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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Algebraic isomorphisms and $\mathcal {J}$-subspace lattices
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by Jiankui Li and Oreste Panaia PDF
Proc. Amer. Math. Soc. 133 (2005), 2577-2587 Request permission

Abstract:

The class of $\mathcal {J}$-lattices was originally defined in the second author’s thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice $\mathcal {L}$ on a Banach space $X$ which is also a $\mathcal {J}$-lattice is called a $\mathcal {J}$-subspace lattice, abbreviated JSL. It is demonstrated that every single element of $Alg\mathcal {L}$ has rank at most one. It is also shown that $Alg\mathcal {L}$ has the strong finite rank decomposability property. Let $\mathcal {L}_1$ and $\mathcal {L}_2$ be subspace lattices that are also JSL’s on the Banach spaces $X_1$ and $X_2$, respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between $Alg\mathcal {L}_1$ and $Alg\mathcal {L}_2$ preserves rank. Finally we prove that every algebraic isomorphism between $Alg\mathcal {L}_1$ and $Alg\mathcal {L}_2$ is quasi-spatial.
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Additional Information
  • Jiankui Li
  • Affiliation: Department of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
  • Email: jli@math.uwaterloo.ca
  • Oreste Panaia
  • Affiliation: School of Mathematics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
  • Email: oreste@maths.uwa.edu.au
  • Received by editor(s): February 4, 2002
  • Received by editor(s) in revised form: April 17, 2003
  • Published electronically: April 15, 2005
  • Communicated by: David R. Larson
  • © Copyright 2005 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 2577-2587
  • MSC (2000): Primary 47L10
  • DOI: https://doi.org/10.1090/S0002-9939-05-07581-7
  • MathSciNet review: 2146201