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Abstract

In this note noncommutative versions of Etemadi's SLLN and Petrov's SLLN are given. As a noncommutative counterpart of the classical almost sure convergence, the almost uniform convergence of measurable operators is used.

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We investigate various aspects of stochastic integration in finite von Neumann algebras. For integration with respect to a bounded L 2 -martingale the idea of treating the integral as a bounded operator is developed. Several classes of integrable processes are defined, it turns out that some of them form a Banach or C *-algebra. We find representations of these algebras and establish relations between the von Neumann algebras generated by these representations. Finally, we characterize the range of the stochastic integration operator.

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Abstract  

In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace τ, and {α t}, a strongly continuous extension to L p(M, τ) of a semigroup of absolute contractions on L 1(M, τ). By means of a non-commutative Banach Principle we prove for a Besicovitch function b and xL p(M, τ), that the averages 1/T0 T b(t)α t(x)dt converge bilateral almost uniformly in L p (M, τ) as T → 0.

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