's game with lies, inKnowledge, Belief and Strategic Interaction, C. Bicchieri and M. L. Dalla Chiara, eds., Cambridge University Press, pp. 275–284.
Mundici, D. (1993). Ulam games, Lukasiewicz logic, and AFC *-algebras,Fundamenta Informaticae,18, 151–161.
Mundici, D. (n.d.a).K 0, relative dimension, and theC *-algebras of Post logic, inProceedings IX Latin American Symposium in Mathematical Logic, Bahia Blanca, Argentina, 1992, M. Abad, ed., Notas de Lògica Mathemàtica, Bahia Blanca, to appear.
Mundici, D. (n.d.-b). Uncertainty measures in MV algebras, and states of AFC *-algebras, to appear.
Mundici, D., and Panti, G. (n.d.). Extending addition in Elliott's local semigroup,Journal of Functional Analysis, to appear.
Sewell, G. L. (1986).Quantum Theory of Collective Phenomena, Clarendon Press, Oxford.
Tarski, A., and Lukasiewicz, J. (1956). Investigations into the sentential calculus, inLogic, Semantics, Metamathematics, Oxford University Press, 1956, pp. 38–59.
Ulam, S. M. (1976).Adventures of a Mathematician, Scribner's, New York.
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Mundici, D. Logic of infinite quantum systems. Int J Theor Phys 32, 1941–1955 (1993). https://doi.org/10.1007/BF00979516
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DOI: https://doi.org/10.1007/BF00979516