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On a pattern of reflexive operator spaces

Author: Lifeng Ding
Journal: Proc. Amer. Math. Soc. 124 (1996), 3101-3108
MSC (1991): Primary 47D15; Secondary 15A30
MathSciNet review: 1343689
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Abstract: A linear subspace $M$ is a separating subspace for an operator space $S$ if the only member of $S$ annihilating $M$ is 0. It is proved in this paper that if $S$ has a strictly separating vector $x$ and a separating subspace $M$ satisfying $Sx \cap [SM] = \{0\}$, then $S$ is reflexive. Applying this to finite dimensional $S$ leads to more results on reflexivity. For example, if dim $S = n$, and every nonzero operator in $S$ has rank $> n^{2}$, then $S$ is reflexive.

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  • 1. B. Aupetit, An improvement of Kaplansky's lemma on locally algebraic operators, Studia Math. 88 (1988), 275-278. MR 89d:47002
  • 2. E. A. Azoff, On finite rank operators and preannihilators, Memoirs Amer. Math. Soc. 357 (1986). MR 88a:47041
  • 3. E. A. Azoff, L. Ding and W. R. Wogen, Separating versus strictly separating vectors, to appear in Proc. Amer. Math. Soc.. CMP 95:11
  • 4. L. Ding, Separating vectors and reflexivity, Lin. Alg. Appl. 174 (1992), 37-52. MR 94a:47075
  • 5. L. Ding, On strictly separating vectors and reflexivity, Integ. Equat. Oper. Th. 19 (1994), 373-380. CMP 94:15
  • 6. W. Gong, D. R. Larson and W. R. Wogen, Two results on separating vectors, preprint.
  • 7. D. Hadwin, Algebraically reflexive linear transformations, Lin. Multilin. Alg. 14 (1983), 225-233. MR 85e:47003
  • 8. D. Hadwin, A general view of reflexivity, Trans. Amer. Math. Soc. 344 (1994), 325-360. MR 95f:47071
  • 9. D. Hadwin and E.A. Nordgren, Reflexivity and direct sums, Acta Sci. Math.(Szeged) 55 (1991), 181-197. MR 92g:47064
  • 10. P. R. Halmos, A Hilbert space problem book, 2nd ed., Springer-Verlag, New York, 1982. MR 84e:47001
  • 11. D. R. Larson, Reflexivity, algebraic reflexivity, and linear interpolation, Amer. J. Math. 110 (1988), 283- 299. MR 89d:47096
  • 12. L. Livshits, Locally finite-dimensional sets of operators, Proc. Amer. Math. Soc. 119 (1993), 165-169. MR 93k:47054
  • 13. B. Magajna, On the relative reflexivity of finitely generated modules of operators, Trans. Amer. Math. Soc. 327 (1991), 221-249. MR 91m:47064

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Additional Information

Lifeng Ding
Affiliation: Department of Mathematics & Computer Science, Georgia State University, Atlanta, Georgia 30303-3083

Keywords: Reflexive operator space, separating vector, separating space
Received by editor(s): October 14, 1994
Received by editor(s) in revised form: March 30, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society

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