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Proceedings of the American Mathematical Society

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The range of operators on von Neumann algebras

Authors: Teresa Bermúdez and N. J. Kalton
Journal: Proc. Amer. Math. Soc. 130 (2002), 1447-1455
MSC (2000): Primary 47A16, 47C15.
Published electronically: October 24, 2001
MathSciNet review: 1879968
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Abstract: We prove that for every bounded linear operator $T:X\to X$, where $X$ is a non-reflexive quotient of a von Neumann algebra, the point spectrum of $T^*$ is non-empty (i.e., for some $\lambda\in\mathbb C$ the operator $\lambda I-T$ fails to have dense range). In particular, and as an application, we obtain that such a space cannot support a topologically transitive operator.

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Additional Information

Teresa Bermúdez
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Canary Islands, Spain

N. J. Kalton
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211-0001

Keywords: Grothendieck space, $L$-embedded space, von Neumann algebra, point spectrum, topologically transitive operator, hypercyclic operator
Received by editor(s): November 20, 2000
Published electronically: October 24, 2001
Additional Notes: The first author was supported by DGICYT Grant PB 97-1489 (Spain)
The second author was supported by NSF grant DMS-9870027
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society