# zbMATH — the first resource for mathematics

Strong limit theorems for orthogonal sequences in von Neumann algebras. (English) Zbl 0601.46058
Let $$\phi$$ be a faithful normal state on a von Neumann algebra A. It is proved that if a sequence $$\{x_ n\}$$ in A is orthogonal relative to $$\phi$$ and $$\sum \phi (x^*_ nx_ n)(\log n/n)^ 2$$ is finite, then the sequence $$\{$$ 1/n$$\sum^{n}_{k=1}x_ k\}$$ of averages converges almost uniformly in A.
Reviewer: E.Azoff

##### MSC:
 46L10 General theory of von Neumann algebras 60F15 Strong limit theorems
Full Text:
##### References:
 [1] L. Acardi, A. Frigerio and J. T. Lewis, Quantum stochastic processes, Comm. Dublin Inst. Adv. Stud. Ser. A 29 (1980). [2] G. Alexits, Convergence problems of orthogonal series, Translated from the German by I. Földer. International Series of Monographs in Pure and Applied Mathematics, Vol. 20, Pergamon Press, New York-Oxford-Paris, 1961. · Zbl 0098.27403 [3] C. J. K. Batty, The strong law of large numbers for states and traces of a \?*-algebra, Z. Wahrsch. Verw. Gebiete 48 (1979), no. 2, 177 – 191. · Zbl 0395.60033 [4] Nghiêm {\Dj}á\textordmasculine ;ng Ngá»\?c, Pointwise convergence of martingales in von Neumann algebras, Israel J. Math. 34 (1979), no. 4, 273 – 280 (1980). · Zbl 0446.60028 [5] M. Sh. Gol$$^{\prime}$$dshteĭn, Theorems on almost everywhere convergence in von Neumann algebras, J. Operator Theory 6 (1981), no. 2, 233 – 311 (Russian). [6] Burkhard Kümmerer, A non-commutative individual ergodic theorem, Invent. Math. 46 (1978), no. 2, 139 – 145. · Zbl 0379.46060 [7] E. Christopher Lance, Ergodic theorems for convex sets and operator algebras, Invent. Math. 37 (1976), no. 3, 201 – 214. · Zbl 0338.46054 [8] A. Łuczak, Some limit theorems in von Neumann algebras, preprint. [9] A. R. Padmanabhan, Convergence in measure and related results in finite rings of operators, Trans. Amer. Math. Soc. 128 (1967), 359 – 378. · Zbl 0166.11503 [10] M. Plancherel, Sur la convergence des series de fonctions orthogonal, C. R. Acad. Sci. Paris 157 (1913), 270-278. [11] Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. · Zbl 0436.46043
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.