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Автор: Grafiati

Опубліковано: 10 грудня 2022

Оновлено: 28 січня 2023

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1

Accardi, Luigi, Abdessatar Souissi, and El Gheteb Soueidy. "Quantum Markov chains: A unification approach." Infinite Dimensional Analysis, Quantum Probability and Related Topics 23, no. 02 (June 2020): 2050016. http://dx.doi.org/10.1142/s0219025720500162.

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Анотація:

In this paper, we study a unified approach for quantum Markov chains (QMCs). A new quantum Markov property that generalizes the old one, is discussed. We introduce Markov states and chains on general local algebras, possessing a generic algebraic property. We stress that this kind of algebras includes both Boson and Fermi algebras. Our main results concern two reconstruction theorems for quantum Markov chains and for quantum Markov states. Namely, we illustrate the results through examples.

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2

Cadavid, Paula, Mary Luz Rodiño Montoya, and Pablo M. Rodriguez. "The connection between evolution algebras, random walks and graphs." Journal of Algebra and Its Applications 19, no. 02 (January 29, 2019): 2050023. http://dx.doi.org/10.1142/s0219498820500231.

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Анотація:

Evolution algebras are a new type of non-associative algebras which are inspired from biological phenomena. A special class of such algebras, called Markov evolution algebras, is strongly related to the theory of discrete time Markov chains. The winning of this relation is that many results coming from Probability Theory may be stated in the context of Abstract Algebra. In this paper, we explore the connection between evolution algebras, random walks and graphs. More precisely, we study the relationships between the evolution algebra induced by a random walk on a graph and the evolution algebra determined by the same graph. Given that any Markov chain may be seen as a random walk on a graph, we believe that our results may add a new landscape in the study of Markov evolution algebras.

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3

Mukhamedov, Farrukh, and Izzat Qaralleh. "Entropy Treatment of Evolution Algebras." Entropy 24, no. 5 (April 24, 2022): 595. http://dx.doi.org/10.3390/e24050595.

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Анотація:

In this paper, by introducing an entropy of Markov evolution algebras, we treat the isomorphism of S-evolution algebras. A family of Markov evolution algebras is defined through the Hadamard product of structural matrices of non-negative real S-evolution algebras, and their isomorphism is studied by means of their entropy. Furthermore, the isomorphism of S-evolution algebras is treated using the concept of relative entropy.

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4

Mukhamedov, Farrukh, and Izzat Qaralleh. "Entropy Treatment of Evolution Algebras." Entropy 24, no. 5 (April 24, 2022): 595. http://dx.doi.org/10.3390/e24050595.

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Анотація:

In this paper, by introducing an entropy of Markov evolution algebras, we treat the isomorphism of S-evolution algebras. A family of Markov evolution algebras is defined through the Hadamard product of structural matrices of non-negative real S-evolution algebras, and their isomorphism is studied by means of their entropy. Furthermore, the isomorphism of S-evolution algebras is treated using the concept of relative entropy.

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5

OHNO, HIROMICHI. "EXTENDABILITY OF GENERALIZED QUANTUM MARKOV CHAINS ON GAUGE INVARIANT C*-ALGEBRAS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 08, no. 01 (March 2005): 141–52. http://dx.doi.org/10.1142/s0219025705001901.

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6

Jenčová, Anna, Dénes Petz, and József Pitrik. "Markov triplets on CCR-algebras." Acta Scientiarum Mathematicarum 76, no. 1-2 (June 2010): 111–34. http://dx.doi.org/10.1007/bf03549824.

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7

Kümmerer, Burkhard. "Markov dilations on W∗-algebras." Journal of Functional Analysis 63, no. 2 (September 1985): 139–77. http://dx.doi.org/10.1016/0022-1236(85)90084-9.

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8

MATSUI, TAKU. "MARKOV SEMIGROUPS ON UHF ALGEBRAS." Reviews in Mathematical Physics 05, no. 03 (September 1993): 587–600. http://dx.doi.org/10.1142/s0129055x93000176.

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Анотація:

We consider a class of Markov semigroups on UHF algebras. We establish the existence of dynamics for long range interactions. Our idea is a non-commutative extension of the argument for classical interacting particle systems. As a by-product we obtain sufficient conditions for unique ergodicity.

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9

Matsumoto, Kengo. "On C*-Algebras Associated with Subshifts." International Journal of Mathematics 08, no. 03 (May 1997): 357–74. http://dx.doi.org/10.1142/s0129167x97000172.

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Анотація:

We construct and study C*-algebras associated with subshifts in symbolic dynamics as a generalization of Cuntz–Krieger algebras for topological Markov shifts. We prove some universal properties for the C*-algebras and give a criterion for them to be simple and purely infinite. We also present an example of a C*-algebra coming from a subshift which is not conjugate to a Markov shift.

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10

Al Harbat, Sadek. "Markov trace on a tower of affine Temperley–Lieb algebras of type Ã." Journal of Knot Theory and Its Ramifications 24, no. 09 (August 2015): 1550049. http://dx.doi.org/10.1142/s0218216515500492.

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Анотація:

We define a tower of affine Temperley–Lieb algebras of type Ã. We prove that there exists a unique Markov trace on this tower, this trace comes from the Markov–Ocneanu–Jones trace on the tower of Temperley–Lieb algebras of type A. We define an invariant of special kind of links as an application of this trace.

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11

Ekhaguere, G. O. S. "Dirichlet forms on partial *-algebras." Mathematical Proceedings of the Cambridge Philosophical Society 104, no. 1 (July 1988): 129–40. http://dx.doi.org/10.1017/s0305004100065300.

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Анотація:

Dirichlet forms and their associated function spaces have been studied by a number of authors [4, 6, 7, 12, 15–18, 22, 25, 26]. Important motivation for the study has been the connection of Dirichlet forms with Markov processes [16–18, 25, 26]: for example, to every regular symmetric Dirichlet form, there is an associated Hunt process [13, 20]. This makes the theory of Dirichlet forms a convenient source of examples of Hunt processes. In the non-commutative setting, Markov fields have been studied by several authors [1–3, 14, 19, 24, 28]. It is therefore interesting to develop a non-commutative extension of the theory of Dirichlet forms and to study their connection with non-commutative Markov processes.

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12

ŁUGIEWICZ, P., R. OLKIEWICZ, and B. ZEGARLINSKI. "NONLINEAR MARKOV SEMIGROUPS ON C*-ALGEBRAS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 16, no. 01 (March 2013): 1350004. http://dx.doi.org/10.1142/s0219025713500045.

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Анотація:

A notion of a nonlinear quantum dynamical semigroup is introduced and discussed. Some sufficient conditions, expressed solely in terms of the duality map, in order that a multivalued mapping on a C*-algebra generates the nonlinear Markov semigroup are proposed.

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13

Duru, Hūlya, and Serkan Ilter. "Weak Markov operators." Filomat 32, no. 15 (2018): 5453–57. http://dx.doi.org/10.2298/fil1815453d.

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Анотація:

Let A and B be f -algebras with unit elements eA and eB respectively. A positive operator T from A to B satisfying T(eA) = eB is called a Markov operator. In this definition we replace unit elements with weak order units and, in this case, call T to be a weak Markov operator. In this paper, we characterize extreme points of the weak Markov operators.

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14

ACCARDI, LUIGI, HIROMICHI OHNO, and FARRUKH MUKHAMEDOV. "QUANTUM MARKOV FIELDS ON GRAPHS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 13, no. 02 (June 2010): 165–89. http://dx.doi.org/10.1142/s0219025710004000.

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Анотація:

We introduce generalized quantum Markov states and generalized d-Markov chains which extend the notion quantum Markov chains on spin systems to that on C*-algebras defined by general graphs. As examples of generalized d-Markov chains, we construct the entangled Markov fields on tree graphs. The concrete examples of generalized d-Markov chains on Cayley trees are also investigated.

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15

MATSUMOTO, KENGO. "RELATIONS AMONG GENERATORS OF C*-ALGEBRAS ASSOCIATED WITH SUBSHIFTS." International Journal of Mathematics 10, no. 03 (May 1999): 385–405. http://dx.doi.org/10.1142/s0129167x99000148.

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Анотація:

We study C*-algebras associated with subshifts that are generalization of Cuntz–Krieger algebras defined for topological Markov shifts. We prove that the C*-algebras associated with subshifts are the universal concrete C*-algebras generated by some partial isometries satisfying certain relations among their support projections and range projections. The relations are coming from the concatenation rules of words of the subshifts.

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16

Rui, Hebing. "Markov traces on cyclotomic Temperley–Lieb algebras." Proceedings of the American Mathematical Society 134, no. 10 (May 5, 2006): 2873–80. http://dx.doi.org/10.1090/s0002-9939-06-08327-4.

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17

Orellana, R. C. "Weights of Markov traces on Hecke algebras." Journal für die reine und angewandte Mathematik (Crelles Journal) 1999, no. 508 (March 12, 1999): 157–78. http://dx.doi.org/10.1515/crll.1999.508.157.

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Анотація:

Abstract We compute the weights, i.e. the values at the minimal idempotents, for the Markov trace on the Hecke algebra of type Bn and type Dn. In order to prove the weight formula, we define representations of the Hecke algebra of type B onto a reduced Hecke algebra of type A. To compute the weights for type D we use the inclusion of the Hecke algebra of type D into the Hecke algebra of type B.

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18

Wagoner, J. B. "Topological Markov chains, C∗-algebras, and K2." Advances in Mathematics 71, no. 2 (October 1988): 133–85. http://dx.doi.org/10.1016/0001-8708(88)90075-8.

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19

Rehren, K. H. "Markov traces as characters for local algebras." Nuclear Physics B - Proceedings Supplements 18, no. 2 (January 1991): 259–68. http://dx.doi.org/10.1016/0920-5632(91)90139-6.

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20

OHNO, H. "FACTORS GENERATED BY C*-FINITELY CORRELATED STATES." International Journal of Mathematics 18, no. 01 (January 2007): 27–41. http://dx.doi.org/10.1142/s0129167x07003947.

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Анотація:

We present several equivalent conditions for C*-finitely correlated states defined on the UHF algebras to be factor states and consider the types of factors generated by them. Subfactors generated by generalized quantum Markov chains defined on the gauge-invariant parts of the UHF algebras are also discussed.

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21

Matsui, Taku. "A Characterization of Pure Finitely Correlated States." Infinite Dimensional Analysis, Quantum Probability and Related Topics 01, no. 04 (October 1998): 647–61. http://dx.doi.org/10.1142/s0219025798000351.

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22

Sumner, Jeremy G. "Multiplicatively closed Markov models must form Lie algebras." ANZIAM Journal 59 (January 3, 2018): 240. http://dx.doi.org/10.21914/anziamj.v59i0.12028.

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23

Matsumoto, Kengo. "Simple $C^*$-algebras arising from certain Markov codes." Acta Scientiarum Mathematicarum 80, no. 12 (2014): 95–120. http://dx.doi.org/10.14232/actasm-012-024-6.

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24

Hermanns, H., U. Herzog, and V. Mertsiotakis. "Stochastic process algebras – between LOTOS and Markov chains." Computer Networks and ISDN Systems 30, no. 9-10 (May 1998): 901–24. http://dx.doi.org/10.1016/s0169-7552(97)00133-5.

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25

Bhat, B. V. Rajarama, and K. R. Parthasarathy. "Kolmogorov’s existence theorem for Markov processes inC* algebras." Proceedings Mathematical Sciences 104, no. 1 (February 1994): 253–62. http://dx.doi.org/10.1007/bf02830889.

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26

SUMNER, JEREMY G. "MULTIPLICATIVELY CLOSED MARKOV MODELS MUST FORM LIE ALGEBRAS." ANZIAM Journal 59, no. 2 (October 2017): 240–46. http://dx.doi.org/10.1017/s1446181117000359.

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Анотація:

We prove that the probability substitution matrices obtained from a continuous-time Markov chain form a multiplicatively closed set if and only if the rate matrices associated with the chain form a linear space spanning a Lie algebra. The key original contribution we make is to overcome an obstruction, due to the presence of inequalities that are unavoidable in the probabilistic application, which prevents free manipulation of terms in the Baker–Campbell–Haursdorff formula.

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27

Rui, Hebing. "Weights of Markov Traces on Cyclotomic Hecke Algebras." Journal of Algebra 238, no. 2 (April 2001): 762–75. http://dx.doi.org/10.1006/jabr.2000.8636.

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28

YAMANAKA, HITOSHI. "WEIGHTS OF MARKOV TRACES OF ALEXANDER POLYNOMIALS FOR MIXED LINKS." Journal of Knot Theory and Its Ramifications 22, no. 07 (June 2013): 1350028. http://dx.doi.org/10.1142/s0218216513500284.

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Анотація:

Using the Fourier expansion of Markov traces for Ariki–Koike algebras over ℚ(q, u1, …, ue), we give a direct definition of the Alexander polynomials for mixed links. We observe that under the corresponding specialization of a Markov parameter, the Fourier coefficients of Markov traces take quite a simple form. As a consequence, we show that the Alexander polynomial of a mixed link is essentially equal to the Alexander polynomial of the link obtained by resolving the twisted parts.

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29

Carbone, Raffaella, and Andrea Martinelli. "Logarithmic Sobolev inequalities in non-commutative algebras." Infinite Dimensional Analysis, Quantum Probability and Related Topics 18, no. 02 (June 2015): 1550011. http://dx.doi.org/10.1142/s0219025715500113.

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Анотація:

We study the relations between (tight) logarithmic Sobolev inequalities, entropy decay and spectral gap inequalities for Markov evolutions on von Neumann algebras. We prove that log-Sobolev inequalities (in the non-commutative form defined by Olkiewicz and Zegarlinski in Ref. 25) imply spectral gap inequalities, with optimal relation between the constants. Furthermore, we show that a uniform exponential decay of a proper relative entropy is equivalent to a modified version of log-Sobolev inequalities. The relations among the mentioned inequalities are investigated and often depend on some regularity conditions, which are also discussed. With regard to this aspect, we provide an example of a positive identity-preserving semigroup not verifying the usually requested regularity conditions (which are always fulfilled for reversible classical Markov processes).

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30

ACCARDI, LUIGI, TAKASHI MATSUOKA, and MASANORI OHYA. "ENTANGLED MARKOV CHAINS ARE INDEED ENTANGLED." Infinite Dimensional Analysis, Quantum Probability and Related Topics 09, no. 03 (September 2006): 379–90. http://dx.doi.org/10.1142/s0219025706002445.

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Анотація:

Entangled Markov chains (EMC) were so baptized on the basis of the conjecture that they provide examples of states, on infinite tensor products of matrix algebras, which are in some sense "entangled".2 In this paper we introduce the notion of multiple (or "many-body") entanglement and extend the two-body criterion of entanglement obtained in Ref. 17 to this case. We then apply this extension to EMC and prove that "generically" they satisfy the entanglement conditions.

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31

SCHMIDT, THOMAS LUNDSGAARD, and KLAUS THOMSEN. "Circle maps and -algebras." Ergodic Theory and Dynamical Systems 35, no. 2 (August 28, 2013): 546–84. http://dx.doi.org/10.1017/etds.2013.64.

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Анотація:

AbstractWe consider a construction of ${C}^{\ast } $-algebras from continuous piecewise monotone maps on the circle which generalizes the crossed product construction for homeomorphisms and more generally the construction of Renault, Deaconu and Anantharaman-Delaroche for local homeomorphisms. Assuming that the map is surjective and not locally injective we give necessary and sufficient conditions for the simplicity of the ${C}^{\ast } $-algebra and show that it is then a Kirchberg algebra. We provide tools for the calculation of the $K$-theory groups and turn them into an algorithmic method for Markov maps.

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32

Bolaños-Servin, Jorge R., and Raffaella Carbone. "Spectral Properties of Circulant Quantum Markov Semigroups." Open Systems & Information Dynamics 21, no. 04 (December 2014): 1450007. http://dx.doi.org/10.1142/s1230161214500073.

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Анотація:

We study the spectral properties of the generators of circulant quantum Markov semigroups. We can find an explicit expression for eigenvalues and eigenvectors of the infinitesimal generator and, in particular, we prove that the spectral gap is strictly positive. By proper techniques, we can reduce the problem on non-commutative algebras to the analogous one for a classical process with a circulant generator.

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33

Kümmerer, Burkhard, and Kay Schwieger. "Diagonal couplings of quantum Markov chains." Infinite Dimensional Analysis, Quantum Probability and Related Topics 19, no. 02 (June 2016): 1650012. http://dx.doi.org/10.1142/s0219025716500120.

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Анотація:

In this paper we extend the coupling method from classical probability theory to quantum Markov chains on atomic von Neumann algebras. In particular, we establish a coupling inequality, which allow us to estimate convergence rates by analyzing couplings. For a given tensor dilation we construct a self-coupling of a Markov operator. It turns out that the coupling is a dual version of the extended dual transition operator studied by Gohm et al. We deduce that this coupling is successful if and only if the dilation is asymptotically complete.

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34

DUTKAY, DORIN ERVIN, and PALLE E. T. JORGENSEN. "Representations of Cuntz algebras associated to quasi-stationary Markov measures." Ergodic Theory and Dynamical Systems 35, no. 7 (June 30, 2014): 2080–93. http://dx.doi.org/10.1017/etds.2014.37.

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Анотація:

In this paper, we answer the question of equivalence, or singularity, of two given quasi-stationary Markov measures on one-sided infinite words, as well as the corresponding question of equivalence of associated Cuntz algebra${\mathcal{O}}_{N}$-representations. We do this by associating certain monic representations of${\mathcal{O}}_{N}$to quasi-stationary Markov measures and then proving that equivalence for a pair of measures is decided by unitary equivalence of the corresponding pair of representations.

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35

Poulain d’Andecy, L., and E. Wagner. "The HOMFLY-PT polynomials of sublinks and the Yokonuma–Hecke algebras." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 6 (September 18, 2018): 1269–78. http://dx.doi.org/10.1017/s0308210518000203.

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Анотація:

We describe completely the link invariants constructed using Markov traces on the Yokonuma–Hecke algebras in terms of the linking matrix and the Hoste–Ocneanu–Millett–Freyd–Lickorish–Yetter–Przytycki–Traczyk (HOMFLY-PT) polynomials of sublinks.

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36

Matsumoto, Kengo. "Topological conjugacy of topological Markov shifts and Ruelle algebras." Journal of Operator Theory 82, no. 2 (September 15, 2019): 253–84. http://dx.doi.org/10.7900/jot.2018apr08.2235.

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Анотація:

We will characterize topological conjugation for two-sided topological Markov shifts (¯¯¯¯¯XA,¯¯¯σA) in terms of the associated asymptotic Ruelle C∗-algebra RA and its commutative C∗-subalgebra C(¯¯¯¯¯XA) and the canonical circle action. We will also show that the extended Ruelle algebra ˜RA, which is a unital and purely infinite version of RA, together with its commutative C∗-subalgebra C(¯¯¯¯¯XA) and the canonical torus action γA is a complete invariant for topological conjugacy of (¯¯¯¯¯XA,¯¯¯σA). The diagonal action of γA has a unique KMS-state on ˜RA, which is an extension of the Parry measure on ¯¯¯¯¯XA.

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37

Kielanowicz, Katarzyna, and Andrzej Łuczak. "Ergodic properties of Markov semigroups in von Neumann algebras." Publicacions Matemàtiques 64 (January 1, 2020): 283–331. http://dx.doi.org/10.5565/publmat6412012.

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38

Paris, Luis, and Loïc Rabenda. "Singular Hecke algebras, Markov traces, and HOMFLY-type invariants." Annales de l’institut Fourier 58, no. 7 (2008): 2413–43. http://dx.doi.org/10.5802/aif.2419.

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39

Chlouveraki, Maria, and Loïc Poulain d'Andecy. "Markov Traces on Affine and Cyclotomic Yokonuma–Hecke Algebras." International Mathematics Research Notices 2016, no. 14 (October 1, 2015): 4167–228. http://dx.doi.org/10.1093/imrn/rnv257.

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40

Kielanowicz, Katarzyna, and Andrzej Łuczak. "Spectral properties of Markov semigroups in von Neumann algebras." Journal of Mathematical Analysis and Applications 453, no. 2 (September 2017): 821–40. http://dx.doi.org/10.1016/j.jmaa.2017.04.029.

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41

Carbone, Raffaella, Emanuela Sasso, and Veronica Umanità. "Decoherence for Quantum Markov Semi-Groups on Matrix Algebras." Annales Henri Poincaré 14, no. 4 (August 12, 2012): 681–97. http://dx.doi.org/10.1007/s00023-012-0199-3.

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42

ACCARDI, LUIGI, and ANILESH MOHARI. "TIME REFLECTED MARKOV PROCESSES." Infinite Dimensional Analysis, Quantum Probability and Related Topics 02, no. 03 (September 1999): 397–425. http://dx.doi.org/10.1142/s0219025799000230.

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Анотація:

A classical stochastic process which is Markovian for its past filtration is also Markovian for its future filtration. We show with a counterexample based on quantum liftings of a finite state classical Markov chain that this property cannot hold in the category of expected Markov processes. Using a duality theory for von Neumann algebras with weights, developed by Petz on the basis of previous results by Groh and Kümmerer, we show that a quantum version of this symmetry can be established in the category of weak Markov processes in the sense of Bhat and Parthasarathy. Here time reversal is implemented by an anti-unitary operator and a weak Markov process is time reversal invariant if and only if the associated semigroup coincides with its Petz dual. This construction allows one to extend to the quantum case, both for backward and forward processes, the Misra–Prigogine–Courbage internal time operator and to show that the two operators are intertwined by the time reversal anti-automorphism.

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43

Erkurşun Özcan, Nazife. "On ergodic properties of operator nets on the predual of von neumann algebras." Studia Scientiarum Mathematicarum Hungarica 55, no. 4 (December 2018): 479–86. http://dx.doi.org/10.1556/012.2018.55.4.1414.

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Анотація:

In this paper, we proved theorems which give the conditions that special operator nets on a predual of von Neumann algebras are strongly convergent under the Markov case. Moreover, we investigate asymptotic stability and existence of a lower-bound function for such nets.

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44

Matsumoto, Kengo. "$C^*$-algebras arising from Dyck systems of topological Markov chains." MATHEMATICA SCANDINAVICA 109, no. 1 (September 1, 2011): 31. http://dx.doi.org/10.7146/math.scand.a-15176.

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Анотація:

Let $A$ be an $N \times N$ irreducible matrix with entries in $\{0,1\}$. We define the topological Markov Dyck shift $D_A$ to be a nonsofic subshift consisting of bi-infinite sequences of the $2N$ brackets $(_1,\dots,(_N,)_1,\dots,)_N$ with both standard bracket rule and Markov chain rule coming from $A$. It is regarded as a subshift defined by the canonical generators $S_1^*,\dots, S_N^*, S_1,\dots, S_N$ of the Cuntz-Krieger algebra $\mathcal{O}_A$. We construct an irreducible $\lambda$-graph system $\mathcal{L}^{{\mathrm{Ch}}(D_A)}$ that presents the subshift $D_A$ so that we have an associated simple purely infinite $C^*$-algebra $\mathcal{O}_{\mathcal{L}^{{\mathrm{Ch}}(D_A)}}$. We prove that $\mathcal{O}_{\mathcal{L}^{{\mathrm{Ch}}(D_A)}}$ is a universal unique $C^*$-algebra subject to some operator relations among $2N$ generating partial isometries.

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45

BLACK, SAMSON. "A STATE-SUM FORMULA FOR THE ALEXANDER POLYNOMIAL." Journal of Knot Theory and Its Ramifications 21, no. 03 (March 2012): 1250008. http://dx.doi.org/10.1142/s0218216511009741.

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Анотація:

We develop a diagrammatic formalism for calculating the Alexander polynomial of the closure of a braid as a state-sum. Our main tools are the Markov trace formulas for the HOMFLY-PT polynomial and Young's semi-normal representations of the Iwahori–Hecke algebras of type A.

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46

KALFAGIANNI, EFSTRATIA. "ON THE G2 LINK INVARIANT." Journal of Knot Theory and Its Ramifications 02, no. 04 (December 1993): 431–51. http://dx.doi.org/10.1142/s0218216593000258.

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Анотація:

We construct a polynomial link invariant as Markov trace on certain one parameter algebras and we prove that it is equal to the invariant corresponding to the exeptional Lie algebra of type G2. We use braid representatives to calculate the invariant for several knots and links.

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47

CECCHINI, CARLO. "INDEPENDENCE AND MARKOVIANITY FOR STATES ON VON NEUMANN ALGEBRAS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 10, no. 01 (March 2007): 1–16. http://dx.doi.org/10.1142/s021902570700266x.

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Анотація:

Several possible generalizations of the classical notion of markovianity are given for states defined on a von Neumann algebras generated on a triple of subalgebras. Their mutual relation is discussed in the particular case in which they mutually commute, and the generalization of the classical; time reversal theorem is proved. A structure theorem for a class of Markov chains is also proved.

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48

Petz, Dénes, and József Pitrik. "Markov property of Gaussian states of canonical commutation relation algebras." Journal of Mathematical Physics 50, no. 11 (November 2009): 113517. http://dx.doi.org/10.1063/1.3253974.

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49

Matsumoto, Kengo. "Orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras." Pacific Journal of Mathematics 246, no. 1 (May 1, 2010): 199–225. http://dx.doi.org/10.2140/pjm.2010.246.199.

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50

Diaconis, Persi, C. Y. Amy Pang, and Arun Ram. "Hopf algebras and Markov chains: two examples and a theory." Journal of Algebraic Combinatorics 39, no. 3 (June 18, 2013): 527–85. http://dx.doi.org/10.1007/s10801-013-0456-7.

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