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.

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 ${L}_{2}={L}_{2}(M,\Phi )$ is the completion of M under the norm ${x\to \left|x\right|}^{2}=\Phi {(x*x)}^{1/2}$.

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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