Passing from Cesàro means to Borel-type methods of summability we prove some ergodic theorem for operators (acting in a Banach space) with spectrum contained in ℂ∖(1,∞).
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Let α be an isometric automorphism of the algebra $𝔹_{p}$ of bounded linear operators in $𝕃^{p}[0, 1]$ (p ≥ 1). Then α transforms conditional expectations into conditional expectations if and only if α is induced by a measure preserving isomorphism of [0, 1].
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We distinguish a class of unbounded operators in $𝕃^{r}$, r ≥ 1, related to the self-adjoint operators in 𝕃². For these operators we prove a kind of individual ergodic theorem, replacing the classical Cesàro averages by Borel summability. The result is equivalent to a version of Gaposhkin's criterion for the a.e. convergence of operators. In the proof, the theory of martingales and interpolation in $𝕃^{r}$-spaces are applied.
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For a sequence $(A_j)$ of mutually orthogonal projections in a Banach space, we discuss all possible limits of the sums $S_n = ∑^n_{j=1} A_j$ in a "strong" sense. Those limits turn out to be some special idempotent operators (unbounded, in general). In the case of X = L₂(Ω,μ), an arbitrary unbounded closed and densely defined operator A in X may be the μ-almost sure limit of $S_n$ (i.e. $S_{n}f → Af$ μ-a.e. for all f ∈ 𝓓(A)).
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A stronger version of almost uniform convergence in von Neumann algebras is introduced. This "bundle convergence" is additive and the limit is unique. Some extensions of classical limit theorems are obtained.
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The Stinespring theorem is reformulated in terms of conditional expectations in a von Neumann algebra. A generalisation for map-valued measures is obtained.
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The paper is devoted to some problems concerning a convergence of pointwise type in the $L_2$-space over a von Neumann algebra M with a faithful normal state Φ [3]. Here $L_2 = L_2(M,Φ)$ is the completion of M under the norm $x → |x|^2 = Φ(x*x)^{1/2}$.
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