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The classical monotone convergence theorem of Beppo Levi fails in noncommutative $L_2$-spaces

Authors: Barthélemy Le Gac and Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 133 (2005), 2559-2567
MSC (2000): Primary 46L53, 46L10
Published electronically: April 8, 2005
MathSciNet review: 2146199
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Abstract: 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

Ferenc Móricz
Affiliation: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary

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
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
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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