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 TMF, 2000, Volume 123, Number 1, Pages 88–93 (Mi tmf588)

Von Neumann algebras generated by translation-invariant Gibbs states of the Ising model on a Bethe lattice

F. M. Mukhamedov

Romanovskii Mathematical Institute of the National Academy of Sciences of Uzbekistan

Abstract: The Ising model on a Bethe lattice of order $k\geq2$ is considered. For maximum or minimum translation-invariant Gibbs states of this model, the relations between the von Neumann algebras generated by these states for the Gelfand–Neimark–Segal representation are found. These algebras can be of types $\mathrm{III}_\lambda$, $\lambda\in(0,1)$, and $\mathrm{III}_1$.

DOI: https://doi.org/10.4213/tmf588

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English version:
Theoretical and Mathematical Physics, 2000, 123:1, 489–493

Bibliographic databases:

Citation: F. M. Mukhamedov, “Von Neumann algebras generated by translation-invariant Gibbs states of the Ising model on a Bethe lattice”, TMF, 123:1 (2000), 88–93; Theoret. and Math. Phys., 123:1 (2000), 489–493

Citation in format AMSBIB
\Bibitem{Muk00} \by F.~M.~Mukhamedov \paper Von Neumann algebras generated by translation-invariant Gibbs states of the Ising model on a Bethe lattice \jour TMF \yr 2000 \vol 123 \issue 1 \pages 88--93 \mathnet{http://mi.mathnet.ru/tmf588} \crossref{https://doi.org/10.4213/tmf588} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1773214} \zmath{https://zbmath.org/?q=an:1066.82505} \transl \jour Theoret. and Math. Phys. \yr 2000 \vol 123 \issue 1 \pages 489--493 \crossref{https://doi.org/10.1007/BF02551055} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000087532100008} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Mukhamedov F, Rozikov U, “On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras”, Journal of Statistical Physics, 114:3–4 (2004), 825–848
2. Mukhamedov, F, “On a factor associated with the unordered phase of lambda-model on a Cayley tree”, Reports on Mathematical Physics, 53:1 (2004), 1
3. Mukhamedov, F, “On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras II”, Journal of Statistical Physics, 119:1–2 (2005), 427
4. Fidaleo F., Mukhamedov F., “On factors associated with quantum Markov states corresponding to nearest neighbor models on a Cayley tree”, Quantum Probability and Infinite Dimensional Analysis, Qp-Pq Quantum Probability and White Noise Analysis, 18, 2005, 237–251
5. L. Accardi, F. M. Mukhamedov, M. Kh. Saburov, “Uniqueness of Quantum Markov Chains Associated with an $XY$-Model on a Cayley Tree of Order $2$”, Math. Notes, 90:2 (2011), 162–174
6. Mukhamedov F., Barhoumi A., Souissi A., “On an Algebraic Property of the Disordered Phase of the Ising Model with Competing Interactions on a Cayley Tree”, Math. Phys. Anal. Geom., 19:4 (2016), 21
7. Accardi L. Mukhamedov F. Souissi A., “On Construction of Quantum Markov Chains on Cayley trees”, Algebra, Analysis and Quantum Probability, Journal of Physics Conference Series, 697, ed. Ayupov S. Chilin V. Ganikhodjaev N. Mukhamedov F. Rakhimov I., IOP Publishing Ltd, 2016, 012018
8. Mukhamedov F., Barhoumi A., Souissi Abdessatar, “Phase Transitions for Quantum Markov Chains Associated with Ising Type Models on a Cayley Tree”, J. Stat. Phys., 163:3 (2016), 544–567
9. Mukhamedov F., Barhoumi A., Souissi A., El Gheteb S., “On Translation Invariant Quantum Markov Chains Associated With Ising-Xy Models on a Cayley Tree”, 37Th International Conference on Quantum Probability and Related Topics (Qp37), Journal of Physics Conference Series, 819, eds. Accardi L., Mukhamedov F., Hee P., IOP Publishing Ltd, 2017, UNSP 012006
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