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Almost sure convergence of iterates of contractions in noncommutative \(L_ 2\)-spaces. (English) Zbl 0739.46048
Recently a remarkable progress has been made in the individual ergodic theory of positive contractions in von Neumann algebras. Many pointwise ergodic theorems have been extended to the operator algebra context. The study of such problematics is motivated by the theory of open (irreversible) quantum dynamical systems. From the physical point of view the most important are (semigroups of) completely positive maps on \(C^*\)- or \(W^*\)-algebras but in the context of this paper it seems to be more natural to consider a larger class of positive contractions. We shall discuss the asymptotic behaviour of Schwarz maps in von Neumann algebras. More exactly, we are going to prove some results concerning the iterates of contractions in \(L_ 2\) over a von Neumann algebra \(M\) (with respect to a faithful normal state \(\Phi\) on \(M\)).
MSC:
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
47A35 Ergodic theory of linear operators
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[1] Alexits, G.: Convergence problems of orthogonal series. New York Oxford Paris: Pergamon Press 1961 · Zbl 0098.27403
[2] Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics, I. New York Heidelberg Berlin: Springer 1979 · Zbl 0421.46048
[3] Burkholder, D.L., Chow, Y.S.: Iterates of conditional expectation operators. Proc. Am. Math. Soc.12, 490–495 (1961) · Zbl 0106.33201
[4] Gaposhkin, V.F.: Criteria of the strong law of large numbers for some classes of stationary processes and homogeneous random fields (Russian). Theory Probab. Appl.,22, 295–319 (1977)
[5] Gaposhkin, V.F.: Individual ergodic theorem for normal operators inL 2 (Russian). Funkts. Anal. Appl.15, 18–22 (1981) · Zbl 0467.22008
[6] Goldstein, M.S.: Theorems in almost everywhere convergence in von Neumann algebras (Russian). J. Oper. Theory6, 233–311 (1981) · Zbl 0488.46053
[7] Hensz, E., Jajte, R.: Pointwise convergence theorems inL 2 over a von Neumann algebra. Math. Z.193, 413–429 (1986) · Zbl 0613.46056
[8] Jajte, R.: Strong limit theorems in non-commutative probability. (Lect. Notes Math., vol. 1110) Berlin Heidelberg New York: Springer 1985 · Zbl 0554.46033
[9] Jajte, R.: Ergodic theorem in von Neumann algebras. Semesterbericht Funktionenanalysis, Tübingen, Sommersemester, pp. 135–144 (1986)
[10] Jajte, R.: Contraction semigroups inL 2 over a von Neumann algebra. Quantum Probability and Applications III. Proceedings, Oberwolfach 1987 (Lect. Notes Math., vol. 1303, pp. 149–153) Berlin Heidelberg New York: Springer 1988
[11] Jajte, R.: Asymptotic formula for normal operators in non-commutativeL 2-spaces. (to appear)
[12] Kümmerer, B.: A non-commutative individual ergodic theorem. Invent. Math.46, 136–145 (1978) · Zbl 0379.46060
[13] Lance, E.C.: Ergodic theorems for convex sets and operator algebras. Invent. Math.37, 201–214 (1976) · Zbl 0338.46054
[14] Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys.48, 119–130 (1970) · Zbl 0343.47031
[15] Neumann, J. von: Functional operators, vol. 2. Princeton: Princeton University Press 1950 · Zbl 0039.28401
[16] Petz, D.: Ergodic theorems in von Neumann algebras. Acta Sci. Math.46, 329–343 (1983) · Zbl 0535.46043
[17] Petz, D.: Quasi-uniform ergodic theorems in von Neumann algebras. Bull. Lond. Math. Soc.16, 151–156 (1984) · Zbl 0535.46042
[18] Sinai, Y.G., Aushelevich, V.V.: Some problems of non-commutative ergodic theory. Russ. Math. Surv.31, 157–174 (1976) · Zbl 0365.46053
[19] Stein, E.M.: Topics in harmonic analysis. Ann. Math Stud. no. 63. Princeton: Princeton University Press 1980
[20] Takesaki, M.: Theory of operator algebras I. Berlin Heidelberg New York: Springer 1979 · Zbl 0436.46043
[21] Watanabe, S.: Ergodic theorems for dynamical semigroups on operator algebras. Hokkaido Math. J.8, 176–190 (1979) · Zbl 0426.46048
[22] Watanabe, S.: Asymptotic behaviour and eigenvalues of dynamical semi-groups on operator algebras. J. Math. Anal. Appl.86, 411–424 (1982) · Zbl 0489.46049
[23] Yeadon, F.J.: Ergodic theorems for semifinite von Neumann algebras I. J. Math. Soc.2, 326–332 (1977) · Zbl 0369.46061
[24] Yeadon, F.J.: Ergodic theorems for semifinite von Neumann algebras II. Math. Proc. Camb. Philos. Soc.88, 135–147 (1980) · Zbl 0466.46056
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