The classical monotone convergence theorem of Beppo Levi fails in noncommutative -spaces |
| |
Authors: | Barthé lemy Le Gac Ferenc Mó ricz |
| |
Institution: | Université de Provence, Centre de Mathématiques et Informatique, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France ; Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary |
| |
Abstract: | Let be a complex Hilbert space and let be a von Neumann algebra over equipped with a faithful, normal state . Then is a prehilbert space with respect to the inner product , whose completion is given by the Gelfand-Naimark-Segal representation theorem, according to which there exist a one-to-one -homomorphism of into the algebra of all bounded linear operators acting on and a cyclic, separating vector such that for all . Given any separable Hilbert space , we construct a faithful, normal state on and an increasing sequence of positive operators acting on such that is bounded, but fails to converge both bundlewise and in -norm. We also present an example of an increasing sequence of positive operators which has a subsequence converging both bundlewise and in -norm, but the whole sequence fails to converge in either sense. Finally, we observe that our results are linked to a previous one by R. V. Kadison. |
| |
Keywords: | von Neumann algebra $\A$ faithful and normal state $\phi$ completion $L_2=L_2 (\A \phi)$ Gelfand--Naimark--Segal representation theorem bundle convergence classical monotone convergence theorem of Beppo Levi increasing sequence of positive operators |
|
| 点击此处可从《Proceedings of the American Mathematical Society》浏览原始摘要信息 |
| 点击此处可从《Proceedings of the American Mathematical Society》下载免费的PDF全文 |
|