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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The classical monotone convergence theorem of Beppo Levi fails in noncommutative $L_2$-spaces
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by Barthélemy Le Gac and Ferenc Móricz PDF
Proc. Amer. Math. Soc. 133 (2005), 2559-2567 Request permission


Let $H$ be a complex Hilbert space and let $\mathfrak {A}$ be a von Neumann algebra over $H$ equipped with a faithful, normal state $\phi$. Then $\mathfrak {A}$ is a prehilbert space with respect to the inner product $\langle A\mid B\rangle := \phi (B^* A)$, whose completion $L_2 = L_2 (\mathfrak {A} ,\phi )$ is given by the Gelfand–Naimark–Segal representation theorem, according to which there exist a one-to-one $*$-homomorphism $\pi$ of $\mathfrak {A}$ into the algebra $\mathcal {L} (L_2)$ of all bounded linear operators acting on $L_2$ and a cyclic, separating vector $\omega \in L_2$ such that $\phi (A) = (\pi (A) \omega \mid \omega )$ for all $A\in \mathfrak {A}$. Given any separable Hilbert space $H$, we construct a faithful, normal state $\phi$ on $\mathcal {L} (H)$ and an increasing sequence $(A_n : n\ge 1)$ of positive operators acting on $H$ such that $(\phi (A^2_n) : n\ge 1)$ is bounded, but $(\pi (A_n) \omega : n\ge 1)$ fails to converge both bundlewise and in $L_2$-norm. We also present an example of an increasing sequence of positive operators which has a subsequence converging both bundlewise and in $L_2$-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.
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Additional Information
  • Barthélemy Le Gac
  • Affiliation: Université de Provence, Centre de Mathématiques et Informatique, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France
  • Email:
  • Ferenc Móricz
  • Affiliation: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary
  • Email:
  • Received by editor(s): September 2, 2002
  • Published electronically: April 8, 2005
  • Additional Notes: This research was started while the second-named author visited the “Centre de Mathématiques et Informatique, Université de Provence, Marseille” during the summer of 2002; it was also partially supported by the Hungarian National Foundation for Scientific Research under Grants T 044782 and T 046192.
  • Communicated by: Jonathan M. Borwein
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 2559-2567
  • MSC (2000): Primary 46L53, 46L10
  • DOI:
  • MathSciNet review: 2146199