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Pointwise convergence theorems in $$L_ 2$$ over a von Neumann algebra. (English) Zbl 0613.46056
Let M be a von Neumann algebra with a faithful normal state $$\phi$$. We consider the Hilbert space $$H=L_ 2(M,\phi)$$ (the completion of M under the norm $$x\mapsto \phi (x^*x)^{1/2})$$. In this space we introduce a notion of the almost sure convergence which coincides with the usual almost everywhere convergence in the case $$M=L_{\infty}$$ (over a probability space).
Using this type of convergence we prove an individual ergodic theorem for unitary operators in H induced by *-automorphisms of M. We also give some (almost sure) asymptotic formulae for the Cesàro means and the ergodic Hilbert transform of a unitary operator in H. A non-commutative extension of the Rademacher-Menshov theorem is proved as well.

##### MSC:
 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 46L55 Noncommutative dynamical systems 28D05 Measure-preserving transformations 81P20 Stochastic mechanics (including stochastic electrodynamics) 60F15 Strong limit theorems
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##### References:
 [1] Alexits, G.: Convergence problems of orthogonal series. New York-Oxford-Paris: Pergamon Press 1961 · Zbl 0098.27403 [2] Cotlar, M.: A unified theory of Hilbert transform and ergodic theorems. Rev. Mat. Cuyana1, 105-167 (1955) [3] Dang-Ngoc, N.: Pointwise convergence of martingales in von Neumann algebras. Isr. J. Math.34, 273-280 (1979) · Zbl 0446.60028 [4] Duncan, R.: Pointwise convergence theorems for selfadjoint and unitary contractions. Ann. Probab.5, 622-626 (1977) · Zbl 0368.40001 [5] 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) [6] Goldstein, M.S.: Theorems in almost everywhere convergence in von Neumann algebras (Russian). J. Oper. Theory6, 233-311 (1981) · Zbl 0488.46053 [7] Goldstein, S., ?uczak, A.: On the Rademacher-Menshov theorem in von Neumann Algebras. To appear in Studia Sci. Math. Hung. · Zbl 0665.46050 [8] Jajte, R.: Strong limit theorems in non-commutative probability, Lect. Notes in Math. No.1110. Berlin-Heidelberg-New York: Springer 1985 · Zbl 0554.46033 [9] Kümmerer, B.: A non-commutative individual ergodic theorem. Invent Math.46, 136-145 (1978) · Zbl 0379.46060 [10] Lance, E.C.: Ergodic theorems for convex sets and operator algebras. Invent. Math.37, 201-214 (1976) · Zbl 0338.46054 [11] Lance, E.C.: Martingale convergence in von Neumann algebras. Math. Proc. Camb. Philos. Soc.84, 47-56 (1978) · Zbl 0398.46051 [12] Petersen, K.: Another proof of the existence of the ergodic Hilberr transform. Proc. Am. Math. Soc.88, 39-43 (1983) · Zbl 0521.28014 [13] Petersen, K.: Ergodic theory. Cambridge-London-New York: Cambridge University Press 1985 · Zbl 0507.28010 [14] Petz, D.: Ergodic theorems in von Neumann algebras. Acta Sci. Math.46, 329-343 (1983) · Zbl 0535.46043 [15] Petz, D.: Quasi-uniform ergodic theorems in von Neumann algebras. Bull. Lond. Math. Soc.16, 151-156 (1984) · Zbl 0535.46042 [16] Sinai, Y.G., Anshelevich, V.V.: Some problems of non-commutative ergodic theory (Russian). Russian Math. Surv.31, 157-174 (1976) · Zbl 0365.46053 [17] Stein, E.M.: Topics in harmonic analysis. Ann. Math. Stud. No.63. Princeton: University Press 1980 [18] Takesaki, M.: Theory of operator algebras I. Berlin-Heidelberg-New York: Springer 1979 · Zbl 0436.46043 [19] Yeadon, F.J.: Ergodic theorems for semifinite von Neumann algebras I. J. London. Math. Soc.2 (16), 326-332 (1977) · Zbl 0369.46061 [20] Yeadon, F.J.: Ergodic theorems for semifinite von Neumann algebras II. Math. Proc. Camb., Phil. Soc.88, 135-147 (1980) · Zbl 0466.46056
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