Palavras-chave
Como Citar
Resumo
In the algebraic treatment of quantum statistical systems, the claim “Nature does not have ideals” is sometimes used to convey the idea that the C*-algebras describing natural systems are simple, i.e., they do not have nontrivial homomorphic images. Using our interpretation of AF C*-algebras as algebras of Lukasiewicz calculus, in a previous paper the claim was shown to be incompatible with the existence of a G¨odel incomplete AF C*-algebra for a quantum physical system existing in nature. In this note we survey recent developments on G¨odel incompleteness and decidability issues for AF C*-algebras.Referências
Behnke, H., Leptin, H. “C*-algebras with a two-point dual”. J. Functional Analysis, 10, pp. 330-335, 1972.
Blackadar, B. K-Theory for Operator Algebras. New York: Springer- Verlag, 1987.
Boyle, M., Marcus, B., Trow, P. “Resolving maps and the dimension group for shifts of finite type”. Mem. Amer. Math. Soc., 70, n. 377, 1987.
Bratteli, O. “Inductive limits of finite dimensional C∗-algebras”.
Trans. Amer. Math. Soc., 171, pp. 195-234, 1972.
Bratteli, O., Robinson, D.W. Operator Algebras and Quantum Statistical Mechanics I, II. Berlin: Springer-Verlag, 1979.
Bratteli, O., Jorgensen, P., Kim, K.H., Roush, F.“Nonstationarity of isomorphism between AF-algebras defined by stationary Bratteli diagrams”’. Ergodic Theory and Dynamical Systems, 20, pp. 1639-1656, 2000.
———–. “Decidability of the isomorphism problem for stationary AF-algebras and the associated ordered simple dimension groups”. Ergodic Theory and Dynamical Systems, 21, pp. 1625-1655, 2001.
———–. Corrigendum to the paper “Decidability of the isomorphism
problem for stationary AF-algebras and the associated ordered simple dimension groups”. Ergodic Theory and Dynamical Systems, 22, p. 633, 2002.
Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.. Algebraic Foundations of many-valued Reasoning. Trends in Logic, vol. 7. Dordrecht: Kluwer Academic Publishers, 2000.
Cignoli, R., Elliott, G.A., Mundici, D. “Reconstructing C∗-algebras from their Murray von Neumann orders”. Advances in Mathematics, 101, pp. 166-179, 1993.
Effros, E.G. Dimensions and C*-algebras. CBMS Regional
Conf. Series in Math., vol. 46, Amer. Math. Soc. Providence, RI,
Elliott, G.A. “On the classification of inductive limits of sequences
of semisimple finite-dimensional algebras”. J. Algebra, 38, pp. 29-44, 1976.
Emch, G.G. Mathematical and Conceptual Foundations of 20thCentury
Physics. Amsterdam: North-Holland, 1984.
Fell, J.M.G. “The dual spaces of C*-algebras”. Trans. Amer. Math. Soc., 94, pp. 365-403, 1960.
Glass, A.M.W., Marra, V. “Embedding finitely generated Abelian lattice-ordered groups: Higman’s Theorem and a realisation of π”. J. London Math. Soc., 68, pp. 545-562, 2003.
Goodearl, K.R. Notes on Real and Complex C*-Algebras. Shiva Math. Series, vol. 5. Boston: Birkh¨auser, 1982.
Haag, D., Kastler, D. “An algebraic approach to quantum field theory”. J. Math. Physics, 5, pp. 848-861, 1964.
Handelman, D. “Positive matrices and dimension groups affiliated to C*-algebras and topological Markov chains”. J. Operator Theory, 6 , pp. 55-74, 1981.
Kastler, D. “Does ergodicity plus locality imply the Gibbs structure?”.
Proc. Symp. Pure Math., II, 38, pp. 467-489, 1982.
Kim, K.H., Roush, F.W. “Some results on decidability of shift equivalence”. J. Combinatorics, Information and System Sci., 4, pp. 123-146, 1979.
———–. “Decidability of shift equivalence”. Lecture Notes in Mathematics, 1342. Springer, pp. 374-424, 1988.
Kitchens, B.P. Symbolic Dynamics: One-sided, Two-sided and Countable State Markov Shifts. Berlin: Springer, 1998.
Mundici, D. “Interpretation of AF C*-algebras in Lukasiewicz sentential calculus”. J. Functional Analysis, 65, pp. 15-63, 1986.
———–. “The Turing complexity of AF C*-algebras with latticeordered K0”. Lecture Notes in Computer Science, 270, pp. 256-264, 1987.
———–. “Farey stellar subdivisions, ultrasimplicial groups, and K0 of AF C*-algebras”. Advances in Mathematics, 68, pp. 23-39, 1988.
———–. “The C*-algebras of three-valued logic”. In: Proceedings Logic Colloquium 1988, Studies in Logic and the Foundations ofn Mathematics. Amsterdam: North-Holland, pp. 61-77, 1989.
———–. “Turing complexity of Behncke-Leptin C*-algebras with a two-point dual”. Annals of Mathematics and Artificial Intelligence, 6, pp. 287-294, 1992.
———–. “Logic of infinite quantum systems”. International Journal of Theoretical Physics, 32, pp. 1941-1955, 1993.
———–. “G¨odel incompleteness and quantum thermodynamic limits”. In: Philosophy of Mathematics Today. Dordrecht: Kluwer Academic Publishers, pp. 287-298, 1997.
———–. “Simple Bratteli diagrams with a G¨odel incomplete isomorphism
problem”. Transactions of the American Mathematical Society, 356, pp. 1937-1955, 2004.
Mundici, D., Panti, G. “Extending addition in Elliott’s local semigroup”. Journal of Functional Analysis, 117, pp. 461-471, 1993.
———–. “Decidable and undecidable prime theories in infinitevalued logic”. Annals of Pure and Applied Logic, 108, pp. 269-278, 2001.
Palis, J., Takens, F. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, 1993. (Cambridge Studies in Advanced Mathematics, 35).
Sewell, G.L. Quantum Theory of Collective Phenomena. Oxford: Clarendon Press, 1986.
Downloads
