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On Fixed Points and Multiparameter Ergodic Theorems in Banach Lattices

We present here multiparameter results about positive operators acting on a weakly sequentially complete Banach lattice. Sections 1, 2 and 3 generalize results obtained by M. A. Akcoglu and the second author in the case of a contraction. Even in that case, the classical L1 theory extends to Banach lattices only under an additional monotonicity assumption (C), introduced in [3], without which the TL (or stochastic) ergodic theorem fails. The example proving this in [4] also shows that, without (C), the decomposition of the space into the “positive” part P, the largest support of a T-invariant element, and the “null” part N on which the TL limit is zero (see, e.g., [22], p. 141), also fails. If T is not a contraction but only mean-bounded, then the space decomposes into the “remaining” part Y, the largest support of a T*-invariant element, and the “disappearing part“ Z (see, e.g., [22], p. 172). Here we obtain, for Banach lattices and in the multiparameter case, a unified proof of both decompositions, and of the TL ergodic theorem.