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.References
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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