×

Weak ergodicity of nonhomogeneous Markov chains on noncommutative \(L^1\)-spaces. (English) Zbl 1276.47012

Let \(M\) be a von Neumann algebra with faithful normal finite trace \(\tau\). Let \(L^1 (M,\tau )\) be an \(L^1 \)-space associated with \(M\). The set of all self-adjoint elements in \(M\) is denoted by \(M_sa \). Let \(T: L^1 (M,\tau ) \rightarrow L^1 (M,\tau )\) be a linear bounded operator, \(X=\{ x\in L^1 (M_sa ,\tau ) : \tau (x)=0 \}\), \(\delta (T)= \sup_{x\in X, x\neq 0} \frac {||Tx||_1}{||x||_1}\). \(\delta (T)\) is called the Dobrushin ergodicity coefficient of \(T\). The author establishes several properties of the Dobrushin ergodicity coefficient in a noncommutative setting. These results allow to investigate the weak ergodicity of nonhomogeneous discrete Markov processes (NDMP) by means of the ergodicity coefficient. Sufficient conditions are given for such processes to satisfy the weak ergodicity. A necessary and sufficient conditions are given for the satisfaction of the \(L^1 \)-weak ergodicity of NDMP. The author gives an example showing that \(L^1 \)-weak ergodicity is weaker that weak ergodicity. He also gives several concrete examples of noncommutative NDMP.

MSC:

47A35 Ergodic theory of linear operators
28D05 Measure-preserving transformations
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] L. Accardi and F. Fidaleo, Non homogeneous quantum Markov states and quantum Markov fields , J. Funct. Anal. 200 (2003), 324-347. · Zbl 1031.46074
[2] S. Albeverio and R. Høegh-Krohn, Frobenius theory for positive maps of von Neumann algebras , Comm. Math. Phys. 64 (1978), 83-94. · Zbl 0398.46054
[3] W. Bartoszek, Asymptotic properties of iterates of stochastic operators on (AL) Banach lattices , Anal. Polon. Math. 52 (1990), 165-173. · Zbl 0719.47022
[4] W. Bartoszek and B. Kuna, Strong mixing Markov semigroups on \({\mathcal C}_1\) are meager , Colloq. Math. 105 (2006), 311-317. · Zbl 1103.46044
[5] W. Bartoszek and B. Kuna, On residualities in the set of Markov operators on \(\mathcal{C}_1\) , Proc. Amer. Math. Soc. 133 (2005), 2119-2129 · Zbl 1093.47009
[6] O. Bratteli and D.W. Robinson, Operator algebras and quantum statistical mechanics, I , New York Heidelberg Berlin, Springer, 1979. · Zbl 0463.46052
[7] M. Bożejko, B. Kümmerer and R. Speicher, \(q\)-Gaussian processes: Noncommutative and classical aspects , Commun. Math. Phys. 185 (1997) 129-154. · Zbl 0873.60087
[8] J.E. Cohen, Y. Iwasa, G. Rautu, M.B. Ruskai, E. Seneta and G. Zbaganu, Relative entropy under mappings by stochastic matrices , Linear Algebra Appl. 179 (1993), 211-235. · Zbl 0764.60068
[9] R.L. Dobrušin, Central limit theorem for nonstationary Markov chains, II , Teor. Veroyatnost. i Primenen. 1 (1956), 365-425.
[10] C.C.Y. Dorea and A.G.C. Pereira, A note on a variation of Doeblin’s condition for uniform ergodicity of Markov chains , Acta Math. Hungar. 110 (2006), 287-292. · Zbl 1098.60071
[11] E.Yu. Emel’yanov and M.P.H. Wolff, Positive operators on Banach spaces ordered by strongly normal cones , Positivity 7 (2003), 3-22. · Zbl 1042.47029
[12] E.Yu. Emel’yanov and M.P.H. Wolff, Asymptotic behavior of Markov semigroups on non-commutative \(L_1\)-spaces , In book: Quantum probability and infinite dimensional analysis, (Burg, 2001), 77-83, QP-PQ: Quantum Probab. White Noise Anal., 15, World Sci. Publishing, River Edge, NJ, 2003. · Zbl 1038.47031
[13] F.Fagnola and R. Rebolledo, On the existance of stationary states for quantum dyanamical semigroups , J. Math. Phys. 42 (2001), 1296-1308. · Zbl 1013.81031
[14] F. Fagnola and R. Rebolledo, Transience and recurrence of quantum Markov semigroups. Probab , Theory Relat. Fields 126 (2003), 289-306. · Zbl 1024.60031
[15] I.C.F. Ipsen and T.M. Salee, Ergodicity coefficients defined by vector norms , SIAM J. Matrix Anal. Appl. 32 (2011), 153-200. · Zbl 1223.15043
[16] R. Jajte, Strong linit theorems in non-commutative probability , Lecture Notes in Math. vol. 1110, Berlin-Heidelberg: Springer 1984. · Zbl 0563.46036
[17] J. Johnson and D. Isaacson, Conditions for strong ergodicity using intensity matrices , J. Appl. Probab. 25 (1988), 34-42. · Zbl 0644.60069
[18] T. Komorowski and J. Tyrcha, Asymtotic properties of some Markov operators , Bull. Polish Acad. Sci. Math. 37 (1989), 220-228. · Zbl 0767.47012
[19] U. Krengel, Ergodic Theorems , Walter de Gruyter, Berlin-New York, 1985. · Zbl 0575.28009
[20] A. Luczak, Qantum dynamical semigroups in strongly finite von Neumann algebras , Acta Math. Hungar. 92 (2001) 11-17. · Zbl 0997.46043
[21] R.W. Madsen and D.L. Isaacson, Strongly ergodic behavior for non-stationary Markov processes , Ann. Probab. 1 (1973), 329-335. · Zbl 0264.60021
[22] F.M. Mukhamedov, On expansion of quantum quadratic stochastic processes into fibrewise Markov processes defined on von Neumann algebras , Izvestiya Math. 68 (2004), 1009-1024. · Zbl 1074.46046
[23] F. Mukhamedov, On \(L_1\)-weak ergodicity of nonhomogeneous discrete Markov processes and its applications , Rev. Mat. Compult. (to appear), DOI 10.1007/s13163-012-0096-9). · Zbl 1334.60145
[24] F. Mukhamedov, S. Temir and H. Akin, On stability properties of positive contractions of \(L^1\)-spaces accosiated with finite von Neumann algebras , Colloq. Math. 105 (2006), 259-269. · Zbl 1121.47004
[25] E. Nelson, Note on non-commutative integration , J. Funct. Anal. 15 (1974), 103-117. · Zbl 0292.46030
[26] C. Niculescu, A. Ströh and L. Zsidó, Noncommutative extensions of classical and multiple recurrence theorems , J. Operator Theory 50 (2003), 3-52. · Zbl 1036.46053
[27] K.R. Parthasarathy, An intoroduction to quantum stochastic calculus , Basel, Brirkäuser, 1992. · Zbl 0751.60046
[28] M. Pulka, On the mixing property and the ergodic principle for nonhomogeneous Markov chains , Linear Algebra Appl. 434 (2011), 1475-1488. · Zbl 1213.60120
[29] M.B. Ruskai, S. Szarek and E. Werner, An analysis of completely positive trace-preserving maps on \(M_2\) , Linear Algebra Appl. 347 (2002), 159-187. · Zbl 1032.47046
[30] T.A. Sarymsakov and G.Ya. Grabarnik, Regularity of monotonically continuous contractions acting on the von Neumann algebra , Dokl. Akad. Nauk UzSSR 1987, No.5, 9-11. · Zbl 0645.47031
[31] T.A. Sarymsakov and N.P. Zimakov, Ergodic principle for Markov semigroups on partially ordered normed spaces with base , Dokl. Akad. USSR 289 (1986), No. 3, 554-558.
[32] E. Seneta, On the historical development of the theory of finite inhomogeneous Markov chains , Proc. Cambridge Philos. Soc. 74 (1973), 507-513 · Zbl 0271.60074
[33] Z. Suchanecki, An \(L^1\) extension of stochastic dynamics for irreversible systems , Lecture Notes in Math. vol. 1391, Berlin-Heidelberg: Springer (1984), 367-374. · Zbl 0684.60042
[34] Ch.P. Tan, On the weak ergodicity of nonhomogeneous Markov chains , Statis. & Probab. Lett. 26 (1996), 293-295. · Zbl 0852.60079
[35] F. Yeadon, Ergodic theorems for semifinite von Neumann algebars I , J. London Math. Soc. (2) 16 (1977), no. 2, 326-332. · Zbl 0369.46061
[36] R. Zaharopol and G. Zbaganu, Dobrushin coefficients of ergodicity and asymptotically stable \(L^{1}\)-contractions , J. Theoret. Probab. 99 (1999), no. 4, 885-902. · Zbl 0965.47010
[37] A.I. Zeifman and D.L. Isaacson, On strong ergodicity for nonhomogeneous continuous-time Markov chains , Stochast. Process. Appl. 50 (1994), 263-273. · Zbl 0802.60069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.