Algebraic isomorphisms and 𝒥-subspace lattices

Li, Jiankui; Panaia, Oreste

doi:10.1090/S0002-9939-05-07581-7

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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