Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. S. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), 384-390. MR 98h:47028a
  • 2. L. Bernal-González, On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), 1003-1010. MR 99f:47010
  • 3. J. Bes and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), 94-112. MR 2000f:47012
  • 4. J. Bonet and A. Peris, Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal. 159 (1998), 587-595. MR 99k:47044
  • 5. K.C. Chan, Hypercyclicity of the operator algebra for a separable Hilbert space, J. Operator Theory 42 (1999), 231-244. MR 2000i:47066
  • 6. R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs no. 64, Longman, London, 1993. MR 94d:46012
  • 7. J. Diestel, Sequences and series in Banach spaces, Springer-Verlag, Berlin, 1984. MR 85i:46020
  • 8. G. Godefroy and J.H. Shapiro,Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229-269. MR 92d:47029
  • 9. K. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36 (1999), 345-381. MR 2000c:47001
  • 10. J. Hagler and W.B. Johnson, On Banach spaces whose dual balls are not weak$^*$-sequentially compact, Israel J. Math. 28 (1977) 325-330. MR 58:2173
  • 11. P. Harmand, D. Werner and W. Werner, M-ideals in Banach spaces and Banach algebras, Springer Lecture Notes, 1547, Springer, Berlin, 1993. MR 94k:46022
  • 12. S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72-104. MR 82b:46013
  • 13. D.A. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), 93-103. MR 95g:47031
  • 14. C. Kitai, Invariant Closed Sets for Linear Operators, Ph.D. thesis, Univ. of Toronto, 1982.
  • 15. R.E. Megginson, An introduction to Banach space theory, Graduate Texts, no. 183, Springer-Verlag, New York, 1998. MR 99k:46002
  • 16. A. Montes-Rodríguez and C. Romero-Moreno, Supercyclicity in the operator algebra, preprint.
  • 17. H. Pfitzner, Weak compactness in the dual of a $C\sp *$-algebra is determined commutatively, Math. Ann. 298 (1994), 349-371. MR 95a:46082
  • 18. H.P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970) 13-36. MR 42:5015
  • 19. S. Shelah, A Banach space with few operators, Israel J. Math. 30 (1978), 181-191. MR 80b:46033
  • 20. S. Shelah and J. Steprans, A Banach space on which there are few operators, Proc. Amer. Math. Soc. 104 (1988) 101-105. MR 90a:46047
  • 21. M. Takesaki, On the conjugate space of an operator algebra, Tohoku Math. J. 10 (1958) 194-203. MR 20:7227
  • 22. P. Wojtaszczyk, Banach spaces for analysts, Cambridge University Press, Cambridge, 1991. MR 93d:46001

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A16, 47C15.

Retrieve articles in all journals with MSC (2000): 47A16, 47C15.

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

American Mathematical Society