The range of operators on von Neumann algebras

Bermúdez, Teresa; Kalton, N.

doi:10.1090/S0002-9939-01-06292-X

The range of operators on von Neumann algebras
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by Teresa Bermúdez and N. J. Kalton

Proc. Amer. Math. Soc. 130 (2002), 1447-1455

DOI: https://doi.org/10.1090/S0002-9939-01-06292-X

Published electronically: October 24, 2001

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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.

References

Shamim I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), no. 2, 384–390. MR 1469346, DOI 10.1006/jfan.1996.3093
Luis Bernal-González, On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1003–1010. MR 1476119, DOI 10.1090/S0002-9939-99-04657-2
Juan Bès and Alfredo Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), no. 1, 94–112. MR 1710637, DOI 10.1006/jfan.1999.3437
José Bonet and Alfredo Peris, Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal. 159 (1998), no. 2, 587–595. MR 1658096, DOI 10.1006/jfan.1998.3315
Kit C. Chan, Hypercyclicity of the operator algebra for a separable Hilbert space, J. Operator Theory 42 (1999), no. 2, 231–244. MR 1716973
Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
Gilles Godefroy and Joel H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229–269. MR 1111569, DOI 10.1016/0022-1236(91)90078-J
Karl-Goswin Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345–381. MR 1685272, DOI 10.1090/S0273-0979-99-00788-0
J. Hagler and W. B. Johnson, On Banach spaces whose dual balls are not weak* sequentially compact, Israel J. Math. 28 (1977), no. 4, 325–330. MR 482086, DOI 10.1007/BF02760638
P. Harmand, D. Werner, and W. Werner, $M$-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993. MR 1238713, DOI 10.1007/BFb0084355
Stefan Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72–104. MR 552464, DOI 10.1515/crll.1980.313.72
Domingo A. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), no. 1, 93–103. MR 1259918
C. Kitai, Invariant Closed Sets for Linear Operators, Ph.D. thesis, Univ. of Toronto, 1982.
Robert E. Megginson, An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, New York, 1998. MR 1650235, DOI 10.1007/978-1-4612-0603-3
A. Montes-Rodríguez and C. Romero-Moreno, Supercyclicity in the operator algebra, preprint.
H. Pfitzner, Weak compactness in the dual of a $C^\ast$-algebra is determined commutatively, Math. Ann. 298 (1994), no. 2, 349–371. MR 1256621, DOI 10.1007/BF01459739
Haskell P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36. MR 270122, DOI 10.4064/sm-37-1-13-36
Saharon Shelah, A Banach space with few operators, Israel J. Math. 30 (1978), no. 1-2, 181–191. MR 508262, DOI 10.1007/BF02760838
Saharon Shelah and Juris Steprāns, A Banach space on which there are few operators, Proc. Amer. Math. Soc. 104 (1988), no. 1, 101–105. MR 958051, DOI 10.1090/S0002-9939-1988-0958051-9
Masamichi Takesaki, On the conjugate space of operator algebra, Tohoku Math. J. (2) 10 (1958), 194–203. MR 100799, DOI 10.2748/tmj/1178244713
P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991. MR 1144277, DOI 10.1017/CBO9780511608735