Relative weak mixing of \(W^*\)-dynamical systems via joinings. (English) Zbl 1425.46046
The authors consider a system \((A,\mu,\alpha)\) where \(A\) is a von Neumann algebra with a tracial state \(\mu\) and an automorphism \(\alpha\) fixing the state, and \((F,\lambda)\) a modular subsystem of \(A\). For this setup, they define the notion of \(A\) being weakly mixing relative to \(F\). This is a natural extension of the classic weak mixing corresponding to the case \(F=\mathbb{C}1_A\). The main result establishes that \(A\) is weakly mixing relative to \(F\) precisely when \(A\odot_F A'\) is ergodic relative to \(F\), with \(A\odot_F A'\) being the relative product system of \(A\) and \(A'\) over \(F\) endowed with the state \(\mu\odot_\lambda\mu'\) that is a joining of \(A\) and \(A'\). The main theorem, when specialised to the central system obtained for \(F=A\cap A'\), has implications for a result due to T. Austin et al. [Pac. J. Math. 250, No. 1, 1–60 (2011; Zbl 1219.46058)] about weakly mixing asymptotically abelian \(W^*\)-dynamical systems.
Reviewer: Nadia Larsen (Oslo)
MSC:
| 46L55 | Noncommutative dynamical systems |
Citations:
Zbl 1219.46058References:
| [1] | B. Abadie and K. Dykema, Unique ergodicity of free shifts and some other automorphisms of C∗-algebras, J. Operator Theory 61 (2009), 279–294. [2] T. Austin, T. Eisner and T. Tao, Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems, Pacific J. Math. 250 (2011), 1–60. [3] J. P. Bannon, J. Cameron and K. Mukherjee, The modular symmetry of Markov maps, J. Math. Anal. Appl. 439 (2016), 701–708. [4] J. P. Bannon, J. Cameron and K. Mukherjee, On noncommutative joinings, Int. Math. Res. Notices 2018, 4734–4779. [5] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed., Springer, New York, 1987. [6] E. Christensen, Subalgebras of a finite algebra, Math. Ann. 243 (1979), 17–29. [7] R. Duvenhage, Joinings of W∗-dynamical systems, J. Math. Anal. Appl. 343 (2008), 175–181. |
| [2] | R. Duvenhage, Ergodicity and mixing of W∗-dynamical systems in terms of joinings, Illinois J. Math. 54 (2010), 543–566. [9] R. Duvenhage, Relatively independent joinings and subsystems of W∗-dynamical systems, Studia Math. 209 (2012), 21–41. [10] R. Duvenhage and F. Mukhamedov, Relative ergodic properties of C∗-dynamical systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17 (2014), 1450005, 26 pp. 84R. Duvenhage and M. King [11] F. Fidaleo, An ergodic theorem for quantum diagonal measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), 307–320. [12] F. Fidaleo, On strong ergodic properties of quantum dynamical systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), 551–564. [13] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. [14] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemer´edi on arithmetic progressions, J. Anal. Math. 31 (1977), 204–256. [15] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures, Princeton Univ. Press, Princeton, NJ, 1981. [16] H. Furstenberg and Y. Katznelson, An ergodic Szemer´edi theorem for commuting transformations, J. Anal. Math. 34 (1978), 275–291. [17] E. Glasner, Ergodic Theory via Joinings, Math. Surveys Monogr. 101, Amer. Math. Soc., Providence, RI, 2003. [18] R. H. Herman and M. Takesaki, States and automorphism groups of operator algebras, Comm. Math. Phys. 19 (1970), 142–160. [19] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1–25. |
| [3] | R. V. Kadison and J. R. Ringrose Fundamentals of the Theory of Operator Algebras Volume II: Advanced Theory, Vol. 2, Amer. Math. Soc., 2015. [21] D. Kerr, H. Li and M. Pichot, Turbulence, representations and trace-preserving actions, Proc. London Math. Soc. (3) 100 (2010), 459–484. [22] S. Popa, Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups, Invent. Math. 170 (2007), 243–295. [23] D. J. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. Anal. Math. 35 (1979), 97–122. [24] J.-L. Sauvageot et J.-P. Thouvenot, Une nouvelle d´efinition de l’entropie dynamique des syst‘emes non commutatifs, Comm. Math. Phys. 145 (1992), 411–423. [25] A. M. Sinclair and R. R. Smith, Finite von Neumann Algebras and Masas, London Math. Soc. Lecture Note Ser. 351, Cambridge Univ. Press, 2008. [26] C. F. Skau, Finite subalgebras of a von Neumann algebra, J. Funct. Anal. 25 (1977), 211–235. |
| [4] | S¸. Str˘atil˘a, Modular Theory in Operator Algebras, Editura Academiei, Bucure¸sti, and Abacus Press, Tunbridge Wells, 1981. [28] M. Takesaki, Theory of Operator Algebras. II, Encyclopaedia Math. Sci. 125, Springer, Berlin, 2003. · Zbl 0504.46043 |
| [5] | T. Tao, Poincar´e’s Legacies, Part I: Pages from Year Two of a Mathematical Blog, Amer. Math. Soc., 2009. · Zbl 1171.00003 |
| [6] | R. J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math. 20 (1976), 555–588. · Zbl 0349.28011 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
