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

DOI:
https://doi.org/10.1090/S0002-9939-05-07976-1

Published electronically:
April 8, 2005

MathSciNet review:
2146199

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Abstract | References | Similar Articles | Additional Information

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

Email:
legac@cmi.univ-mrs.fr

**Ferenc Móricz**

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

Email:
moricz@math.u-szeged.hu

Keywords:
von Neumann algebra $\mathfrak {A}$,
faithful and normal state $\phi$,
completion $L_2=L_2 (\mathfrak {A} ,\phi )$,
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.